Optimal control of a single-link flexible manipulator

This paper deals with an exact state space dynamic model for manipulators with flexible links. We use the Bernoulli-Euler beam equations to derive a frequency domain matrix transfer function. This transfer function is then used to compute the Laplace transform of the state vector as a function of the lateral position along a single link manipulator. The problem of optimal end point control of the beam is then addressed. A sixth-order state space model is derived for the manipulator and the controller is based on this model. Several control laws are studied for this model. Next, the manipulator is modeled as eighth order but the control law based on the sixth-order model is retained. We then estimate the six states from the output of the eighth-order model and feed these states back to the controller to derive the control torque used to drive the manipulator. A filter is introduced to compensate for spillover. The results are very satisfactory, and are illustrated by simulated case studies.     

Experimental control of a single-link flexible robot arm using energy shaping

Flexible-link robotic manipulators are mechanical devices whose control can be rather challenging, among other reasons because of their intrinsic under-actuated nature. This paper presents the application of an energy-based control design methodology (the so-called IDA-PBC, interconnection and damping assignment passivity-based control) to a single-link flexible robotic arm. It is shown that the method is well suited to handle this kind of under-actuated device not only from a theoretical viewpoint but also in practice. A Lyapunov analysis of the closed-loop system stability is given and the design performance is illustrated by means of a set of simulations and laboratory control experiments, comparing the results with those obtained using conventional control schemes for mechanical manipulators.     

Near-minimum time control of flexible manipulators with a payload

In this paper near-minimum time controllers for coordinating flexible two-link robots carrying an object in a workspace are developed. Bang-bang control theory in conjunction with synchronization of execution time for each joint is used to derive the near-minimum lime controller. The near-minimum time control law is implemented for two distinct cases. One is for a single flexible robot grasping a payload white the other is for a master/slave configuration for the motion of two flexible robots and their load. Simulation results indicate the feasibility of the proposed schemes.     

Tracking control of small-scale helicopters using explicit nonlinear MPC augmented with disturbance observers

Small-scale helicopters are very attractive for a wide range of civilian and military applications due to their unique features. However, the autonomous flight for small helicopters is quite challenging because they are naturally unstable, have strong nonlinearities and couplings, and are very susceptible to wind and small structural variations.A nonlinear optimal control scheme is proposed to address these issues. It consists of a nonlinear model predictive controller (MPC) and a nonlinear disturbance observer. First, an analytical solution of the MPC is developed based on the nominal model under the assumption that all disturbances are measurable. Then, a nonlinear disturbance observer is designed to estimate the influence of the external force/torque introduced by wind turbulences, unmodelled dynamics and variations of the helicopter dynamics. The global asymptotic stability of the composite controller has been established through stability analysis. Flight tests including hovering under wind gust and performing very challenging pirouette have been carried out to demonstrate the performance of the proposed control scheme.     

Synthesized sliding mode control of a single-link flexible robot

This paper proposes a synthesized sliding mode controller using frequency shaping and terminal attractor methods for single-link r-mode flexible robotic manipulators. The frequency shaping technique is used to suppress the inherent resonance modes of the flexible manipulator, while terminal attractor is used to expedite the convergence of the sliding mode near the equilibrium and to reduce steady state error. The asymptotical stability of the synthesized sliding mode is proven by using Popov's criterion. Numerical simulations confirm the effectiveness and robustness of the proposed sliding mode controller.     

Robust nonlinear control of a three-tank system using finite-time disturbance observers

A finite-time disturbance observer-based robust control method is proposed for output tracking of the Inteco threetank system in the presence of mismatched uncertainties. The controller is designed in a backstepping manner. At each step of the virtual controller design, a robust feedback controller with some effective nonlinear damping terms is designed so that the system states remain in the feasible domain. The nonlinear uncertainty is compensated by a finite-time disturbance observer. And to avoid the shortcoming of “explosion of terms”, the dynamic surface control technique which employs a low-pass filter is adopted at each step of the virtual controller design. Attention is paid to reducing the measurement noise effects and to initialization technique of the system states and reference output trajectory. Theoretical analysis is performed to clarify the control performance. And the theoretical results are verified based on the experimental studies on the real Inteco three-tank system.     

Shear force feedback control of a single-link flexible robot with arevolute joint

In this paper we present a shear force feedback control method for a single-link flexible robot arm with a revolute joint for which it has been shown that direct bending strain feedback can suppress its vibration. Our primary concern is the stability analysis of the closed-loop equation which has not appeared in the literature. We show the existence of a unique solution and the exponential stability of this solution by doing spectral analysis and estimating the norm of the resolvent operator associated with this equation. Some experiments are also conducted to verify these theoretical developments     

Position/vibration control of two-degree-of-freedom arms having one flexible link with artificial pneumatic muscle actuators

A manipulator with a light and thus flexible link would be advantageous over a rigid link in the sense that it is physically safer when it comes into contact with its environment than a manipulator with a rigid and thus heavy link, even though it is harder for a flexible link manipulator to be robustly controlled. On the other hand, if an actuator can deliver enough force while maintaining proper compliance, it would be advantageous for the sake of safety. An artificial pneumatic muscle-type actuator is an adequate choice in this case. In this work, position control problem of a two-degree-of-freedom arm system having a flexible second link with artificial pneumatic muscle-type actuators is addressed. A composite controller design method is proposed in the framework of the singular perturbation method. Various robust control schemes are designed in order to meet with payload variation, parameter uncertainty, unmodeled vibration mode, actuator dynamics both in the slow and the fast subsystems.     

Direct tip regulation of a single-link flexible manipulator by adaptive variable structure control

Direct tip position regulation of flexible manipulators is one of the most challenging control tasks. There are mainly three problems to be addressed in order to achieve good performance. The first two control problems arise owing to the unstable zeros and complex poles in the system nominal part which is dominated by a transfer function. The third problem is the existence of unstructured uncertainties owing to the truncation of high-order resonance modes and system nonlinearities. Because of the above difficulties and in particular the non-minimum-phase nature, tip regulation task of flexible manipulators is usually solved indirectly: direct control of joint angle and suppression of the flexible link vibration. The aim of this study is to investigate the direct approach for tip regulation. Since the tip transfer function contains unstable zeros and the first few dominant flexible modes (complex poles), a reference model of the same order is selected which does not have any finite zero but all negative real poles. In order to force the system to follow the reference model in the presence of the unstructured uncertainties, a variable structure controller is adopted in which the switching surface is derived from the reference model. When in sliding mode, the system performs as the reference model. Hence there will be no vibration and the tip position regulation can be achieved when the system approaches steady state. To improve the system responses further, an adaptation law with dead-zone scheme is combined with the variable structure controller. Simulation results show that link vibrations have been eliminated and the control profile is fairly smooth.     

On tracking control of flexible robot arms

A controller for solving the tracking problem of flexible robot arms is presented. In order to achieve this goal, the desired trajectory for the link (flexible) coordinates is computed from the dynamic model of the robot arm and is guaranteed to be bounded, and the desired trajectory for the joint (rigid) coordinates can be assigned arbitrarily. The case of no internal damping is also considered, and a robust control technique is used to enhance the damping of the system     

Application of discrete-time model reference adaptive control to a flexible single-link robot

This article addresses the feasibility of applying discrete-time model reference adaptive control techniques to the flexible link of robot mechanisms. The method of separation of variables is used to represent the deflection of the link. A nonlinear model is obtained using a Lagrangian equation, and the candidate frequencies and the associated mode functions are obtained using Bernoulli-Euler beam theory. By considering the effect of flexibility as an internal disturbance torque acting on the rigid body motion of the system, a discrete-time MRAC is determined for a single non-rigid link. The control algorithm is implemented for a collocated sensor and actuator system, and for a noncollocated end-point sensor and actuator system. Results of computer simulation show the feasibility of this approach and the advantage of using an end-point sensing system.     

Nonlinear cascade control of an electro-hydraulic actuator with large payload variation

Electro-hydraulic actuators have been widely used in industrial production, but the unknown variable payload seriously affects its position control accuracy. Therefore, a radial basis function neural network disturbance observer is designed to estimate the lumped disturbance force through strong online learning ability in the absence of force sensor. Besides, a nonlinear cascade controller with double loop structure is proposed in this paper. A global fast terminal sliding mode control method is firstly applied in the outer loop position system, which can eliminate chattering and improve convergence speed comparing to traditional sliding mode control. The inner loop force system adopts a backstepping control method to calculate the actual input of the whole system. Theoretical analysis indicates that the proposed controller is stable even if existing time-variant disturbance. Moreover, three comparative controllers are designed and tested in both simulations and experiments. Comparative results show that the developed method has absolute average errors of 1.14 and 0.49 mm in different position tracking, which means more satisfactory tracking performance compared to the contrast controllers.     

Virtual passive control of flexible arms with collocated and noncollocated feedback

A novel approach to the control of flexible manipulators is proposed. The controller includes both joint‐variable and tip‐deflection feedback. It is shown that tip‐deflection feedback transforms the original structure into new system in which the structure parameters are virtually scaled up or down. The new system can hence be easily stabilized via a strictly passive feedback law. A co‐hub, lumped‐parameter structure with multiple massless links is first investigated and stability conditions are developed. The results are then applied to a distributed‐parameter flexible arm, which is decomposed into an equivalent lumped‐parameter structure via a set of modal functions normalized in a particular way. Tip‐deflection feedback is shown to be capable of enhancing control performance on a flexible arm, and stability is ensured as long as the gain associated with the noncollocated feedback satisfies a simple inequality. The stability criteria re valid independent of high‐order flexible modes. © 2001 John Wiley & Sons, Inc.     

A finite-element approach to control the end-point motion of a single-link flexible robot

A structural finite-element technique based on Bernoulli-Euler beam theory is presented which will permit the finding of the torques (or forces) that are necessary to apply at one end of a flexible link to produce a desired motion at the other end. This technique is suitable for the open loop control of the tip motion. It may also provide a good control law for feedback control. The finite-element method is used to discretize the equations of motion. This method has a major advantage in the fact that different material properties and boundary conditions like hubs, tip loads, changes in cross sections, etc., can be handled in a very simple and straightforward manner. The resulting differential equations are integrated via the frequency domain. This allows for the expansion of the desired end motion into its harmonic components and helps to visualize the complex wave propagation nature of the problem. The performance of the proposed technique is illustrated in the solution of a practical example. Results point out the potential that this technique has in the study of the dynamics and control not only of flexible robots, but also of any other flexible mechanisms like those used in biomechanics, where high precision at the tip of very light flexible arms is required.     

A hybrid actuator scheme for robust position control of a flexible single-link manipulator

This article presents a novel hybrid actuator scheme to actively and robustly control the endpoint position of a very flexible single-link manipulator. The control scheme consists of two actuators; a motor mounted at the beam hub and a piezoceramic bonded to the surface of the flexible link. The control torque of the motor, which produces a desired angular motion, is determined by employing the sliding mode control theory on the equation of motion of the rigid link having the same mass as that of the proposed flexible link. The torque is then applied to the flexible manipulator to activate the commanded motion. During the motion, the undesirable oscillation caused by the torque, based on the rigid link dynamics, is actively suppressed by applying a feedback control voltage to the piezoceramic actuator. Consequently, desired tip motion is achieved. Both regulating and tracking control responses are analyzed through experimental implementation to demonstrate high performance characteristics to be accrued from the proposed methodology. © 1996 John Wiley & Sons, Inc.     

Feedforward control of multimode single-link flexible manipulators based on an optimal mechanical design

This paper proposes a method to modify the dynamics of single-link flexible arms in order to allow the design of a control system which is more robust to changes in the payload value. This is achieved by attaching some small lumped masses at some points of the link. Therefore, connections between the dynamics and the feedforward control of single-link flexible manipulators are considered to simplify the control and make the system faster and more robust. The feedforward controller design is based on the so-called dynamic model inversion technique derived from a discretization of the system dynamic model. A comparative assessment between the discrete model inversion based on feedforward control and the command preshaping technique is presented. Numerical examples and experimental results thus changing the payload values are carried out.     

Designing feedforward command shapers with multi-objective genetic optimisation for vibration control of a single-link flexible manipulator

This paper presents investigations into the design of a command-shaping technique using multi-objective genetic optimisation process for vibration control of a single-link flexible manipulator. Conventional design of a command shaper requires a priori knowledge of natural frequencies and associated damping ratios of the system, which may not be available for complex flexible systems. Moreover, command shaping in principle causes delay in system's response while it reduces system vibration and in this manner the amount of vibration reduction and the rise time conflict one another. Furthermore, system performance objectives, such as, reduced overshoot, rise time, settling time, and end-point vibration are found in conflict with one another due to the construction and mode of operation of a flexible manipulator. Conventional methods can hardly provide a solution, for a designer-oriented formulation, satisfying several objectives and associated goals as demanded by a practical application due to the competing nature of those objectives. In such cases, multi-objective optimisation can provide a wide range of solutions, which trade-off these conflicting objectives so as to satisfy associated goals. A multi-modal command shaper consists of impulses of different amplitudes at different time locations, which are convolved with one another and then with the desired reference and then used as reference (for closed loop) or applied to system (for open loop) with the view to reduce vibration of the system, mainly at dominant modes. Multi-objective optimisation technique is used to determine a set of solutions for the amplitudes and corresponding time locations of impulses of a multi-modal command shaper. The effectiveness of the proposed technique is assessed both in the time domain and the frequency domain. Moreover, a comparative assessment of the performance of the technique with the system response with unshaped bang–bang input is presented.     

Inversion techniques for trajectory control of flexible robot arms

A general framework is given for computing the torques that are needed for moving a flexible arm exactly along a given trajectory. This torque computation requires a dynamic generator system, as opposed to the rigid case, and can be accomplished both in an open- or in a closed-loop fashion. In the open-loop case, the dynamic generator is the full or reduced order inverse system associated to the arm dynamics and outputs. In order to successfully invert the arm dynamics, the torque generator should be a stable system. The stability properties depend on the chosen system output, that is on the robot variables (e.g., joint or end-effector) to be controlled. The same inversion technique can be applied for closed-loop trajectory control of flexible robots. A simple but meaningful nonlinear dynamic model of a one-link flexible arm is used to illustrate different feasible control strategies. Simulation results are reported that display the effects of the system output choice on the closed-loop stability and on the overall tracking performance.     

Position control of a flexible gantry robot arm using smart material actuators

This article presents new feedback actuators that achieve accurate position control of a flexible gantry robot arm. Translational motion in the plane is generated by two dc motors and controlled by applying electric fields to electro‐rheological (ER) clutch actuators. On the other hand, during control action of translational motion, a flexible arm attached to the moving part produces undesirable oscillations due to its inherent flexibility. Oscillations are actively suppressed by employing feedback voltage to the piezoceramic actuator attached to the surface of the flexible arm. Consequently, an accurate position control at the end‐point of the flexible arm can be achieved. To accomplish this control goal, governing equations of the proposed system are derived and written as transfer functions. Transfer functions are used in design of a set of robust H_∞ controllers. Electric fields to be applied to ER clutch and control voltage for the piezoceramic actuator are determined via H_∞ methodology which is incorporated with classical loop shaping design technique. To evaluate effectiveness of the proposed control system, experiments for both regulating and tracking controls are undertaken. ©1999 John Wiley & Sons, Inc.     

Modeling and robust force control of constrained one-link flexible arms

In this article modeling and robust force control of constrained flexible one-link arms on the basis of a distributed parameter model are discussed. Since the tip of the flexible arm contacts a given constraint surface, a constraint condition should be satisfied. By using the Lagrange multiplier method and the Hamilton's principle, we derive dynamic equations of the joint angle, the vibration of the flexible arm, and the constraint force. The boundary condition of the derived distributed parameter system is related to the contact force and is nonhomogeneous. We introduce a change of variables to derive a homogeneous boundary condition. On the basis of a finite-dimensional modal model of the distributed parameter system, we analyze the stability of the force feedback by using the root locus technique and the compliance control. To compensate the spillover instability an optimal controller with low-pass property and a robust H_∞ controller are constructed. Experiments have been carried out and results confirm that the controllers perform remarkably well. © 1998 John Wiley & Sons, Inc.