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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.

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 [

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 [

The DRC idea has been first proposed in [

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 [

Starting from a linearized model briefly described in Section

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 [

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

The scalars

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 [

The equations of motion in (

The DRC idea is to make

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

The

The control problem is therefore stated as finding

The control problem in (

Such a problem can be modified by weighing each equation of the system in (

Finally, the poles and zeros in

The DRC architecture is schematically described by the block diagram in Figure

DRC multiloop architecture.

The real-time computation of

A four-bar planar linkage with flexible steel bars has been developed (see Figure

Inertial and geometrical parameters of the four-bar linkage.

Link | Frame | Crank | Coupler | Follower |
---|---|---|---|---|

Length, m | 0.365 | 0.390 | 0.535 | 0.640 |

Flexural stiffness, Nm^{2} | 20.16 | |||

Cross-sectional area, m^{2} | | |||

| ||||

Joint | | | | |

| ||||

Mass, kg | 0 | | | - |

Inertia, kgm^{2} | | 0 | 0 | |

Picture of the mechanism.

The motion control of the crank is performed by a standard position-velocity-current multiloop controller, as it is represented in Figure

Inner control loop.

The DRC specification is to reduce the deformation of both the crank and the follower. Resistive strain gauges are therefore adopted (link 1,

An effective and simple choice for

The linearized model matrices in (

Finally, Table

Closed-loop system eigenvalues.

Eigenvalues |
---|

(1) −8.92 |

(2) −38.17 |

(3) −9123.7 |

(4) −2.75 ± 70.436 |

(5) −2.89 ± 324.10 |

(6) −21.74 ± 956.52 |

(7) −35.50 ± 229.33 |

(8) −56.16 ± 1347.6 |

(9) −45.91 ± 1438.7 |

(10) −131.3 ± 2309.3 |

(11) −287.9 ± 3554.2 |

(12) −391.2 ± 4133.2 |

(13) −948.3 ± 6400.8 |

(14) −1824.4 ± 8787.7 |

(15) −2953.8 ± 11027 |

(16) −6612.7 ± 15751 |

(17) −23441.4 ± 22030 |

(18) −40601.7 ± 12000 |

The motion reference is a motion law with piecewise constant acceleration (trapezoidal speed profile), aimed at moving the crank from the horizontal configuration

Figure

Reference and actual values of the crank angle versus time.

Figure

Time delay and action reference parameter versus time.

Figure

The measured strains are finally shown in Figure

Comparison of the crank strains.

Comparison of the follower strains.

As for the steady values of the link strains, they are due to the static deformations induced by gravity and therefore they just depend on the mechanism configuration (and on the presence of small disturbances that justify the negligible differences). Therefore, they cannot be eliminated through the DRC, as well as through any other controller. The presence of the aforementioned high-pass filter prevents from uncontrolled growth of the time delay due to such steady-state values.

A control strategy for reducing elastic deformation in rest-to-rest motion has been proposed and experimentally validated in this paper. The technique, named Delayed Reference Control (DRC), takes advantage of the dynamic coupling between rigid-body motion and elastic motion to reduce the link deformation by suitably shaping the time history of the position reference on the basis of the sensed link strains. Being based on the measured strains, the proposed feedback scheme allows overcoming one of the main limitations of the open-loop optimal planning techniques, whose performances are affected by the presence of model uncertainty, unknown external forces, or unknown initial conditions.

Basically, the proposed DRC scheme reduces the elastic deformations by delaying or speeding up the planned reference input through an action reference parameter, which is computed in a feedback loop devoted to deformation control and closed outside a standard position control loop. The outer loop also includes the trajectory planner, which calculates the time history of the actuator position reference to be tracked by the inner position loop. Overall, this strategy allows ensuring the correct tracking of the desired spatial path while reducing the unwanted elastic deformations that usually downgrade precision and accuracy of machines. Additionally, the multiloop architecture makes the DRC implementation straightforward and suitable to be applied to manufacturing systems with proprietary controllers. Indeed, the proposed scheme can be achieved by simply adding a new loop outside proprietary position controllers.

The experimental results obtained by applying the DRC to a four-bar linkage with flexible links demonstrate the effectiveness of the approach and its stability over a wide range of gains. In particular, a significant reduction in the link unwanted elastic deformation has been achieved with a small time delay. Additionally, the results prove the correctness of some minor simplifications assumed in the formulation of the optimization problem leading to the analytical and straightforward computation of the suitable time delays. Finally, the low computational effort of the analytical solution proposed makes the DRC suitable for hard real-time implementation.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research has been funded by the University of Padova through the Progetto di Ateneo 2015 CPDA157149, “efficient modeling of flexible link manipulators for real-time state estimation.”