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The design and tuning of a simple feedback strategy with delay to stabilize a class of underactuated mechanical systems with dead time are presented. A linear time-invariant (LTI) model with time delay of fourth order and a Proportional Retarded (PR) controller are considered. The PR controller is shown as an appealing alternative to the application of observer-based controllers. This paper gives a step forward to obtain a better understanding of the effect of output delays and related phenomena in mechatronic systems, making it possible to design resilient control laws under the presence of uncertain time delays in measurements and obtain an acceptable performance without using a derivative action. The Furuta pendulum is a standard two-degrees-of-freedom benchmark example from the class of underactuated mechanical systems. The configuration under study includes an inherent output delay due to wireless communication used to transmit measurements of the pendulum’s angular position. Our approach offers a constructive design and a procedure based on a combination of root loci and Mikhailov methods for the analysis of stability. Experiments over a laboratory platform are reported and a comparison with a standard linear state feedback control law shows the advantages of the proposed scheme.

Underactuated mechatronic systems have been a subject of intensive research during the past three decades, where partial feedback linearization, normal forms, and energy-based methods have offered successful results as well as breakthrough paradigms [

The present paper focuses on the delay phenomena in an underactuated system called Furuta pendulum. The main result introduces a methodology for the optimal selection of the controller gains and a procedure based on Mikhailov method for the study of complete stability under the presence of measurement delays. Here, it is assumed that only angular positions are available for measurement. For comparison purposes, a design of a state feedback controller, which includes the on-line estimation of velocities, is presented. All the proposed algorithms are verified with simulations and experiments over a laboratory test bench, obtaining the desired performance.

The outline of this paper is distributed as follows. In Section

The Furuta pendulum test bench consists of a passive pendulum attached to the end of an actuated rotating arm as shown in Figure

Scheme of the Furuta pendulum model by [

The external torque that allows the control of the arm is produced by a DC motor. The motor induces a torque

There are two control problems commonly associated with the Furuta pendulum: stabilization around one of the two equilibrium points and the swing-up [

In this context, the equation of motion (

Physical parameters of Furuta pendulum.

Mass of the horizontal rod ( |
0.431 kg |

Length of the horizontal rod ( |
0.262 m |

Distance to center of gravity ( |
0.131 m |

Moment of inertia of the rod ( |
0.012 kg m^{2} |

Mass of the pendulum ( |
0.128 kg |

Length of the pendulum ( |
0.47 m |

Distance to center of gravity ( |
0.185 m |

Moment of inertia of pendulum ( |
0.0035 kg m^{2} |

Friction on |
0.0032 kg m^{2}/s |

Friction on |
0.004 kg m^{2}/s |

Motor constant ( |
1.07 rad kg m^{2}/s^{2} |

Using classical techniques, stabilization of the Furuta pendulum requires measurements of variables

Consider the following control law:

As a second step, the Mikhailov analysis will provide the stability conditions under the presence of an inherent sufficiently small delay

In this section, the tuning of the controller gains proposed in the previous section and an analysis of stability of closed-loop system (

Consider the second-order quasi polynomial, roots of which cover all the roots of (

Root of quasi polynomial (

Roots of (

To determine stability of quasi polynomial (

Root locus of (

Next, the root locus of quasi polynomial (

Root locus of (

Now, Mikhailov’s criterion [

Mikhailov criterion for quasi polynomial (

On the other hand, in Figure

Root locus of (_{2} = 0.008 and

Therefore, we have the following result.

Consider a quasi polynomial of the form (

The experimental results were obtained in the Robotics and Control Lab at the Department of Applied Physics and Electronics, Umeå University, Umeå, Sweden. The system used for the experiments is from the PendCon Advanced series [

The control goal is to stabilize the pendulum at its unstable equilibrium point

Angular positions (rad)

Control actions (

The design and tuning of the controller are done as follows:

Consider a system of the form (

Factor the quasi polynomial of 4th order in one quasi polynomial of 2nd order and one polynomial of 2nd order, when

Use Lemma

Obtain

It is important to remark that there is an inherent unknown delay

In order to compute the estimation of the first derivative for both position variables,

These methods are designed to be used on-line. For comparison purposes, it is obvious that better velocity estimation can be achieved with an off-line method when both previous and future values of the position are used. Then, we propose to postprocess the measured position to obtain an off-line estimation. For this purpose, the measured signal is fitted with a smoothing spline and next the off-line estimation of velocity is obtained as an analytical differentiation of the spline (see, e.g., [

Table of error comparison.

Differentiators | ||||
---|---|---|---|---|

HG1 | HG2 | ST | ||

Error |
Norm 2 | 1.4167 | 1.1556 | 1.349 |

Norm inf | 24.9879 | 25.0273 | 10.0679 | |

Error |
Norm 2 | .3888 | .1292 | .0634 |

Norm inf | .8410 | 3.2624 | .5597 |

With the proposed PR controller, the use of the estimation of

An additional experiment was performed in order to test the robustness against external perturbations. For this aim, several impacts were applied to the pendulum at the time instants

Angular positions (rad)

Control action (

From the experimental results, the recovery of performance with the proposed PR controller is verified. Besides, it is important to highlight that a better transient, a smoother response, and chattering attenuation in the control signal are observed with the proposed approach under the presence of delays in measurements. A video of the recorded experiments is available in the following link:

Presence of time delays, quantization errors of encoders, and errors of the wireless communication make the design of a robust control law difficult. The design and stability analysis of a PR action-based controller for stabilization of the Furuta pendulum under the presence of time delays in measurements is introduced in this work. With the proposed approach, the design of observers for the passive variable is avoided simplifying the control algorithm and recovering an acceptable performance without relying on accurate estimates of the derivative for feedback. Since a retarded action is included in the control law, the closed-loop dynamics are represented by a characteristic quasi polynomial and the stability analysis was successfully done with Mikhailov’s criterion and a root locus of the corresponding quasi polynomial. Furthermore, the stability analysis includes an inherent time delay due to the wireless communication. Besides, a two-step constructive procedure for tuning the four parameters of a controller is presented. Experimental results and a comparison with a state feedback controller show the advantages of the proposed methodology. We are aware that a possible extension of the proposed controller is the addition of a PR controller in the actuated variable as well; however, this is considered as a future work, since the order of the quasi polynomial will increase significantly, losing the scope of the actual contribution. Additionally, extension for a wider class of underactuated mechanical systems and the inclusion of nonlinearities is considered for future work.

Consider the next second-order system,

Let

In Figure

Also, note that these regions collapse into a single point (red asterisk) which correspond to the unique maximum achievable decay

Let

Stability conditions stated in terms of hodographs (frequency plots) have a long history. In their most general form, these conditions were given by Cauchy’s principle of the argument around 1829. The graphical use of the argument principle was introduced to the engineering community by Nyquist in 1932 and Mikhailov in 1938 for linear time-independent systems; see [

Consider the following quasi polynomial for a time-delay system:

Immediately, the above theorem is used to determine stability or instability of quasi polynomials of Section

Let

Example of

It should be noted, however, that application of Mikhailov’s method in the case of quasi polynomials must be done with great care, since there is typically no monotonicity of rotation of the curve along

Angular position of the pendulum

Angle of the rotational rod

Mass of the rotational rod

Pendulum mass

Moment of inertia of the rod

Moment of inertia of the pendulum

Length of the horizontal rod

Pendulum length

Distance to rod center of gravity

Distance to pendulum center of gravity

Gravity acceleration

Induced torque.

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