Experimental Platform for Analyzing Friction Models Applied to Mechanical Systems with Revolute Joints

Abstract

This article presents an experimental platform for testing friction models used by control strategies on a one-degree-of-freedom mechanical motion control system. This platform aims to carry out experiments to estimate dry friction parameters and related motion control strategies. The presented device can be built using low-cost components available in most laboratories. The platform enables both a correct friction parameters estimation and the experimental validation of related motion control strategies. The proposed platform can be applied to the validation of a wide spectrum of parameter identification and motion control procedures. Experimental results illustrate the usability of the proposed device for research purposes. However, the platform could be used as an educational device to illustrate the performance of specific friction models with various control strategies.

1. Introduction

Friction plays an important role in the motion control of mechanical systems since it is a resistive and non-conservative force. Dry friction is the force that appears due to contact between two surfaces. It is usually transformed into heat, and noise and can even generate wear of materials causing deformations. This topic has been addressed and analyzed by many researchers because it is difficult to predict its real impact on mechanical systems during their motion [1,2,3].

Several studies have analyzed how friction and wear alters the physical integrity of mechanical systems, and how this affects control performance (e.g., [4,5,6,7]). On the other hand, increasingly complex models have been proposed to model the effect of friction, such as those from LuGre and Dahl [1,8,9]. This effect is usually difficult to model, and due to this, the controller commonly compensates friction by adding the system-estimated friction forces. Such a procedure allows the system to reach the desired state when the estimated friction parameters are correct. The effect of friction may also be considered in the system model. However, this raises the difficulty of solving its related evolution equations because of the discontinuities that arise in most friction models.

Friction testing machines have been developed to study friction effects on diverse objects and interfacing components. These friction testing machines are commonly used to study wear and deformation generated by friction on certain materials [10,11,12]. These studies are presented without providing details on the design and construction of the testing machine. Additionally, these prototypes are usually aimed at characterizing friction and search for the causes of this phenomenon.

In the field of robotics and mechatronics, friction is not generally used as a design parameter. Instead, it is an observed phenomenon that requires compensation [13]. The overall friction effect affecting the system is estimated based on the selected friction model and is compensated afterwards. The selected friction model defines the parameters to be identified and these parameters are identified according to some estimation method (e.g., [14,15,16,17,18]). The approaches that follow this methodology typically rely on industrial robots for experimental validation [19,20,21,22]. Therefore, laboratory platforms exclusively dedicated to the validation of control theories involving friction, in the general context of dynamical systems (such as robotics and mechatronics), are not generally built to this end. As an alternative, this work proposes an experimental setup dedicated to the study of friction effects that affect robotic and mechatronic systems.

This paper presents and describes a platform originally developed for the experimental validation of a geometric optimal control methodology, developed by the authors, where friction forces are considered at the modeling level [23]. The motivation for developing this experimental platform came from the observation that the traditional feedback-level friction compensation degrades the optimal control performance in real-life conditions because unmodeled friction forces are added by the controller and modify the optimal signal (see the experimental results in [24,25] for example). Therefore, an experimental platform where the theoretical analysis relating to the modeling of mechanical systems with revolute joints, subject to friction, was required. The platform shall enable experimentation with a wide range of friction models. The general idea is to evaluate the performance of both a selected friction model and control strategy, by assessing how close or how far does the system ends up with respect to a set goal position.

Following this brief introduction, the next section describes the design of the experimental platform and its components. Section 3 details the construction of the platform. The main steps to operate the experimental platform are described in Section 4. The experiments described in Section 5 validate the experimental platform. The manuscript ends with the conclusive Section 6. Appendix A describe an in-house designed current sensor.

2. Design

The main idea was to build a platform that would allow us to carry out motion control experiments on a dynamical system affected by joint-level dry friction. With this in mind, a single-degree-of-freedom (one-DOF) mechanical device equipped with a rotating actuator was built. The device is equipped with a braking element in order to enhance the effect friction on the system. This feature brings in the possibility to regulate the intensity of joint-level friction forces during motion.

The components of the experimental platform are: an actuator, a power supply, a position sensor, a current sensor, a control system, and a pendulum with a brake. The actuator is a direct current (DC) brushed motor. This choice ensures control ease as compared to an alternative brushless DC motor or any alternating current motor. This is due to a more specific power stage and control strategy requirement for either one of these two alternatives [26]. Since the DC motor is controlled by direct current, a linear programmable power supply that allows for low voltage DC motor control was chosen. This is necessary because the maximum DC motor voltage feed is of 90 $\mathrm{V}$, so a high power stage is usually required.

An optical encoder is used to measure the angular position of the pendulum. The selected encoder has good resolution and performs well at low speeds when compared with other alternatives such as magnetic, hall-effect, or capacitive encoders. Note that these alternative encoders often require serial communication to reach their maximum resolution. An in-house designed current sensor ensures current measurement. It is composed of a shunt resistor (low-value resistor) and an amplifier. The schematic can be found in Appendix A.

The control system consists of two main parts. The first part is formed of a CPU with all the necessary software for running the experiment in real-time. In our case, the Simulink Matlab software that ensures real-time operation is used. The second part deals with data acquisition (DAQ) and is composed of a card that sends the required voltage to the DC motor and handles the received current and position information. In our case, the Quanser Q4 DAQ card was selected. It has analog and digital inputs and outputs and can handle most devices without additional conditioning stages.

The following components constitute the proposed experimental platform:

• a McMillan Electric Company C3350B3006 permanent magnet DC motor with a metallic disc and a brake on its rim (see Figure 1a), from McMillan Electric Company, 400 Best Road, Woodville, WI 54028, USA;
• a metallic pendulum bar on the actuator load axle (see Figure 1a);
• a UPM-2405 linear powersupply (see Figure 1d);
• a 12 bit resolution (4096 steps per revolution) optical encoder (see Figure 2b);
• an in-house designed current sensor;
• a control system, in this case, composed of a computer running Matlab Simulink software along with a Quanser Q4 DAQ card, operating in real-time (QuaRC software version 1.1, sample period of $0.001$ $\mathrm{s}$) (see Figure 1c), from Quanser Consulting Inc., 119 Spy Ct, Markham, ON L3R 5H6, Canada.

Most of these materials and components were available in our laboratory (TecNM/ITEnsenada Advanced Robotics Laboratory).

Table 1. Bill of materials.

Quantity	Component	Source of Materials	Material Type	Cost
1	1 $\mathrm{k}\Omega$ resistor (Stackpole Electronics, Inc., 3110 Edwards Mill Rd, Suite 207, Raleigh, NC 27612, USA)	https://www.digikey.com/en/products/detail/stackpole-electronics-inc/CF14JT1K00/1741314 (accessed on 25 January 2025)	Through hole resistor axial carbon film $\pm 5$ %, $0.25$ $\mathrm{W}$	$0.10
6	10 $\mathrm{k}\Omega$ resistor (KOA Speer Electronics, Inc., 199 Bolivar Dr, Bradford, PA 16701, USA)	https://www.digikey.com/en/products/detail/koa-speer-electronics-inc/CF1-4CT52R103J/13537366 (accessed on 24 January 2025)	Through hole resistor axial carbon film $\pm 5$ %, $0.25$ $\mathrm{W}$	$0.60
1	Optical encoder (SKY Devices LLC, 1348 Washington Ave STE 350, Miami Beach, FL 33139-4212, USA)	https://www.sameskydevices.com/product/motion-and-control/rotary-encoders/incremental/modular/amt13a-series (accessed on 24 June 2025)	Incremental, 12 bit resolution (4096 steps per turn) or higher	$41.84
1	20 $\mathrm{k}\Omega$ resistor (Stackpole Electronics, Inc., 3110 Edwards Mill Rd, Suite 207, Raleigh, NC 27612, USA)	https://www.digikey.com/en/products/detail/stackpole-electronics-inc/CF14JT20K0/1741340 (accessed on 25 January 2025)	Through hole resistor axial carbon film $\pm 5$ %, $0.25$ $\mathrm{W}$	$0.10
1	21 $\mathrm{k}\Omega$ resistor (YAGEO Corporation, 3F, 233-1, Baoqiao Rd., Xindian District, New Taipei City, Taiwan 23145)	https://www.digikey.com/en/products/detail/yageo/MFR-25FBF52-21K5/13282 (accessed on 20 December 2024)	Through hole resistor axial carbon film $\pm 5$ %, $0.25$ $\mathrm{W}$	$0.10
4	741 Operational Amplifier (Texas Instruments, Post Office Box 655303, Dallas, TX 75265, USA)	https://www.digikey.com/en/products/detail/texas-instruments/UA741CP/382197 (accessed on 3 March 2025)	Standard operational amplifier	$2.28
1	Brake	In-house designed and built	Plastic bar ($200\mathrm{mm}\times 50\mathrm{mm}\times 10\mathrm{mm}$)	Not available
1	Current sensor	In-house designed and built	See Appendix A	The price is according to the required components
1	McMillan Electric Company C3350B3006 Permanent magnet DC motor (McMillan Electric Company 400 Best Road, Woodville, WI 54028, USA)	https://mcmillanelectric.com/permanent-magnet-motors/ (accessed on 8 November 2024)	Brushed, 18 Amperes, 1.3 HP continuos duty at 95 $\mathrm{V}$	Legacy device (current price not available)
1	Metallic bar (pendulum)	In-house designed and built	Structural steel bar ($170\mathrm{mm}\times 50\mathrm{mm}\times 4\mathrm{mm}$, 200 $\mathrm{g}$)	Not available
1	Quanser Q4 DAQ card (Quanser Consulting Inc., 119 Spy Ct, Markham, ON L3R 5H6, Canada)	https://docs.quanser.com/quarc/documentation/q4.html (accessed on 20 November 2024)	Data Acquisition Card with 14-bit analog inputs , simultaneous sampling of A/D and encoder inputs, simultaneous output to D/A channels, 16 digital input outputs, and two 32-bit multipurpose counters	Legacy device (current price not available)
1	Current Sense Resistor GMR100HJBFA10L0 (ROHM Co., Ltd., 21 Saiin Mizosaki-cho, Ukyo-ku, Kyoto 615-8585, Japan)	https://www.mouser.mx/ProductDetail/ROHM-Semiconductor/GMR100HJBFA10L0?qs=wT7LY0lnAe1Xj5RsER0nqQ%3D%3D (accessed on 21 November 2024)	Shunt Resistor $0.5$ $\Omega$	$2.83
1	UPM-2405 linear power supply	Not available	Power Amplifier with $\pm 12\mathrm{V}$ power supply, 4 analog sensor inputs and power amplified analog output	Legacy device (current price not available)

reports a list of specifications of the components.

The functional capabilities of the proposed platform are the following.

• 360° of motion range around its rotation axis (in both directions).
• Current sensor range between $-5$ $\mathrm{A}$ and 5 $\mathrm{A}$ (10 bit resolution).
• Maximum torque of $0.8$ $\mathrm{N}$ $\mathrm{m}$ with a voltage supply of 24 $\mathrm{V}$.
• Compatible with real-time experimentation.
• Adjustable brake to increase or decrease friction effects.

3. Build Instructions

For the construction of the experimental platform, it was necessary to make some previous modifications to the components and manufacture some parts and devices. First, a pendulum was manufactured with a mass based on the torque that our motor can deliver with our power supply, in this case, approximately 200 $\mathrm{g}$ (see Figure 2a). Classical design and actuator dimensioning procedures can be followed [27]. This pendulum is fixed to the load axis of the motor using two screws with nuts. Also, it was necessary to directly couple the encoder to the load axis of the motor. To do this, the internal components of the encoder were removed and directly coupled to the load axis by the rear part of the DC motor (see Figure 2). In Figure 2b it can be observed that for the coupling, a custom-made bushing was used that joins the rear part of the motor with the internal axis of the encoder. Additionally, a brake was fitted on the rim of the metal disc. This brake consists of a metal bar held over the rim of the metal disc with a plastic material at the tip. The bar exerts pressure on the metal disk that is coupled to the motor shaft to increase friction. This brake can be set up in different ways in order to increase or decrease friction effects.

Figure 3 shows the drawing of the actuated pendulum system and highlights its main components. The front view features the metallic disc used by the braking system. The brake is a plastic bar pushed toward the edge of the metallic disc surface through an adjusting screw. The pendulum described earlier is also shown. The back view displays the encoder and the bushing coupling as previously described. The encoder is mounted on a bracket, and the codewheel is coupled to the motor shaft by a bushing. The technical drawings detailing the design of each part of the proposed experimental platform are provided as Supplementary Material.

Since we require a current of about 5 $\mathrm{A}$ (maximum current delivered by the source used), a current sensor (see Appendix A) was specifically designed and built for this project. This current sensor is based on a shunt resistor for measurement and an amplification stage which is classical in the literature [28].

The power supply takes in a low-voltage sent by the DAQ card Quanser Q4 (in a range between $-5 \mathrm{V}$ to $5 \mathrm{V}$), and outputs a voltage in the range of $-24 \mathrm{V}$ to $24 \mathrm{V}$. The current sensor measures the current consumed by the DC motor and converts it to a proportional voltage, which is then delivered to the DC motor. This voltage is then processed by the Quanser Q4 DAQ card.

The optical encoder measures position. The Quanser Q4 card has encoder inputs that require minimal configuration steps. This is sufficient to process the data coming from the encoder within Simulink. Note that according to its reference manual, the Quanser Q4 DAQ card and its handling QuaRC software require Matlab Simulink version 7.0. Depending on the selected data acquisition card, a different software may be required.

The following steps summarize the building process:

• Install Matlab software with real-time Simulink version 7.0 (or an alternative software that handles a DAQ card of choice).
• Install the Quanser Q4 DAQ card (or an alternative DAQ card of choice) along with the terminal board (see Figure 1b) and necessary drivers (for more information on this card see the official page of Quanser [29]).
• Fix the DC motor to a base and connect it to the programmable power supply.
• Connect the current sensor input to the DC motor and the output to one of the DAQ card analog inputs.
• Connect the UPM-2405 programmable power supply, optical encoder, and current sensor output to the terminal board (analog output, encoder input, and analog input, respectively).
• Use Matlab Simulink to design the control system.
• Turn on the programmable power supply and power the current sensor.
• Run the control system using Matlab Simulink set up as “External Mode”.

4. Operating Instructions

The experimental platform is intended to serve as a testing machine for robotics, where the overall friction of the system is commonly compensated for at the controller level. Experiments can be performed using any methodology that involves offsetting the total friction of the system, regardless of the specific friction model, parameter estimation technique, or selected controller. The shown examples are given without loss of generality for similar procedures. Therefore, in this section, the operating instructions will be described following the methodology presented in [23]. This experimental platform was specifically designed to carry out experiments for that work, so these experiments will be explained. Note however that the proposed platform can be used for a wide variety of motion control experiments.

The procedure to perform these experiments consists of three main parts:

• Estimation of electrical parameters of the DC motor.
• Estimation of the mechanical parameters of the prototype.
• Control system design.

4.1. Estimation of Electrical Parameters

The electrical parameters of the DC motor require estimation in the absence of a torque sensor. Following this approach, an inner control loop is used to ensure the desired torque. The desired torque is reached by calculating the current and multiplying it by a torque constant. An experiment is carried out to obtain the electrical parameters, and these parameters are then used in the inner control loop.

Let us consider the motor model where the armature inductance is neglected (due to its low value) [30]:

$$v=k_b\dot{q}+R_a{i}_a,$$

(1)

where v is the armature voltage, $R_a$ the armature resistance, $i_a$ the current y $k_b$ a constant of back electromotive force. The output torque of the DC motor $\tau$ is proportional to the current $i_a$, so that

$$\tau =k_a{i}_a,$$

(2)

where $k_a$ is the torque constant of the DC motor. Considering $k_a=k_b$ [31] and dividing (2) by $i_a$ on both sides of the equation we obtain

$${\displaystyle \frac{v}{i_a}}=k_a{\displaystyle \frac{\dot{q}}{i_a}}+R_a.$$

(3)

From the above, the electrical DC motor parameters, $k_a$ y $R_a$ can be estimated by measuring v (through a voltmeter), $i_a$ (through the designed current sensor), and $\dot{q}$ (differentiating the position read from the optical encoder. Let us remark that Equation (3) is that of a straight line, where ${\displaystyle \frac{v}{i_a}}$ is viewed as the dependent variable and ${\displaystyle \frac{\dot{q}}{i_a}}$ is viewed as the independent variable. Therefore, an experiment is performed where the voltage v is varied while the current $i_a$ and the speed $\dot{q}$ are measured. The results of this experiment are shown in Figure 4 which shows a first-order linear interpolation. The motor torque constant $k_a$ corresponds to the slope of the regression line and $R_a$ corresponds to the height at the origin. According to (3), the electrical parameters are $k_a=0.1762\mathrm{N}\mathrm{m}/{\mathrm{A}}^{-1}$ (slope) and $R_a=5.54\Omega$ (ordered at the origin).

It is important to consider that the experiments aimed at the DC motor parameters estimation were all conducted without any load or braking element element attached. Note that varying loads can affect the outcome of the parameter estimation process but this analysis is beyond the scope of the presented work. Parameter estimation experiments were conducted under an uncontrolled environment in terms of temperature and other ambient conditions. To reduce the impact of the environmental conditions, the experiments described in the next section were done briefly after completing the parameter estimation process. Finally, the voltage source used and the Quanser Q4 data acquisition card have filtering stages which minimize electrical noise.

The input to the system is given by a reference torque ${\tau }_{\mathrm{ref}}$. For this, an inner control loop is used to ensure that ${\tau }_{\mathrm{ref}}=\tau$ ($\tau$ is the exact system torque). This inner loop is a current PI controller as shown in Figure 5. This is the controller that is used for open-loop control experiments. The following gains were used in our experiments are $k_p=5$ and $k_i=5000$. Following (2),

$$i_{\mathrm{ref}}={\displaystyle \frac{{\tau }_{\mathrm{ref}}}{k_a}}$$

(4)

where $i_{\mathrm{ref}}$ is the current which ensures that ${\tau }_{\mathrm{ref}}$. The used controller is

$${\displaystyle \frac{\tau }{k_a}}=i=k_p(i_{\mathrm{ref}}-i)+k_i\int (i_{\mathrm{ref}}-i)$$

(5)

4.2. Estimation of Mechanical Parameters

The mechanical parameters of the pendulum also require estimation. If the machine elements were previously designed and machined with precision, the parameter estimation will bring only minimal adjustments to the theoretical data. Mechanically speaking, these adjustments usually arise from the use of cables and other non-modeled machine elements. Then, the system parameters are used to obtain the optimal trajectories and torques that will be tracked by the controller according to the optimal control methodology presented by the authors in [23].

Considering the pendulum model that includes the smooth Coulomb-tanh friction model [1], that is

$$I\ddot{q}+mglsinq+f\left(\dot{q}\right)=\tau ,$$

(6)

where I is the moment of inertia, m is the mass, g is the gravitational acceleration, and l is the distance from the axis of rotation to the center of mass. The function $f\left(\dot{q}\right)=f_{c}tanh\beta \dot{q}$ is the smooth Coulomb-tanh friction model. For the experiments where friction is compensated at feedback-level, constants $f_c>0$ and $\beta >0$ ($\beta =100\mathrm{s} {\mathrm{rad}}^{-1}$ is fixed). Finally, $\tau$ is the torque applied by the DC motor; and q, $\dot{q}$ and $\ddot{q}$ the position, velocity, and angular acceleration of the pendulum, respectively.

Rewriting (6) to take only a set of 3 parameters; we have

$${\theta }_1\ddot{q}+{\theta }_{2}gsinq+{\theta }_{3}tanh\beta \dot{q}=\tau ,$$

(7)

where ${\theta }_1=I$, ${\theta }_2=ml$ and ${\theta }_3=f_c$ are the unknown mechanical parameters of the pendulum.

To obtain the mechanical parameters of the pendulum, the “Parameter Estimation” tool of Matlab Simulink is used. To use this tool, it is necessary to design the dynamic model of the pendulum (7) in Simulink. Subsequently, an open-loop experiment is needed to obtain the goal reference position and so the input is given by

$$\tau \left(t\right)=0.3(1-e^{-1.8t})+0.1sin(26t+0.08)+0.1sin(12t+0.34).$$

(8)

To perform this experiment, follow the block diagram shown in Figure 5. This diagram can also be followed to perform open-loop experiments.

The procedure consists of running the designed pendulum model in Simulink with approximate parameters ${\theta }_1$, ${\theta }_2$, and ${\theta }_3$ (small values around ${10}^{-3}$ are suggested). Then, the reference position coming from the motion planning guides the system toward the one obtained in the experiment using the “Parameter Estimation” tool. Then following the optimal control methodology described in [23] the estimated parameters are used to obtain the optimal trajectories, velocities, and torques. The parameter estimation procedure is provided without loss of generality. To work with more complex friction models, more parameters would require estimation to meet the goals of the new friction model.

4.3. Control System Design

The estimated system electric and mechanical parameters are used in the controller design (inner control loop and external control loop). We will proceed to design the One-DOF mechanical system on Matlab Simulink. As mentioned above, the Real-Time Simulink software is required since it enables the “External Mode” to connect the DAQ card. The DAQ card collects the required data during motion control (current measurement, pendulum position, etc.). This data is made available through blocks in Matlab Simulink.

The blocks are enabled by the HIL library. HIL library allows the use of the inputs and outputs of the DAQ target Quanser Q4 card, within the Matlab Simulink environment, to be used in real-time.

The designed controller is of the Proportional-Derivative PD type. Reference positions and velocities plus torques are required. The one-DOF mechanical system on Simulink is built based on blocks. Figure 6 shows the block diagram that describes this system with the controller in a closed loop. Note that this controller does not use any type of compensation at the feedback level. In the following experiments, the position, velocity, and torque of reference come from the optimal control methodology presented in [23]. The controller is

$$\tau =K_p\tilde{q}+K_v\dot{\tilde{q}}+u_{\mathrm{ref}},$$

(9)

where $\tilde{q}=q_{\mathrm{ref}}-q$ is the position error; $\dot{\tilde{q}}=\dot{q}-q$ is the velocity error; $u_{\mathrm{ref}}$ is the reference torque (optimal torque); $K_p$ and $K_v$ are proportional and derivative gains (for our experiments $K_p=1.2$ and $K_v=0.005$.

The Simulink block design for this experiment is of the closed loop type and is represented by Figure 7. which shows the control system built in Simulink. The external control loop is intended to reach the reference states. The requested torque is delivered to the inner control loop that ensures the desired torque using the current measured by the current sensor. The DC motor is fed with the voltage corresponding to the desired current. Position and current sensors inform on current and position status.

5. Validation

This section presents some results obtained with the described experimental platform. The following shows the type of experiments for which the presented device was built, but keep in mind that a wider variety of motion control experiments can be analyzed with the proposed platform.

Following the optimal control methodology that incorporates friction within the model [23], some experiments are carried out. For the open-loop experiment, the diagram shown in Figure 5 was followed. For the closed-loop experiment the diagram shown in Figure 6 was followed instead. A word on the notation used in this section:

• an open-loop variable is marked with a tilde on top ($\tilde{q}$ is the open-loop position);
• a closed-loop variable is marked with a bar on top ($\overline{q}$ is the closed-loop position).

Figure 8 shows a comparison between open-loop and closed-loop experiments where friction is incorporated into the model, as in [23]. Figure 8a showcases the appropriate performance of the optimal control methodology proposed in [23] because the system reaches the goal even when controlled as an open loop. This means that the selected friction model is appropriate for this system. The rest of the subfigures in Figure 8 show some other quantities of interest: experimental velocity (Figure 8b), torque (Figure 8a) and current (Figure 8d) values as compared with their respective references. The experimental system behavior is close to the simulated one even in open-loop.

Noise on the experimental data can be observed. However, this is a quite common issue that usually arises due to parametric uncertainties and data measurement. It is important to mention that even if the physical components of the system (voltage supply, DAQ card or encoder) incorporate filtering stages, the presence of electrical noise is inevitable. Specialized software such as Simulink offer the possibility to include filtering blocks such as low-pass or Kalman filters, which should be used to improve the signal quality. In our particular scenario, the detected noise was not large enough and either the low-pass or the Kalman filter gave unsatisfactory results leading to important data loss. Also, results can be improved by using a more complete friction model or by using electrical components with higher precision and resolution. Based on the experimental data, it can be confirmed that the platform fulfills its purpose. To run the real-time experiments, a real-time kernel was used. No latency problems were detected while conducting experiments.

Let us consider another experiment where friction is compensated at the feedback level. The results are compared with those from Figure 8 where friction is incorporated into the optimal control model. The diagram shown in Figure 9 is followed to carry out this experiment. The diagram shows that friction is added to a PD position controller. This is common practice in modern control and usually achieves acceptable results.

Figure 10 shows the experimental results of this new experiment, as compared with the data coming from Figure 8 (where friction is incorporated into the optimal control model). The goal position is achieved either way (Figure 10a), however, the trajectory differs greatly. In the feedback-level friction compensation case (denoted as FLFC in Figure 10), oscillations are induced by the online addition of the friction forces, to a signal that did not consider these effects. This can also be observed on the velocity and torque curves of Figure 10b and Figure 10c, respectively. The proposed platform enables this kind of analysis.

6. Conclusions

The proposed experimental platform here fulfills its main purpose: it enabled the experimental validation of an optimal control procedure that incorporates friction effects into the system model. However, the device can also be used to analyze and validate a wide variety of parameter identification and motion control methodologies where joint-level friction arises. Note that our approach focuses on robotic systems where the total system friction is typically estimated and then compensated for in the controller. Therefore, we do not analyze the specific effects caused by individual components, such as the roughness of the metal disc or the friction at the disc edge with the plastic partas friction-model construction is beyond the scope of this work. The estimated parameters are derived based on the chosen friction model, as described in previous sections. Although we do not consider how each component influences the overall friction, we obtain the parameters that allow the selected friction model to match the friction effect that affects the system. This procedure has allowed the validation of optimal control methods that take friction into account within the system model.

The intended area of application for the proposed device is research, as shown by reference [23], where an earlier version of this device was used to validate complex optimal control theories that include friction within the system model. The targeted device is a laboratory platform aimed at testing and validating friction models used in practical robotics and mechatronics applications. In this sense, this work may serve as reference to develop more elaborate prototypes for the laboratory. However, the presented prototype may serve as an educational tool for dynamical systems friction models testing. To this end, a practical test involves driving the system to a goal position using several friction models and comparing the performance of each model during motion.

The design process and main steps to build and operate a similar system have been described. Two main experiments have been included to illustrate the kinds of analysis that can be carried out. Experimental results show that the device operates correctly under the controllers that were used. Reference trajectories were correctly followed thanks to adequate friction and system parameters estimation. Let us remark that the Coulomb-tanh friction model that is used for experimentation is just one example among many more (see e.g., [1,2,3]) and is used as a starting point. However, the described platform is not restricted to using this particular model. Therefore, experimentation with more complex friction models could be carried out with ease. These experimentations will be left for future work, along with the improvement of the platform that shall be achieved by using devices and conditioning stages with better characteristics.