1. Introduction
Haptics is the technology of touch. It has found many applications in robotics such as accurate positioning, mobile robot navigation, robotic medical exploration and surgery, and collaborative robotics. These tactile sensors are complex and exhibit a distributed nature. Then, issues such as placement, sensor robustness, or wiring complexity have to be addressed. Often, sensors used in haptics try to mimic nature. In particular, haptic sensors inspired by insect antennae or mammal whiskers (see
Figure 1) have been built since the early 1990s.
The “sensing antenna” has become the most popular among these sensors because it is a robust and compact device. It is an active sensor composed of a very slender flexible beam, which is moved by two servo-controlled motors, and a load cell placed between the beam and the motors. This device replicates the touch sensors that some animals possess and performs an active sensing strategy in which the servomotor system moves the beam back and forth until it hits an object. At this instant, information on the motor angles, combined with force and torque measurements, allows us to calculate the positions of the contact points, providing valuable information about the object surface. Thus, a 3D map of the surface of an object can be obtained using this device, enabling its recognition. Two strategies can be applied to obtain such 3D maps. The first one is to keep moving the beam back and forth in order to hit the object at different points, allowing for the determination of their 3D coordinates and, subsequently, extracting the map of the object surface. This procedure is carried out by some insects that use their antennae for this purpose (e.g., [
1]). The other strategy is to slide the beam across the object, exerting a controlled force on the surface of the object in order to maintain contact, and collect the 3D coordinates of points on the object surface during this movement. This approach is utilized by some mammals that have whiskers as sensors (e.g., [
2]). Both of these strategies can be implemented using the aforementioned sensing antennae.
We are developing a sensing antenna system to aid the navigation of the prototype of a mobile robot shown in
Figure 2. Our system will localize obstacles by carrying out the first of the aforementioned strategies. We obtain the 3D information about the contact point using the method reported in [
3]. This method is based on the fact that the frequencies of the vibrations of the antenna change from the free motion case to the constrained motion case and, in this last case, also change as a function of the location of the contact point. Then, the torque measurements of the vibrations produced in the beam of the antenna during its movement in a recognition task are processed to estimate the first two vibration frequencies and, thereafter, estimate the contact point.
The dynamics of these antennae is quite complicated, which hinders fast and accurate movements: beam elasticity produces residual vibrations during the antenna movement and when the target has been reached. Therefore, the design of the controllers of the motors must consider these dynamics to achieve accurate and fast approaches to the target points to be investigated. Moreover, an inefficient cancellation of the beam vibrations could produce permanent collisions with the object, in which the antenna hits back and forth. These introduce delays in the recognition, diminish the quality of the estimates of the 3D points on the object surface and, consequently, reduce the efficacy of the device.
Active control can be used to remove these beam vibrations. It allows us to search for special points in a precise manner in such a way that the maximum amount of task-relevant information is provided. In order to design such high-performance control systems that drive the beam and position its tip with high precision while preventing vibrations, very accurate models of the dynamics of the antenna in free movement are required.
This article is devoted to obtaining analytic models of the dynamics of an antenna rotating in a horizontal plane, i.e., having azimuthal movements. This system is linear but has an infinite number of nearly undamped vibration modes. In particular, modeling damping phenomena in this antenna is of the utmost importance in order to achieve faster control systems, since damping increases the relative stability of the system but slows down its response. We will show that using fractional derivatives to model the damping of the antenna enhances the accuracy of the dynamic model and, based on this, we will develop new fractional-order models that describe better the antenna’s behavior.
The remainder of the paper is organized as follows.
Section 2 presents the state of the art in modeling flexible link antennae.
Section 3 describes the experimental setup considered in this work, consisting of a flexible sensing antenna.
Section 4 deals with the dynamic model that describes the behavior of the flexible antenna. In
Section 5, the methods used to identify the dynamics of the system is detailed.
Section 6 shows and discusses the results obtained after identification, highlighting the models that describe more accurately the system behavior. Finally, concluding remarks are drawn in
Section 7.
2. State of the Art in Modeling Flexible Link Antennae
The dynamics of a sensing antenna is mostly characterized by the dynamics of its flexible link. Two approaches to modeling flexible links have been developed: assuming a massless link with an equivalent mass lumped at its tip, and assuming mass distributed along the entire beam. In the first approach, the dynamics includes a single vibration mode, exhibiting a minimum phase behavior. For the latter approach, the dynamics of non-collocated input-output pairs have non-minimum phase behaviors whose models are described by high-order coupled non-linear differential equations. In addition, the system is driven by small servomotors in which the limited torques, intermittent operation, and the strong non-linearity owing to the static friction constrain the motor control design. In these systems, spillover effects may easily appear, i.e., instability of the closed-loop control system caused by non-modeled high-frequency dynamics and parameter uncertainties.
Taking into account the distributed mass approach, the dynamic models of flexible links can be obtained using a wide variety of methods and considering different types of damping terms. As explained in [
4], the response of a flexible link can be obtained with exact methods, i.e., modal analysis or Laplace transformations, and approximate methods, i.e., assumed modes or Galerkin’s method. Regarding damping phenomena, many varied models have been considered in other works, both linear and non-linear. When integer-order is considered, examples can be found in [
5,
6]. In [
6], two of the damping models proposed are viscous air damping, assumed to be proportional to velocity, and Kelvin–Voigt damping, which is proportional to the fourth spatial derivative of the deflection of the link. If the flexible link behavior is extended to the fractional order, different kinds of fractional vibrations can be contemplated. In [
7], six classes of fractional vibrators can be found based on what terms (inertia, friction, and/or position) are defined as fractional. Only three of these classes considered fractional friction terms explicitly. For example, the authors in [
8,
9] define models in which the inertia and friction terms are fractional.
Within fractional vibration models that only include fractional friction terms, two kinds of common linear damping can be considered. The first one defines Kelvin–Voigt damping as fractional-order damping [
10,
11,
12]. In [
10], the response of a Euler–Bernoulli beam under quasi-static and dynamic loads is studied. In [
11], the Euler–Bernoulli theory and modal analysis are used to model a slewing flexible beam with a mass at its tip and an external load. In addition, the model of a non-linear simply supported beam with harmonic excitation is derived by the Galerkin approximation method in [
12]. On the other hand, a fractional-order viscous damping (dependent only on the time derivative) is considered in [
13,
14,
15,
16,
17,
18]. In [
13], a model of a continuous beam is obtained with modal analysis, setting the fractional derivative order of the damping to
. In [
14], the Adomian decomposition method is utilized to solve a simply supported beam with a load. In [
15,
16], an Euler–Bernoulli beam with moving load is solved by combining the modal analysis method with the Laplace transform. In [
17] and [
18], the homotopy perturbation method and the homotopy analysis method, respectively, are used to obtain the dynamic response of a beam with external loads. To the authors’ knowledge, no model of a flexible link with both fractional-order dampings has been published yet. This model is studied in this work, and an exact transfer function is obtained through the Laplace transform, following the methodology of [
19], which was developed for an undamped beam.
For identification of the model of flexible structures, frequency-domain methods are generally preferred over others in the time domain because the computational load required is lower, and for certain classes of systems, they are usually less sensitive to sensor noise and, consequently, may be more accurate [
20]. Furthermore, identification algorithms in the frequency domain are traditionally founded on Levy’s method, which is a least-squares-based formulation whose results are not equally good at all frequencies, leading to the introduction of weights in some cases so as to handle this frequency dependence and improve the results [
21]. Several methods can be found in the literature to identify a transfer function from a frequency response [
22,
23]. Some of them have been extended to fractional orders to describe some complicated real systems more adequately than integer-order models (see, e.g., [
24] for further reading). In this respect, numerous studies have shown that many physical phenomena can be described more concisely and precisely by fractional-order models from frequency-domain data, such as in [
25] for a permanent magnet synchronous motor, in [
26] for the human arm dynamics, and in [
27] for a fuel cell. Likewise, fractional-order models of flexible links were also identified in [
28,
29,
30] in the frequency domain.
Given this motivation, the aim of this paper is threefold:
Measure, record, and characterize the intrinsic dynamics of the flexible link by extending the model proposed in [
19] to fractional order, with the measured motor angle as model input and the measured motor coupling torque as output.
Consider different effects in the model in terms of damping. In particular, six different model structures will be considered, including internal and external damping, both of integer and fractional orders.
Analyze the variability in the identified models, as well as the performance of the fitting.
Then, the main contribution of this paper is to develop a general model, of fractional order, able to accurately describe the dynamics of flexible links including both internal damping, which is related to the link material, and external damping, which characterizes the environment in which the link is located. For illustration purposes, an application example is given for a carbon fiber antenna that moves freely in the air using frequency-domain identification methods.
3. Experimental Setup
The experimental setup is a version of one degree of freedom (DOF) of the robotic system with a single flexible link shown in
Figure 3a, which was used as a sensing antenna in haptic applications (refer to, e.g., [
31] for more details). It is a 2DOF platform, namely, it allows azimuth and elevation movement for the flexible link. From this system, 1DOF was removed (i.e., the elevation motor and its supporting structure) to allow only the link rotation on the horizontal plane. This modification aimed to avoid, as much as possible, the coupling of the resonant frequencies of the aforementioned motor structure set over the frequency response of the flexible link and, consequently, reduce noise in the experiments. Hence, the experimental platform used for this work is represented in
Figure 3b, where it can be seen that a simpler component is used to attach the link to the azimuth motor.
To ensure stability, the structure has three legs made of stainless steel. The link is attached at one of its ends to one Harmonic Drive mini servo DC motor PMA-5A set, which includes zero-backlash reduction gears () and an incremental encoder. The encoder provides us with a precision of rad outside the gear reduction. The system also has a force-torque (F-T) sensor, an ATI Mini 40, between the flexible link and the motor, allowing us to measure the coupling torque with a precision of Nm. The structures that fix the F-T sensor to the motor and the link to the sensor are made of polylactide (PLA) with a 3D printer.
Table 1 shows the characteristics of the flexible link, which is made of carbon fiber. Here,
L is the length of the link,
the linear mass density,
E the Young’s modulus, and
I the area moment of inertia about the bending axis.
Moreover, the motor is controlled in order to provide precise and fast motor positioning responses. The control structure proposed in [
32] is used. It is a 2DOF proportional-integral-derivative (PID) controller with low-pass filters that ensures good trajectory tracking, compensates for disturbances such as unmodeled components of the friction, and is robust to parameter uncertainties.
For the data acquisition and control algorithms, the program LabVIEW 2010 is used. The sampling time is ms for data acquisition (measurements, control signals, and written data). In addition to this, MATLAB R2022b is utilized to implement the methods described below.
4. Dynamic Model
In this section, the behavior of a flexible sensing antenna is modeled using fractional-order models for the damping phenomena. With this aim, we obtain the exact transfer function between the motor angle (measured by the encoder) and the coupling torque of the antenna. This allows us to study only the dynamics of the antenna, without considering the motor behavior (which includes non-linear and time-varying friction components).
Therefore, we develop the dynamic model of a slewing flexible beam with fractional-order damping.
Figure 4a details the scheme of the uniform flexible beam mentioned in the previous section. The following frames of reference are defined: an inertial frame
and a non-inertial frame
, which rotates attached to the motor. The rotation angle of
with respect to
is called the angle of the motor
,
is the transversal flexible deflection of the beam with respect to the frame
, and
is the coupling torque between the beam and the motor.
The deflection
is described by the Euler–Bernoulli beam theory [
33] under the following hypotheses:
The deflection is small enough with respect to x to approximate by .
The lateral deflection of the beam (out of the plane) is assumed to be zero.
The cross-sections of the slender beam remain plane and orthogonal to the beam axis even after deformation. Moreover, rotatory inertia and shear deformation can be neglected because our antenna is very slender.
Furthermore, damping phenomena are classically categorized as internal and external. Usually, when integer-order damping is taken into account, both dampings are incorporated into a single one. In other works, this single term is modeled as a fractional-order (see, e.g., [
7]). In our model, both dampings are independently considered: internal damping is proposed to model the energy dissipated by the internal friction of the beam, which is related to the beam material, whereas external damping describes the beam vibrating in the environment.
Hence, the equation of motion is calculated from the equilibrium of moments in a differential segment of the beam. In this way, the two kinds of damping are included by using the appropriate force or moment [
6]. Therefore, by considering the differential segment of the beam represented in
Figure 4b, the following equation is obtained by the equilibrium of moments
where
is the bending moment, and
is the shear force [
7].
The bending moment
is correlated with the deflection
. When internal damping is considered, as Kelvin–Voigt damping [
6], the bending moment
is the union of two moments that model the elastic and viscous behavior. Moreover, if a fractional-order model for viscoelastic materials [
34] is used, the bending moment can be defined as
letting
and
(
) be the damping coefficient and the internal damping order, respectively.
The shear force
is calculated from the force equilibrium of the differential segment as
where the right-hand term is the inertial force, and
is the distributed force over the beam. Since we include external damping in the model, the distributed force
is equal to the damped force exercised by the environment over the beam. This kind of damping is usually modeled as a damping force proportional to velocity when considering the air [
6]. In our case, we assume it as a fractional-order damping force as follows
where
and
(
) are the damping coefficient and the external damping order, respectively. It should be highlighted that the total velocity of a differential segment of the beam is the addition of two terms: the deflection velocity and the linear velocity in
x generated by the rotation of the motor.
Remark 1. Typically, the fractional order α is confined to the range of , as noted in [11,34,35]. Nonetheless, we have elected to expand this definition to encompass the range of , similarly to [7]. This assumption is also considered for the order λ. Remark 2. Fractional derivatives are defined in the Caputo sense [35]. Finally, the complete equation of motion, also called the Euler–Bernoulli equation, is obtained from Equations (
1)–(
4) as
where (
) represents a spatial derivative with respect to
x. In order to solve it, two initial conditions and four boundary conditions are needed. The four boundary conditions are the same as for a clamped-free beam, where the beam is fixed to the motor in
and it is free in
, i.e.:
Additionally, the relation between the coupling torque
(which is equal to the moment at the link base) and the deflection
is given by
Since we want to obtain the exact transfer function between
and
, the differential Equation (
5), the boundary conditions (
6)–(
9), and Equation (
10) are Laplace transformed with respect to the time derivative [
19]. Then, Equation (
5) results in
where
and
are the Laplace transforms of
and
, respectively, and the damping coefficients
and
are defined. Furthermore, making
, we have
Additionally, the boundary conditions (
6)–(
9) become
and Equation (
10) is
with
the Laplace transform of
.
In the first place, the system of partial differential equations formed by (
11)–(
15) is solved. This system can be expressed as a state-space representation with the space variable
x as the independent one by
So, Equation (
17) is
where we define
For the sake of simplifying the notation, the dependency of
s on matrices
and
will be omitted from now on.
Despite Equation (
18) not looking like a conventional state-space equation, it can be solved as a state-space model continuous and space-invariant. With that aim, we use the state-transition matrix. Then, the general solution is
where the state-transition matrix, keeping in mind that
is constant for all
x, is
Moreover, the integral of Equation (
19) can be solved taking into account that
is constant and using the form of
. Then, the solution of
is
where the value of
needs to be known. With that purpose, the particular solution of (
21) in
is combined with the boundary conditions (
12)–(
15), resulting in a set of eight linear algebraic equations. By solving them, we can obtain the values of
and
.
Furthermore, the transition matrix can be evaluated using the inverse Laplace transform as
(converting a function dependent on
p to another in the space domain). Since
the exponential matrix is
with
.
Finally, taking Equation (
16) with
from the solution of
, the exact transfer function between the coupling torque
and the motor angle
can be obtained as:
The above transfer function is expressed in terms of transcendental functions of
, and differs from the result of [
19] in the inclusion of the fractional damping, which affects the form of expressions (
25) and
.
7. Conclusions
This paper has addressed the dynamic modeling of a flexible antenna that is used for haptic navigation of mobile robots. The fast and vibration-free non-contact movement of the antenna is of utmost importance in order to obtain efficient haptic systems. This requires a high-performance controller, whose design must be based on a very precise dynamic model of the system.
We have developed a fractional-order general model that allows us to accurately describe the dynamics of flexible links involving internal and external damping, which are related to the link material and the environment where it is located, respectively. Likewise, we have investigated its adequacy for a carbon fiber antenna that moves freely in the air. In particular, six models with different combinations of dampings (of integer or fractional order) have been identified using frequency-domain methods from experiments conducted with our antenna. In total, 240 models have been obtained (20 experiments performed, six structures of the dynamic model, and two forms of the cost function considered for the non-linear optimization identification procedure). All the identified models have been assessed in terms of fitting accuracy to experimental curves and robustness (i.e., the variation in the estimated parameters when the identification is repeated several times).
After this process, we concluded that the best tradeoff between fitting accuracy and identification robustness is given by a model that includes an internal damping of integer order and an external damping of fractional order close to 0.5. Since damping has a small value in our antenna that moves freely in the air, the improvement attained by introducing fractional-order derivatives in the model has a low absolute value. However, the relative value of that improvement is noticeable.
Likewise, the proposed general model, of fractional order, will be valuable to accurately describe the dynamics of flexible links made of different materials that move within different environments by means of both the included internal and external damping, respectively. Our future work will be in such a direction: using the model to characterize the behavior of an antenna with different properties (e.g., made of carbon fiber and coated with a silicone pseudo-skin or of a polymer) immersed in water or other fluids.