Shock and Vibration

Volume 2018 (2018), Article ID 9016028, 9 pages

https://doi.org/10.1155/2018/9016028

## Deformation Control in Rest-to-Rest Motion of Mechanisms with Flexible Links

Correspondence should be addressed to Alberto Trevisani

Received 31 August 2017; Revised 12 January 2018; Accepted 8 February 2018; Published 11 March 2018

Academic Editor: Kumar V. Singh

Copyright © 2018 Roberto Caracciolo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper develops and validates experimentally a feedback strategy for the reduction of the link deformations in rest-to-rest motion of mechanisms with flexible links, named Delayed Reference Control (DRC). The technique takes advantage of the inertial coupling between rigid-body motion and elastic motion to control the undesired link deformations by shifting in time the position reference through an action reference parameter. The action reference parameter is computed on the fly based on the sensed strains by solving analytically an optimization problem. An outer control loop is closed to compute the references for the position controllers of each actuator, which can be thought of as the inner control loop. The resulting multiloop architecture of the DRC is a relevant advantage over several traditional feedback controllers: DRC can be implemented by just adding an outer control loop to standard position controllers. A validation of the proposed control strategy is provided by applying the DRC to the real-time control of a four-bar linkage.

#### 1. Introduction

The increasing effort towards improving dynamic performances and energy efficiency in machines and manufacturing systems is leading to the reduction of the moving masses. However, decreasing link masses also reduces their stiffness and causes elastic vibrations and deformations, which reduce system accuracy and precision [1] and cause large residual vibrations. Hence, motion planning and control of lightweight mechanisms and manipulators are two challenging issues. Traditional control schemes, such as standard PID (proportional-integral-derivative) regulators widely used in industrial controllers, have been demonstrated to be unsatisfactory to ensure effective compensation of unwanted elastic deformations. Therefore, attention should be paid to the synthesis of either optimal command generation techniques [2–4] or active vibration control schemes [5–11].

Optimal command generation techniques are feedforward approaches aimed at preplanning the control input on the basis of the system model, in such a way that the link elastic deformations are prevented. Their main advantage is the ease of implementation, since they just rely on the system model and no additional sensors are required. On the other hand, open-loop approaches usually have poor disturbance rejection and high sensitivity to model uncertainty and unknown initial conditions. Designing closed-loop active control systems is therefore an effective way to cope with disturbances and model uncertainty. Several different techniques have been presented over the years. Most of these approaches take advantage of additional actuators, such as smart actuators embedded into flexible links [5, 6], or impose the modification of the torque, speed, or position control loops of the servo axes driving the systems [7–9]. However, these modifications are hard to implement in industrial controllers that are usually based on proprietary and closed architectures that cannot be modified. Among the few control techniques that do not impose modification of servo axis controllers, wave-based control [10, 11] and Delayed Reference Control [12] (DRC) should be mentioned. They both rely on the feedback-based modification of the position reference.

The DRC idea has been first proposed in [12] for vibrating linear systems with a single resonant mode and then extended in [13] to multi-DOF (Degree of Freedom) linear systems. The basic idea of DRC is to shape the desired position reference on the fly based on the measured values of the variables to be controlled, namely, the elastic deformations or oscillations. Vibration control is obtained by delaying or anticipating the original preplanned position reference if it causes elastic deformation. The preplanned desired input reference is not therefore defined explicitly as a function of time but rather as a function of a scalar time-varying “action reference parameter,” which is in practice an environment-sensitive parameter computed on the basis of the measured deformations. DRC schemes can be adopted in those applications, where the major goal is performing a prescribed displacement accurately, with negligible elastic deformations, rather than a trajectory in a given time, since the motion time can be modified in a variable way. For the same reason, DRC schemes are sometimes referred to as “non-time-based” controllers, since the position reference is not explicitly defined as a function of the time. The concept of non-time-based control for dynamic systems, also referred to as “event-based” control, has been introduced in the 90s, and several successful controllers have been proposed thereafter, for instance, in neurosurgery [14], manufacturing processes [15], coordinated motion control [16], and robot control [17].

A peculiar feature of DRC schemes is that the trajectory planner assumes a primary role in the feedback closed loop, since it shapes the position references on the fly. Consequently, a multiloop control scheme is obtained, consisting of an inner position control loop and an outer loop. The former can be any standard position controller and is fed with the position reference computed by the outer loop. The latter includes the trajectory planner and is introduced to compute the action reference parameter for generating the suitable delayed position reference accomplishing the control goal. This architecture allows for the straightforward implementation of DRC schemes by simply adding an outer loop to standard position controllers, such as standard servo regulators employed in industrial automatic machines and manipulators. This is a relevant strength of the proposed method compared to most of the control techniques proposed in literature for deformation and vibration control, which impose the modification of the inner loop to perform vibration control.

In the numerical study [18], the DRC theory has been applied to the control of flexible link mechanisms in repetitive motion through numerical simulations. Starting from such a preliminary work, this paper proposes the first extension of the DRC to deformation control in rest-to-rest motion of flexible link mechanisms. The aim of the controller is to reduce the elastic deformations of the links, and therefore the method is suitable for those applications where this is a major requirement. The method is here developed with reference to an arbitrary flexible link mechanism in the presence of small deformations and then experimentally validated through a four-bar planar linkage with all the links being flexible. In particular, the crank and the follower deformations are controlled simultaneously by computing a scalar value of the action reference parameter, optimally trading between different control specifications.

Starting from a linearized model briefly described in Section 2, which provides a simplified but adequately accurate representation of the system dynamics, the method for synthesizing DRC schemes for curvature control is explained in Section 3. The experimental results validating the proposed approach, as well as the description of the test-bed and of the method implementation, are discussed in Section 4. Finally, concluding remarks are stated in Section 5.

#### 2. System Modelling

Although the dynamics of flexible link mechanisms and manipulators is nonlinear and needs large dimensional models, the design of motion and vibration control schemes often relies on simplified, reduced-order, and linearized models. Indeed, it is widely recognized that, in the case of small deformations, the accuracy of linearized models about operating points is usually satisfactory enough to make their use successful in the synthesis of effective and stable control schemes. The neglected nonlinear and high-frequency dynamics can be treated, for example, as model uncertainty, which can be effectively tackled by paying attention to the controller robustness. All these considerations justify the use of a linearized model in the synthesis of the DRC scheme, which also makes the controller implementation easier by drastically reducing its computational effort and allowing for hard real-time control.

The model adopted is obtained by linearizing a nonlinear model accounting for the mutual coupling between large rigid-body motion and small elastic displacements. The model is valid for an arbitrary flexible link mechanism with holonomic and scleronomous constraints in the presence of small elastic displacements. The reader should refer to [9] and to the references therein for a more detailed description of such a model.

Independent coordinates have been adopted to represent the rigid-body motion, and the model is directly formulated through ordinary differential equations (ODE). As a matter of fact, this formulation allows for a more straightforward use of the control theory, which usually assumes this kind of model formulation.

The total motion of each flexible link is separated into the large motion of an equivalent rigid-link system (ERLS) behaving as a rigid moving reference mechanism whose number of DOFs is denoted as* r* and the small elastic deflections of the links with respect to the ERLS named** u** whose size is denoted by . Among the possible definitions of the rigid moving reference, the ERLS is defined in this work by imposing the fact that its generalized coordinates** q** coincide with the joint variables, which are also those measured and actuated to drive the mechanism. In this way, the actuators set the motion of the ERLS directly. The sum of** u** and the vector of the rigid positions of the node of the ERLS, denoted as** r**, describes the motion of each node of the finite element model. By linearizing the nonlinear equations of motion obtained through the application of the Principle of Virtual Work, the following linear model is obtained by linearizing the nonlinear model about an equilibrium configuration set by the equilibrium values of the rigid displacement vectors and of the external force vector** v** (resp, ):In (1), the following definitions have been adopted:(i) and are the matrices obtained by assembling the mass and stiffness matrices of the finite elements.(ii)**S** is the ERLS sensitivity coefficient matrix for all the nodes, relating the velocities of the ERLS generalized coordinates, , to the velocities of all the nodes of the ERLS, .(iii)**I** and** 0** are the identity and the null matrices.(iv)The scalars and are the Rayleigh damping coefficients.(v) denotes the inner product between arbitrary matrix** N** and vector** b**, for all the subscripts , , .

Admittedly, the linearized model holds only in a finite neighborhood of the working point; however, when large displacements are tackled, piecewise-linear models can be successfully employed to approximate the nonlinear system’s dynamics better, as it has been proven in [9].

#### 3. Synthesis of the Delayed Reference Regulator

##### 3.1. Controller Specifications

The equations of motion in (1) are partitioned and expressed in the following more convenient form:The subsystem of (2) comprising the first equations can be rewritten as the equations of motion of an* e*-DOF elastic system excited by the external nodal forces, collected in vector** v**, and by the inertial coupling with the rigid-body motion, :The right-hand side term of (3) can be therefore thought of as an equivalent force vector, henceforth referred to as :If the most critical vibrational modes are controllable, then can excite and control the link elastic displacements. In order to control the deformation, it is here suggested to make behave as a damping force. Hence, is required to be a function of the first derivatives of the measured strains of the link, ( denotes the vector of the measured strains):In (5), represents both the DRC gains and the filters adopted to remove undesired signal components. Besides low-pass filters, adopted to remove noise, high-pass filters are employed to remove static components of strains.

The DRC idea is to make assume the desired values, defined in (5), by shaping (i.e., the time history of the rigid-body motion) rather than . Hence, no active deformation control is explicitly introduced in the force vector . This is similar to the approach followed by optimal planning techniques, where rigid-body motion is properly shaped to reduce oscillations. Compared with such methods, however, DRC is a feedback scheme and the optimal motion law is computed on the fly based on sensor measurements. The control problem is therefore stated as shaping the time history of the rigid-body motion in order to exert the desired active damping force defined through (5):

Besides exerting an equivalent damping force, it is also required that the controller ensure correct tracking of the desired displacement (which sets the path in the case of systems with more rigid DOFs), defined regardless of the time of execution, rather than a trajectory in time as it happens in classical control schemes. To satisfy this second specification, the position reference of the rigid-body control loop is not made an explicit function of just the time but of a real scalar time-varying parameter named “action reference parameter” [12]. The position reference will be henceforth referred to as . The action reference parameter is computed on the basis of the measured strains to exert the desired damping force . The following definition of is chosen:The real and scalar is named “time delay,” although it can assume also negative values (in other words, it can anticipate the execution of trajectory whenever necessary).

The* l*-domain value of the position reference should be known explicitly, since it describes the ideal desired path to be tracked in the case of rigid bodies. In contrast, the time history of the position reference, referred to as , is not known a priori, since it depends on the time history of* l*, which in turn is computed on the fly through the measured strains.

The control problem is therefore stated as finding such that the desired equivalent damping force in (5) is exerted and the actuator motion (which can be seen as the rigid-body motion) does not drift away from the desired displacement:

##### 3.2. Problem Solution

The control problem in (8) has no exact solution in the general case. However, some reasonable problem simplifications lead to an analytical solution. First of all, by assuming that the mechanism position controller ensures precise dynamic responses, in (6), the actual ERLS acceleration can be approximated by its reference value in time . This assumption is common in most of the literature on optimal motion planning and therefore should not be considered restrictive. Secondly, the position reference may be linearized over a finite interval: where the rigid-body reference velocity vector is assumed to be finite and constant over such an interval. Because of this assumption on , step inputs as position references cannot be employed if a DRC is used. Again, this assumption is not a severe drawback of the DRC, since step position references are not of interest in practice: indeed, only continuous position references, such as polynomials, are adopted in industrial applications. Hence, by replacing (9) and its first and second derivatives in (8), it holds thatThe boundary conditions in (10) are set to ensure that if , then 0 and 0 (i.e., ); this implies that if no undesired oscillations arise, no control force is exerted and therefore no delay is applied. Hence, if , then . These boundary conditions lead to an overdetermined system made by independent differential equations in the scalar unknown :Equation (11) can be written in the frequency domain through the Laplace transforms of , , and (denoted as , , and , resp.):A unique, scalar time delay satisfying simultaneously all these equations does not exist in general. The controller synthesis is therefore formulated as a least-square problem by looking for the scalar minimizing the norm of the residual in (12):

Such a problem can be modified by weighing each equation of the system in (13) through a scalar-product matrix to reflect different levels of concern about each control requirement. Additionally, an extra term is included in the minimization problem to avoid large and fast varying delays by penalizing its magnitude in the cost function:The scalar trades off between the cost of missing the control specification and the cost of using large control effort. Since is defined as a positive-definite matrix, . Hence, the scalar exists for any and and for any . The solution is thereforeThe extra term included in (14) is often denoted as “regularization term” [19]. The regularization problem significantly improves the numerical conditioning of the problem solution and increases its robustness with respect to the model variability and uncertainty [19].

Finally, the poles and zeros in due to the high-pass filters adopted to get rid of the steady-state values of the strains are separated and made explicit through to obtain a more convenient form. In order to keep the controller implementation as simple as possible, it is here suggested to employ a first-order high-pass filter, that is, , whose pole lies in . The final formulation of the optimal time delay is therefore

##### 3.3. The DRC Multiloop Architecture

The DRC architecture is schematically described by the block diagram in Figure 1. The DRC scheme is based on a multiloop architecture made by an inner position loop (gray lines in Figure 1) and an outer loop (black lines) which run simultaneously. The “inner loop position controller” that can have any arbitrary architecture (such as those usually adopted in industrial controllers, as shown in Section 4.1) regulates the actuator to make track as computed at each time step by the “position planner.” The latter belongs to the outer loop, which implements the DRC equations. Hence, the outer loop is aimed at performing active control of the link deformation by computing suitable values of (which is calculated in the so-called action reference block) and therefore .