Trajectory Tracking of a 2-Degrees-of-Freedom Serial Flexible Joint Robot Using an Active Disturbance Rejection Controller Approach

Abstract

This paper presents the development of an Active Disturbance Rejection Controller (ADRC) to address the trajectory tracking problem of a 2DOF (Degrees of Freedom) Serial Flexible Robot. The proposed approach leverages differential flatness theory to determine the system’s flat output, simplifying the trajectory tracking problem into a linear state feedback control with disturbance rejection. A set of a Generalized Proportional Integral Observer (GPIO) and Luenberger observers is employed to estimate the derivatives of the flat output and both internal and external disturbances in real time. The control law is experimentally validated on a 2DOF Serial Flexible Robot prototype developed by Quanser. Quantitative results demonstrate that the ADRC achieves superior performance compared to a partial state feedback control scheme, with a Mean Squared Error (MSE) as low as 1.0651 × 10⁻⁵ rad² for trajectory tracking. The ADRC effectively suppresses oscillations, minimizes high-frequency noise and reduces saturation effects, even under external disturbances. These findings underscore the robustness and efficiency of the proposed method for underactuated flexible systems.

1. Introduction

In the past few decades, the interest of researchers in human–robot interaction systems has increased more than ever due to the need to have shared spaces between robots and humans where both can collaborate in a safe manner [1]. For this reason, researchers have been looking for strategies that minimize the risks associated with physical contact during the robot’s interaction with the environment [2,3]. To minimize these risks, multiple strategies have been proposed, particularly focusing on the mechanical design of the robots. In recent years, researchers have put special emphasis on Flexible Joint Robots (FJRs) [4].

Flexible Joint Robots (FJRs) have been widely studied because they provide multiple advantages, like variable mechanical impedance, which minimizes the risks associated with physical contact of the robot with other structures [5,6], and also the capacity to store mechanical energy due to flexible joints, which is useful in some applications, like semi-active systems [4,7]. Due to these advantages, FJRs have been used in a wide range of applications that require high performance and a soft interaction between the robot and its environment [3]. Some of these applications include flexible manipulators for aerospace applications [8,9,10], robots that handle dynamic loads in machining [11] and milling applications [12], medicine [13] and industrial flexible robots [14,15,16].

Unlike traditional rigid robots, where the problems of position control and trajectory tracking are well known and understood [3], one of the main difficulties when working with Flexible Joint Robots (FJRs) is the development of controllers that provide accurate trajectory tracking and positioning control while suppressing the vibrations and oscillations that occur during movement due to the flexible joints of the robot [17]. With this in mind, it is possible to highlight two challenges that contribute to this difficulty.

The first challenge is the difficulty in modeling FJRs [1,18] due to the oscillatory disturbances caused by the nature of their mechanical structure. These disturbances, if not taken into account, can greatly affect the accuracy and stability of the robot [19]. The second challenge is the selection and design of the control scheme in order to obtain accurate trajectory tracking while suppressing the vibrations [20,21]. Related to the first challenge, this difficulty also lies in the need to take into consideration the disturbances caused by the flexible joints.

In the last few years, researchers have put a lot of effort into solving these challenges and have come up with new methodologies and strategies for both modeling and controlling FJRs. With respect to the first challenge, multiple modeling methods have been proposed in the literature. One of the most common modeling methods is using the Lagrangian principle [22]. In the case of FJRs, the Lagrangian is generally approximated using the Constant Mode Shapes Technique (CMST) to obtain a set of Ordinary Differential Equations (ODEs) that describe the system dynamics [23]. Other authors have proposed modeling methods based on the Hamilton approach, which is useful for designing more accurate and stable control systems [24,25]. In addition, other methods include the use of identification procedures based on the Generalized Maxwell-Slip friction model [26]; a bisection method-based algorithm for rapid convergence during inverse dynamics analysis [27]; the use of multi-scale neural networks [28] and the formulation of the dynamic model of the FJR using Partial Differential Equations (PDEs) based on the Hamilton principle [29,30].

Regarding the second challenge, multiple authors have proposed different control strategies to deal with the disturbances present in FJRs. Among these, it is possible to highlight the use of backstepping control [31]; the development of an Adaptive Noise Cancellation-based Tracking Controller using self-generated sinusoidal disturbances (ANC-TC) [32] and the development of a Robust Constrained Moving-Horizon $H_{\infty }$ controller based on a Model Predictive Control scheme (MPC) [21]. A simulation of human gait movements using the activity of the central nervous system and the muscles is presented in [33]. An internal MPC scheme was set up and played as the human CNS to estimate the trajectory and to calculate the required signal for muscles to perform the movement. MPC computes the required torques for each joint and generates optimal trajectories subject to human physical constraints for muscles. The results of the simulation are analyzed and compared to real human gait motions captured by a real motion capture system (VICON). In comparison, ADRC and MPC both allow accurate trajectory tracking, but MPC needs the mathematical model, and ADRC could neglect part of the model and estimate it online as a disturbance. MPC can predict the optimal trajectory, but its prediction depends on information on the system’s constraints and the environment provided to the algorithm.

Active Disturbance Rejection Control (ADRC) has emerged as a powerful technique in various control applications, demonstrating robustness against internal and external disturbances. Recent advancements include its application in flexible aerocrafts, where a modified Extended State Observer (ESO) successfully suppresses vibrations and enhances stability [34]. Modularized ADRC solutions have also been developed for motion control systems, achieving improved set-point tracking despite unknown mass variations and disturbances [35]. In torsional systems, ADRC has been shown to effectively compensate for harmonic uncertainties and suppress vibrations with minimal noise amplification [36,37,38]. Furthermore, ADRC has proven its versatility in industrial settings, including robotic arm stabilization and vibration control in civil engineering structures [39,40]. These studies highlight ADRC’s potential for trajectory tracking and vibration attenuation in underactuated and uncertain systems, forming the foundation for its application in this work.

ADRC needs an external trajectory reference generator. MPC and ADRC schemes can solve problems of different natures with excellent results: the use of PI controllers based on fuzzy parameter adjustment [20] and BP Neural Network real-time parameter adjustment for vibration suppression [41]; the use of fuzzy adaptive controllers [42,43]; the development of a Sliding Mode Control (SMC)-based Output Feedback Controller using High-Gain Observers [44]; a Cascade Sliding Mode Controller [45,46] and the development of control schemes based on the Active Disturbance Rejection Control (ADRC) paradigm [47,48,49,50].

This paper presents a novel controller for a 2DOF Serial Flexible Joint Robot based on a combination of differential flatness and Active Disturbance Rejection Control (ADRC). To apply this method, the Flexible Joint Robot is divided in two independent stages; the Flat Output of each stage is calculated using a simplified mathematical model, obtaining a disturbed fourth-order linear model in which only the angular positions are measured and both the angular speed and disturbances are estimated by using a Generalized Proportional Integral Observer (GPIO). The proposed control method allows accurate trajectory tracking while completely removing the robot oscillations.

This paper is organized as follows: Section 2 presents the theoretical foundations and methodology conducted in this research. Section 3 presents the implementation of the main results on a 2DOF Serial Flexible Joint Robot, and the results are analyzed and discussed. Finally, Section 4 presents the main conclusions of this work and possible future studies.

2. Methodology

2.1. The 2DOF Serial Flexible Joint Robot

The platform studied in this article is a 2DOF Serial Flexible Joint Robot experimental setup (as shown in Figure 1), developed by Quanser. It utilizes MATLAB-Simulink 2023b for control implementation, interfaced via a data acquisition device. The following sections describe the mathematical model and the controller to be implemented in this study.

2.1.1. Schematic Representation

Figure 2 presents a serial robot with flexible joints (SRFJ). The system is composed of two flexible joints that are connected in series. Each joint is actuated by a dedicated drive system, represented by a motor. From Figure 2, we have:

• $I_{mi}$: ${\mathrm{Motor}}_i$ currents.

• ${\tau }_i$: ${\mathrm{Motor}}_i$ torque and ${\tau }_i=K_{mi}{I}_{mi}$, where $K_{mi}$ is the torque constant for motor i.

• $J_{ij}$: Load moments inertia for the links $ij$.

• $d_{ij}$: Intermediary load viscous damping coefficients for the links $ij$.

• $k_{si}$: Flexible joint torsional stiffness constants for the joint i.

• ${\theta }_{i2}$: Angular position for the links i.

• ${\theta }_{i1}$: Angular position for motor i.

Here, $i=1,2$ represents the number of the stage, $j=1$ the motor and $j=2$ the link.

2.1.2. Space State Modeling of the System

In order to establish a dynamic model for the SRFJ, the system was decoupled and analyzed in two independent stages, as shown in Figure 3; stage 1 is shown in Figure 3a, and stage 2b is shown in Figure 3b.

Stage 1 Space State Model

The dynamic model is obtained from the Lagrange methodology [1]. The system’s state vector is represented by $x_1$, and it includes the generalized coordinates and their first-order time derivatives. In addition, it is assumed that the current is considered the system input and the viscous friction is neglected in the mathematical model. Stage 1 represents a fourth-order system with $n=4$; it is underactuated due to the motor driving the link via a flexible joint, and it is perturbed by the Stage 2. The state space representation is given by

(1)$x_1^{\prime }=A_1{x}_1+B_1{u}_1,$

where:

$x_1=\left[\begin{array}{c}{\theta }_{11}\\ {\theta }_{12}\\ {\theta }_{11}^{\prime }\\ {\theta }_{12}^{\prime }\end{array}\right],u_1=I_{m1},A_1=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -{\displaystyle \frac{k_{s1}}{J_{11}}}& {\displaystyle \frac{k_{s1}}{J_{11}}}& 0& 0\\ {\displaystyle \frac{k_{s1}}{J_{12}}}& -{\displaystyle \frac{k_{s1}}{J_{12}}}& 0& 0\end{array}\right],B_1=\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{K_{m1}}{J_{11}}}\\ 0\end{array}\right].$

Stage 2 Space State Model

Similar to the previous stage, the state vector of stage 2 is denoted by $x_2$, which contains the generalized coordinates and the first-order time derivatives. The current is considered input, the viscous friction is neglected, and the system is perturbed by stage 1. The state space model is presented as follows:

(2)$x_2^{\prime }=A_2{x}_2+B_2{u}_2,$

where:

$x_2=\left[\begin{array}{c}{\theta }_{21}\\ {\theta }_{22}\\ {\theta }_{21}^{\prime }\\ {\theta }_{22}^{\prime }\end{array}\right],u_2=I_{m2},A_2=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -{\displaystyle \frac{k_{s2}}{J_{21}}}& {\displaystyle \frac{k_{s2}}{J_{21}}}& 0& 0\\ {\displaystyle \frac{k_{s2}}{J_{22}}}& -{\displaystyle \frac{k_{s2}}{J_{22}}}& 0& 0\end{array}\right],B_2=\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{K_{m2}}{J_{21}}}\\ 0\end{array}\right].$

Note that stage 1 and stage 2 have the same simplified mathematical models, which are considered decoupled from each other; each stage has its own constant coefficients because each stage is built with a different motor, length, inertia and flexible joint. The coupled dynamics presented in the 2DOF Serial Flexible Joint Robot, such as the Coriolis force and the dynamics due to inertia, are considered external disturbances that affect the entire system and impact the oscillatory phenomenon presented on the stages.

2.2. Flatness Controller-Based ADRC Design

Each joint parameter and matrix is assumed with the notation i to compute the flatness controller. The general structure for each stage is the same, and their results are generalized to simplify the theoretical results.

Flat Output Computation

In order to compute the flat output, we must first compute the Kalman controllability matrix for the system $C_{Oi}$ (see [51,52]). The Kalman controllability matrix is computed using the system’s matrices $A_i$ and $B_i$ as follows:

(3)$C_{Oi}=\left[\begin{array}{ccccc}{B}_i& A_i{B}_i& A_i^2{B}_i& \dots & A_i^{n-1}{B}_i\end{array}\right],$

Due to stages 1 and 2 being a fourth-order system with $n=4$, $C_{Oi}$ is simplified as:

(4)$C_{Oi}=\left[\begin{array}{cccc}{B}_i& A_i{B}_i& A_i^2{B}_i& A_i^3{B}_i\end{array}\right],$

Computing the each term for $C_{Oi}$ gives:

$A_i{B}_i=\left[\begin{array}{c}{\displaystyle \frac{K_{mi}}{J_{i1}}}\\ 0\\ 0\\ 0\end{array}\right],A_i^2{B}_i=\left[\begin{array}{c}0\\ 0\\ -{\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}^2}}\\ {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}\end{array}\right],A_i^3{B}_i=\left[\begin{array}{c}-{\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}^2}}\\ {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}\\ 0\\ 0\end{array}\right].$

Then, the controllability matrix for stage i is of the form:

(5)$C_{Oi}=\left[\begin{array}{cccc}0& {\displaystyle \frac{K_{mi}}{J_{i1}}}& 0& -{\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}^2}}\\ 0& 0& 0& {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}\\ {\displaystyle \frac{K_{mi}}{J_{i1}}}& 0& -{\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}^2}}& 0\\ 0& 0& {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}& 0\end{array}\right],$

and its determinant is

$\det \left(C_{Oi}\right)=-{\displaystyle \frac{k_{si}^2{K}_{mi}^4}{J_{i1}^4{J}_{i2}^2}}.$

The controllably matrix of each stage is non-singular (controllable), so the differential flatness methodology can be applied to control this system.

Now, the flat output is given by ${\tilde{y}}_i$ and computed as (see [53]),

(6)${\tilde{y}}_i=\beta C_{Oi}^{-1}{x}_i,$

where

$\beta =\left[\begin{array}{cccc}0& 0& 0& 1\end{array}\right],$

Then,

${\tilde{y}}_i={\displaystyle \frac{J_{i1}{J}_{i2}}{k_{si}{K}_{mi}}}{\theta }_{i2},$

Multiplying ${\tilde{y}}_i$ by the gain $ϵ={\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}$, it follows that:

(7)$y_i=ϵ{\tilde{y}}_i={\theta }_{i2}.$

The flat output $y_i$ coincides with the angular position of links ${\theta }_{i2}$. To compute the flat output and its derivatives, we choose the output vector $C_i=\left[\begin{array}{cccc}0& 1& 0& 0\end{array}\right]$ to select the state that corresponds to the flat output; then, the flat output and its time derivatives are calculated using the matrices $C_i$, $A_i$ and $B_i$ as follows:

(8)$y_i=C_i{x}_i={\theta }_{i2}{y}_i^{\prime }=C_i{A}_i{x}_i={\theta }_{i2}^{\prime },$

(9)$y_i^{″}=C_i{A}_i^2{x}_i={\displaystyle \frac{k_{si}}{J_{i2}}}\left({\theta }_{i1}-{\theta }_{i2}\right)y_i^{\prime \prime \prime }=C_i{A}_i^3{x}_i={\displaystyle \frac{k_{si}}{J_{i2}}}\left({\theta }_{i1}^{\prime }-{\theta }_{i2}^{\prime }\right).$

Through the computation of the fourth-order time derivative, we integrate the input system into the model, that is:

(10)$y_i^{\left(4\right)}=C_i{A}_i^4{x}_i+C_i{A}_i^{3}B{u}_i=-{\displaystyle \frac{\left(J_{i1}+J_{i2}\right)k_{si}^2}{J_{i1}{J}_{i2}^2}}\left({\theta }_{i1}-{\theta }_{i2}\right)+{\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}{I}_{mi}.$

Combining (8)–(10), the matrix representation is given by

(11)$\left[\begin{array}{c}{y}_i\\ y_i^{\prime }\\ y_i^{″}\\ y_i^{‴}\\ y_i^{\left(4\right)}\end{array}\right]=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 0& 1& 0\\ {\displaystyle \frac{k_{si}}{J_{i2}}}& -{\displaystyle \frac{k_{si}}{J_{i2}}}& 0& 0& 0\\ 0& 0& {\displaystyle \frac{k_{si}}{J_{i2}}}& -{\displaystyle \frac{k_{si}}{J_{i2}}}& 0\\ -{\displaystyle \frac{\left(J_{i1}+J_{i2}\right)k_{si}^2}{J_{i1}{J}_{i2}^2}}& {\displaystyle \frac{\left(J_{i1}+J_{i2}\right)k_{si}^2}{J_{i1}{J}_{i2}^2}}& 0& 0& {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}\end{array}\right]\left[\begin{array}{c}{\theta }_{i1}\\ {\theta }_{i2}\\ {\theta }_{i1}^{\prime }\\ {\theta }_{i2}^{\prime }\\ I_{mi}\end{array}\right].$

Therefore, the system states can be expressed in terms of the flat output and its time derivatives as follows:

(12)$\left[\begin{array}{c}{\theta }_{i1}\\ {\theta }_{i2}\\ {\theta }_{i1}^{\prime }\\ {\theta }_{i2}^{\prime }\\ I_{mi}\end{array}\right]=\left[\begin{array}{c}{y}_i+{\displaystyle \frac{J_{i2}}{k_{si}}}{y}_i^{″}\\ y_i\\ y_i^{\prime }+{\displaystyle \frac{J_{i2}}{k_{si}}}{y}_i^{‴}\\ y_i^{\prime }\\ {\displaystyle \frac{J_{i1}{J}_{i2}}{k_{si}{K}_{mi}}}{y}_i^{\left(4\right)}+{\displaystyle \frac{J_{i1}+J_{i2}}{K_{mi}}}{y}_i^{″}\end{array}\right].$

2.3. Control Design-Based ADRC

Let us consider (10) as a linear perturbed system as follows:

(13)$y_i^{\left(4\right)}={\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}{I}_{mi}+{\xi }_i\left(t\right),$

where the disturbance ${\xi }_i\left(t\right)$ is of the form:

${\xi }_i\left(t\right)=-{\displaystyle \frac{k_{si}^2}{J_{i1}{J}_{i2}^2}}\left(J_{i1}+J_{i2}\right)\left({\theta }_{i1}-{\theta }_{i2}\right)+{\psi }_i\left(t\right),$

and ${\psi }_i\left(t\right)$ represents the neglected dynamics as external disturbances, viscous and Coulomb frictions, among others.

Now, proposing the tracking error as

$e_{yi}=y_i-y_i^{*}\left(t\right),$

its dynamics are of the form

$e_{yi}^{\left(4\right)}=y_i^{\left(4\right)}-y_i^{\left(4\right)\ast }\left(t\right),$

which, according to (13), can be written as

(14)$e_{yi}^{\left(4\right)}={\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}{I}_{mi}+{\xi }_i\left(t\right)-y_i^{\left(4\right)\ast }\left(t\right).$

Since the total disturbance $F_i\left(t\right)$ is expressed as

(15)$F_i\left(t\right)={\xi }_i\left(t\right)-y_i^{\left(4\right)\ast }\left(t\right),$

it follows that the error dynamics (14) can be rewritten in the form

$e_{yi}^{\left(4\right)}={\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}{I}_{mi}+F_i\left(t\right),$

and by defining $z_i=e_{yi}$ for $i=1,...,4$ and $z_{i5}=F_i\left(t\right)$, noting that $z_{i3}=e_{yi}^{″}={\displaystyle \frac{k_{si}}{J_{i2}}}\left({\theta }_{i1}-{\theta }_{i2}\right)-y_i^{″}\left(t\right)$, from (11), we can express the above system in an extended state space representation as follows:

(16)$\begin{array}{ccc}{z}_{i1}^{\prime }& =& z_{i2}\\ z_{i2}^{\prime }& =& z_{i3}\\ z_{i3}^{\prime }& =& z_{i4}\\ z_{i4}^{\prime }& =& {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i2}}}{I}_{mi}+z_{i5}\\ z_{i5}^{\prime }& =& 0.\end{array}$

Now, two decoupled observers are proposed to estimate the variables related to the flat outputs and the total disturbance: a Luenberger observer for variable $z_{i2}$

(17)$\begin{array}{ccc}{\widehat{z}}_{i1}^{\prime }& =& {\widehat{z}}_{i2}+{\phi }_{i1}\left(z_{i1}-{\widehat{z}}_{i1}\right)\\ {\widehat{z}}_{i2}^{\prime }& =& {\widehat{z}}_{i3}+{\phi }_{i0}\left(z_{i1}-{\widehat{z}}_{i1}\right),\end{array}$

and a Generalized Proportional Integral Observer (GPIO) for variable $z_{i4}$ and the total disturbance $z_{i5}$; thus,

(18)$\begin{array}{ccc}{\widehat{z}}_{i3}^{\prime }& =& {\widehat{z}}_{i4}+{\lambda }_{i2}\left(z_{i3}-{\widehat{z}}_{i3}\right)\\ {\widehat{z}}_{i4}^{\prime }& =& {\displaystyle \frac{k_{si}{K}_{mi}}{J_{i1}{J}_{i1}}}{I}_{mi}+{\widehat{z}}_{i5}+{\lambda }_{i2}\left(z_{i3}-{\widehat{z}}_{i3}\right)\\ {\widehat{z}}_{i5}^{\prime }& =& {\widehat{z}}_{i6}+{\lambda }_{i0}(z_{i3}-{\widehat{iz}}_{i3}).\end{array}$

The aforementioned is illustrated in Figure 4.

The characteristic polynomial of the linear dominant dynamics for the Luenberger observer is

(19)$p_{o1}={(s+{\omega }_{o1i})}^2,$

while for the GPIO it is

(20)$p_{o2}={(s+{\omega }_{o2i})}^3.$

It should be noted that the parameters ${\omega }_{o1i}$ and ${\omega }_{o2i}$ represent the bandwidths of the observers.

Combining the above, the controller can be computed as

$I_{mi}=-{\displaystyle \frac{J_{i1}{J}_{i2}}{k_{si}{K}_{mi}}}\left({\alpha }_{i0}{e}_{y_i}+{\alpha }_{i1}{\widehat{e}}_{y_i}^{\prime }+{\alpha }_{i2}{\widehat{e}}_{y_i}^{″}+{\alpha }_{i3}{\widehat{e}}_{y_i}^{\prime \prime \prime }+{\widehat{F}}_i\left(t\right)\right),$

or in a state space variables representation, the proposed control law with disturbance cancellation can be rewritten as

(21)$I_{mi}=-{\displaystyle \frac{J_{i1}{J}_{i2}}{K_{si}{K}_{mi}}}\left({\alpha }_{i0}{z}_{i1}+{\alpha }_{i1}{\widehat{z}}_{i2}+{\alpha }_{i2}{\widehat{z}}_{i3}+{\alpha }_{i3}{\widehat{z}}_{i4}+{\widehat{z}}_{i5}\right).$

Here, the control parameters ${\alpha }_{ik}$$k=0,...,3$ are selected in such a way that follows the dynamics of a Hurwitz polynomial, namely,

(22)$p_c\left(s\right)={(s^2+2{\xi }_i{\omega }_{ci}s+{\omega }_{ci}^2)}^2.$

2.4. Experimental Design

To validate the calculated model and the ADRC with differential flatness, a series of experiments are proposed:

1. Partial State Feedback Controller: This controller aims to represent the oscillatory nature of the system and demonstrate that position control cannot be achieved using this control scheme.

2. Full State Feedback Controller with LQR Gains: This experiment seeks to compare the designed controller with another optimal control scheme based on the LQR methodology.

3. ADRC with Differential Flatness: This experiment demonstrates the results obtained from the controller designed specifically for this study.

4. ADRC with Differential Flatness under External Disturbances: Three masses are distributed across the two stages of the robot to evaluate the robustness of the controller in the presence of unknown external disturbances.

3. Experimental Results

In this section, the proposed ADRC control law, as defined in (21), is implemented on the platform described in Section 2.1. The effectiveness of the proposed control law is assessed by means of tracking a smooth rest-to-rest angular position reference trajectory. For comparative purposes, a partial feedback state controller and Full State Feedback Controller with LQR Gains are designed and implemented in the experimental platform.

3.1. Experimental Setup

A block diagram of the experimental platform is presented in Figure 5. The experimental platform used comprises the following elements, all manufactured and supplied by Quanser (Markham, ON, Canada):

• Power amplifier: Quanser AMPAQ current amplifier.

• Data acquisition device: Q8-USB, QPID/QPIDe.

• Experimental platform: Quanser 2DOF Serial Flexible Joint Robot.

• Computer with Matlab-Simulink.

Figure 5

Quanser 2DOF Serial Flexible Joint Robot.

[Figure omitted. See PDF]

This robotic system features two DC motors (Maxon 273759 and Maxon 118752, manufactured by Maxon Motor, Sachseln, Suiza) that drive harmonic gearboxes (CS-14-100-1U-CC-SP and CS-8-50-1U-CC-SP from Harmonic Drive LLC, Beverly, MA, USA) and a two-bar serial linkage with rigid links. The primary link connects to the first motor through a flexible joint, supporting the second harmonic drive at its end. This drive, in turn, connects to the second rigid link via another flexible joint. Both motors and flexible joints are equipped with quadrature optical encoders for precise measurement (1024 lines per revolution from US Digital, Vancouver, WA, USA). The control law is implemented in a Matlab-Simulink QUARC real-time model, where the sampling period is fixed to be $0.001$ $\mathrm{s}$. Table 1 shows the parameters of a 2DOF Serial Flexible Robot prototype found in the manufacturer’s manual.

3.2. Problem Formulation: Trajectory Tracking

The initial conditions considered for the prototype are ${\theta }_{11}={\theta }_{12}={\theta }_{21}={\theta }_{22}=0$. The desired trajectory consists of a fast smooth rest-to-rest maneuver, which is defined as follows: The path is initialized at the position 0 rad, and after two seconds, the mechanism moves to the position $-{\displaystyle \frac{\pi }{4}}$ rad in a time interval of 2 s and maintains this position for 3 s. After this time interval, the mechanism moves to the position ${\displaystyle \frac{\pi }{4}}$ rad with a time interval of 3 s. Finally, after remaining in this position for 3 s, the mechanism moves to its initial position 0 rad in 2 s, where it remains until the end of the experiment.

3.3. Partial State Feedback Controller

In order to show the oscillatory nature of the system and the difficulty of designing a controller that avoids oscillations when the system moves with fast trajectories, a partial state feedback (PSF) control strategy is developed using only information of the actuators (motors) to control all the system. In this process, the system’s viscous friction is explicitly considered to ensure accurate modeling and performance evaluation.

3.3.1. Stage 1

As mentioned earlier, the state vector of the system, denoted by $x_1$, comprises the generalized coordinates along with their first-order time derivatives. We define the error vector as follows: $e_{x1}=x_1^{*}-x_1$, where the vector with the desired trajectory for the first stage is defined as $x_1^{*}$. Additionally, the system assumes that the input variable is the current $u_1=I_{m1}$. Then, by considering (1), where:

(23)$A_{s1}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -{\displaystyle \frac{k_{s1}}{J_{11}}}& {\displaystyle \frac{k_{s1}}{J_{11}}}& -{\displaystyle \frac{B_{11}}{J_{11}}}& 0\\ {\displaystyle \frac{k_{s1}}{J_{12}}}& -{\displaystyle \frac{k_{s1}}{J_{12}}}& 0& -{\displaystyle \frac{B_{12}}{J_{12}}}\end{array}\right],B_{s1}=\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{K_{m1}}{J_{11}}}\\ 0\end{array}\right],$

the proposed control law for stage 1 consists of a partial state feedback control law of the form:

(24)$I_{m1}=-K_{s1}{D}_P{e}_{x1},$

where the vector gains

(25)$K_{s1}=\left[\begin{array}{cccc}76.57& 81.55,& 2.86& 23.02\end{array}\right],$

The gain vector and its values are determined through the LQR tuning algorithm, and $D_P\in {\mathbb{R}}^{4\times 4}$ is of the form

(26)$D_P=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$

and is used to obtain the partial state feedback.

Figure 6a shows the performance of stage 1 of the 2DOF-SFJ plant, ${\theta }_{11}$ and ${\theta }_{12}$. It can be observed that while ${\theta }_{11}$ is able to follow the desired trajectory, there are unwanted oscillations of considerable magnitude in ${\theta }_{12}$. The control current $I_{m1}$, depicted in Figure 6b, exhibits saturation, which clearly reveals the presence of high-frequency noise.

3.3.2. Stage 2 Space State Model

Let us consider (2). As in the previous stage, the current is considered the input $u_2=I_{m2}$ and error vector $e_{x2}=x_2^{*}-x_2$ with

$A_{s2}=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -{\displaystyle \frac{k_{s2}}{J_{21}}}& {\displaystyle \frac{k_{s2}}{J_{21}}}& -{\displaystyle \frac{B_{21}}{J_{21}}}& 0\\ {\displaystyle \frac{k_{s2}}{J_{22}}}& -{\displaystyle \frac{k_{s2}}{J_{22}}}& 0& -{\displaystyle \frac{B_{22}}{J_{22}}}\end{array}\right],B_{s2}=\left[\begin{array}{c}0\\ 0\\ {\displaystyle \frac{K_{m2}}{J_{21}}}\\ 0\end{array}\right].$

As in stage 1, the proposed control law consists of a state feedback control law, namely:

(27)$I_{m2}=-K_{s2}{D}_P{e}_{x2}.$

Here, the vector gains

(28)$K_{s2}=\left[\begin{array}{cccc}47.95& -7.13,& 0.67& 2.90\end{array}\right],$

is the gain vector, which is also determined using the LQR algorithm. The partial state feedback controller is derived using matrix D, defined in (26).

Figure 7a illustrates the performance of the variables corresponding to stage 2, namely, ${\theta }_{21}$ and ${\theta }_{22}$. In this case, we can see that ${\theta }_{21}$ closely follows the reference trajectory. In contrast, ${\theta }_{22}$ exhibits oscillations, preventing it from accurately achieving the reference. Unlike the previous stage, the control current has no saturation or significant noise.

3.4. State Feedback Controller with LQR Gains

Using an LQR controller provided by the robot’s manufacturer, trajectory tracking is performed with the outputs defined as the angles ${\theta }_{12}$ and ${\theta }_{22}$, corresponding to the angular positions of the links. The following sections present the results for each stage under a smooth trajectory.

3.4.1. LQR Controller Stage 1

Now, the proposed LQR control law with full state feedback for stage 1 is obtained using Equations (24) and (25):

(29)$I_{m1}=-K_{s1}{D}_F{e}_{x1},$

with the matrix $D_F\in {\mathbb{R}}^{4\times 4}$ allowing the full state feedback

(30)$D_F=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

In Figure 8a, the trajectory tracking for the output ${\theta }_{12}$ is shown. Figure 8b depicts the trajectory tracking error, with a Mean Squared Error (MSE) of 0.0054 rad². It can be observed that the system attempts to follow the trajectory but exhibits a delay in its transitory response, confirming that the LQR controller is primarily designed for regulation purposes.

In Figure 9a, both motor 1 angular position ${\theta }_{11}$ and link 1 angular position ${\theta }_{12}$ are depicted and compared with the desired trajectory. In both cases, no oscillations are presented in the resting position but the delay with respect to the desired trajectory is observed. Finally, Figure 9b shows the control current, which exhibits significant noise and a highly saturated behavior.

3.4.2. LQR Controller Stage 2

In the same way as the previous stage, the LQR control law with full state feedback is obtained using Equations (27), (28) and (30):

(31)$I_{m2}=-K_{s2}{D}_F{e}_{x2},$

In Figure 10a, the trajectory tracking for the output ${\theta }_{22}$ is shown. Figure 10b depicts the trajectory tracking error, with a Mean Squared Error (MSE) of 0.0016 rad. It can be observed that the system attempts to follow the trajectory but exhibits a delay in its response, confirming that the LQR controller is primarily designed for regulation purposes. Additionally, the system presents a steady-state error that the controller is unable to compensate for, as it lacks integral action.

In Figure 11a, both ${\theta }_{21}$ and ${\theta }_{22}$ are depicted. In both cases, the described behavior can be observed. Finally, Figure 11b shows the control current, which exhibits an acceptable behavior.

3.5. Active Disturbance Rejection Control

In the previous section, it was observed that the inherent oscillations of the system could not be eliminated; therefore, in this section, the ADRC is implemented.

3.5.1. ADRC Stage 1

To control the oscillations of the first stage, the following considerations are taken into account: For the observers, parameters (19) and (20) are designated as ${\omega }_{o11}=95$ for the Luenberger observer (17) and ${\omega }_{o21}=25$ for the GPIO (18). Meanwhile, the controller parameters (22) are selected as ${\xi }_1=0.7$ and ${\omega }_{c1}=20$ for the ADRC (21).

Figure 12a illustrates the trajectory tracking of flat output ${\theta }_{12}$. Figure 12b shows the trajectory error; the MSE is 1.0651 × 10⁻⁵ rad². It is evident that the flat output follows the reference trajectory smoothly and accurately, without any noticeable oscillations, despite the presence of disturbances that are estimated and compensated for by the ADRC. In Figure 13a, both ${\theta }_{11}$ and ${\theta }_{12}$ are depicted. Observe the behavior of ${\theta }_{11}$, which anticipates or advances its motion to allow ${\theta }_{12}$ to follow the desired trajectory effectively. The control current $I_{m1}$ is presented in Figure 13b, where some saturation and the presence of high-frequency noise are observed; however, the effect is more attenuated than that presented in the previous section. Finally, the effect’s total disturbance is estimated by means of the GPIO ${\widehat{z}}_{15}$ and is presented in Figure 14.

3.5.2. ADRC Stage 2

In the second stage, the presence of steady-state error becomes evident, so an integral term needs to be added to the controller to compensate for it. Thus, the ADRC takes the form:

$I_{m2}=-{\displaystyle \frac{J_{21}{J}_{22}}{K_{s2}{K}_{m2}}}\left({\alpha }_{20}\int z_1+{\alpha }_{21}{z}_1+{\alpha }_{22}{\widehat{z}}_2+{\alpha }_{23}{\widehat{z}}_3+{\alpha }_{24}{\widehat{z}}_4+{\widehat{z}}_5\right),$

and the control parameters ${\alpha }_{2k}$$k=0,\dots ,4$ are selected in such a way that follows the dynamics of the Hurwitz polynomial

$p_{c2}\left(s\right)={(s^2+2{\xi }_i{\omega }_{c2}s+{\omega }_{c2}^2)}^2(s+{\omega }_{c2}),$

where ${\xi }_1=0.7$ and ${\omega }_{c1}=25$, and for the observers, parameters (19) and (20) are designated as ${\omega }_{o12}=95$ for the Luenberger observer (17) and ${\omega }_{o22}=10$ for the GPIO (18).

Figure 15a shows the trajectory tracking of flat output ${\theta }_{22}$. Even with accurate trajectory tracking without any noticeable oscillations, along with disturbance estimation and rejection using the GPIO and the ADRC, an overshoot can still be observed. Figure 15b shows the trajectory error; the MSE is 5.2692 × 10⁻⁵ rad². Both variables ${\theta }_{21}$ and ${\theta }_{22}$ are depicted in Figure 16a, where the effectiveness of the scheme is evident in trajectory tracking tasks, where the reference trajectory was followed without noticeable oscillations in both tracked variables. In Figure 16b, the control current $I_{m2}$ is presented, and there is no noticeable saturation, although some high-frequency noise is present. The effect’s total disturbance is estimated by means of the GPIO ${\widehat{z}}_{15}$ and is presented in Figure 17. Overall, while oscillations are inherent to this type of system, the ADRC effectively estimates and rejects the disturbances, resulting in a significant reduction in system oscillations and exceptional performance in trajectory tracking tasks.

3.6. Active Disturbance Rejection Control—External Disturbance

Once the ADRC control is tested, an external disturbance test is conducted. For this purpose, masses are added to each arm to observe the behavior of the control system. The distribution of masses is shown in Figure 18.

3.6.1. ADRC Stage 1 Perturbed

For stage 1 in Figure 19, two masses are positioned: the first one weighs $0.4$ $\mathrm{k}$$\mathrm{g}$ and is placed at a distance of 11 $\mathrm{c}$$\mathrm{m}$, while the second one weighs $0.5$ $\mathrm{k}$$\mathrm{g}$ and is placed at a distance of 20 $\mathrm{c}$$\mathrm{m}$.

Figure 20a illustrates the trajectory tracking of flat output ${\theta }_{12}$. Figure 20b shows the trajectory error; the MSE is 1.1123 × 10⁻⁵ rad². It is evident that the flat output follows the reference trajectory smoothly and accurately, without any noticeable oscillations, despite the presence of disturbances that were estimated and compensated for by the ADRC. In Figure 20a, both ${\theta }_{11}$ and ${\theta }_{12}$ are depicted. Observe the behavior of ${\theta }_{11}$, which anticipates or advances its motion to allow ${\theta }_{12}$ to follow the desired trajectory effectively. The control current $I_{m1}$ is presented in Figure 20b, where some saturation and the presence of high-frequency noise are observed; however, the effect is more attenuated than that presented in the previous section. Finally, the effect’s total disturbance is estimated by means of the GPIO ${\widehat{z}}_{15}$ and is presented in Figure 21.

3.6.2. ADRC Stage 2 Perturbed

For stage 2, one mass is positioned with a weight of $0.2$ $\mathrm{k}$$\mathrm{g}$, placed at a distance of 23 $\mathrm{c}$$\mathrm{m}$.

Figure 22a shows the trajectory tracking of flat output ${\theta }_{22}$. Even with accurate trajectory tracking without any noticeable oscillations, along with disturbance estimation and rejection using the GPIO and the ADRC, an overshoot can still be observed. Figure 22b shows the trajectory error; the MSE is 7.3216 × 10⁻⁵ rad². Both variables ${\theta }_{21}$ and ${\theta }_{22}$ are depicted in Figure 23a, where the effectiveness of the scheme is evident in trajectory tracking tasks, where the reference trajectory was followed without noticeable oscillations in both tracked variables. In Figure 23b, the control current $I_{m2}$ is presented, and there is no noticeable saturation, although some high-frequency noise is present. The effect’s total disturbance is estimated by means of the GPIO ${\widehat{z}}_{15}$ and is presented in Figure 24. Overall, while oscillations are inherent to this type of system, the ADRC effectively estimates and rejects the disturbances, resulting in a significant reduction in system oscillations and exceptional performance in trajectory tracking tasks.

3.7. Comparison of Results

This section presents a comparison of the results obtained from the different control strategies applied to the system. Key metrics such as trajectory tracking accuracy, steady-state error, response delay and robustness to external disturbances are analyzed to evaluate the performance of each controller. The comparison aims to highlight the strengths and limitations of the controllers, providing insights into their suitability for this type of underactuated system.

With respect to trajectory tracking precision, the proposed ADRC achieves satisfactory performance despite the presence of external disturbances, although it exhibits a phase shift. For applications involving Cartesian plane tracking, this phase shift impacts the real-world results. Table 2 presents a comparison of the MSE values for the implemented controllers.

Regarding steady-state error, the ADRC with differential flatness effectively corrects this issue compared to the other controllers. Analyzing the LQR controller, it is observed that while it exhibits a delay in trajectory tracking, it provides proper regulation with smooth behavior.

Finally, the external disturbance test for the ADRC demonstrates its robustness. Despite the increased weight on the links, it continues to perform trajectory tracking with low error values.

4. Conclusions and Future Work

In this paper, an Active Disturbance Rejection Controller is proposed to solve the trajectory tracking problem of a 2DOF Serial Flexible Robot, a differential flatness approach is used to simply the trajectory tracking problem to a linear controller with a cancellation of disturbance, and the effectiveness of the proposed controller is evaluated in a 2DOF Serial Flexible Robot prototype of Quanser. A partial feedback state controller is applied to show the oscillations inherent to this type of system. The proposed ADRC effectively estimates and rejects the disturbances, resulting in a significant reduction in system oscillations and exceptional performance in trajectory tracking tasks, reducing saturation and minimizing high-frequency noise. Future work could focus on using the ADRC approach to solve the trajectory tracking problem in Cartesian coordinates by applying direct and inverse kinematics.

Author Contributions

Conceptualization, M.R.-N. and G.O.-O.; Data curation, J.C.T.; Formal analysis, M.R.-N. and G.O.-O.; Funding acquisition, M.R.-N. and E.G.H.-M.; Investigation, M.R.-N., G.O.-O. and J.C.T.; Methodology, M.R.-N. and G.O.-O.; Project administration, M.R.-N. and A.L.-G.; Resources, A.T.-O.; Software, J.C.T.; Supervision, M.R.-N., G.O.-O., A.T.-O., E.G.H.-M. and J.C.T.; Validation, A.T.-O.; Visualization, G.O.-O.; Writing—original draft, M.R.-N.; Writing—review and editing, M.R.-N. and G.O.-O. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The original contributions presented in the study are included in the article; further inquiries can be directed to the corresponding authors.

Conflicts of Interest

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

Abbreviations

The following abbreviations are used in this manuscript:

FJR	Flexible Joint Robots
CMST	Constant Mode Shapes Technique
ODE	Ordinary Differential Equation
PDE	Partial Differential Equation
ANCTC	Adaptive Noise Cancellation-based Tracking Controller
MPC	Model Predictive Control
CNS	Central Nervous System
SMC	Sliding Mode Control
ADRC	Active Disturbance Rejection Control
GPIO	Generalized Proportional Integral Observer
DOF	Degrees of Freedom
SRFJ	Serial Robot with Flexible Joints
LQR	Linear Quadratic Regulator
MSE	Mean Squared Error

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Quanser 2DOF Serial Flexible Joint Robot.

Figure 2. Schematic representation for SRFJ.

Figure 3. Decoupled schema for SRFJ.

Figure 4. Luenberger observer and GPIOs.

Figure 6. Partial state feedback controller. (a) [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] stage 1 with the PSF controller. (b) Control current of stage 1 with the PSF controller.

Figure 7. Partial state feedback controller. (a) [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 2 with PSF controller. (b) Control current of stage 2 with PSF controller.

Figure 8. LQR controller. (a) Trajectory tracking of [Forumla omitted. See PDF.] of stage 1 with LQR controller. (b) Trajectory error of stage 1 with LQR controller.

Figure 9. LQR controller. (a) Trajectory tracking of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 1 with LQR controller. (b) Control current [Forumla omitted. See PDF.] of stage 1 with LQR.

Figure 10. LQR controller. (a) Trajectory tracking of [Forumla omitted. See PDF.] of stage 2 with LQR controller. (b) Trajectory error of stage 2 with LQR controller.

Figure 11. LQR controller. (a) Trajectory tracking of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 2 with LQR controller. (b) Control current [Forumla omitted. See PDF.] of stage 2 with LQR.

Figure 12. ADRC. (a) Trajectory tracking of flat output [Forumla omitted. See PDF.] of stage 1 with ADRC. (b) Trajectory error of stage 1 with ADRC controller.

Figure 13. ADRC. (a) Trajectory tracking of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 1 with ADRC. (b) Control current [Forumla omitted. See PDF.] of stage 1 with ADRC.

Figure 14. Disturbance estimation [Forumla omitted. See PDF.] of stage 1 with ADRC.

Figure 15. ADRC. (a) Trajectory tracking of flat output [Forumla omitted. See PDF.] of stage 2 with ADRC. (b) Trajectory error of stage 2 with ADRC controller.

Figure 16. ADRC. (a) Trajectory tracking of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 2 with ADRC. (b) Control current [Forumla omitted. See PDF.] of stage 2 with ADRC.

Figure 17. Disturbance estimation [Forumla omitted. See PDF.] of stage 2 with ADRC.

Figure 18. The distribution of masses in the experiment with the external disturbances.

Figure 19. ADRC. (a) Trajectory tracking of flat output [Forumla omitted. See PDF.] of stage 1 with ADRC. (b) Trajectory error of stage 1 with ADRC controller.

Figure 20. ADRC. (a) Trajectory tracking of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 1 with ADRC. (b) Control current [Forumla omitted. See PDF.] of stage 1 with ADRC.

Figure 21. Disturbance estimation [Forumla omitted. See PDF.] of stage 1 with ADRC.

Figure 22. ADRC. (a) Trajectory tracking of flat output [Forumla omitted. See PDF.] of stage 2 with ADRC. (b) Trajectory error of stage 2 with ADRC controller.

Figure 23. ADRC. (a) Trajectory tracking of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] of stage 2 with ADRC. (b) Control current [Forumla omitted. See PDF.] of stage 2 with ADRC.

Figure 24. Disturbance estimation [Forumla omitted. See PDF.] of stage 2 with ADRC.

Table 1

Parameters of 2DOF Serial Flexible Robot prototype of Quanser.

Parameter	Value		Parameter	Value
$K_{m1}$	$0.1190$	(N-m)/A	$K_{m2}$	$0.0232$	(N-m)/A
$k_{s1}$	$7.78$	(N-m)/rad	$k_{s2}$	$3.16$	(N-m)/rad
$J_{11}$	$0.0111$	kg m2	$J_{21}$	$0.0015$	kg m2
$J_{12}$	$0.2304$	kg m2	$J_{22}$	$0.0107$	kg m2
$B_{11}$	$4.5$	(N-m)s/rad	$B_{12}$	$0.028211$	(N-m)s/rad

Table 2

Mean Squared Error (MSE) for implemented controllers.

Controller	MSE ${\mathit{\theta }}_{12}$ [rad2]	MSE ${\mathit{\theta }}_{22}$ [rad2]
Partial state feedback controller	0.0088	3.2765 × 10−4
State Feedback Controller with LQR Gains	0.0054	0.0016
Active Disturbance Rejection Control	1.0651 × 10−5	5.2692 × 10−5
Active Disturbance Rejection Control—External disturbance	1.1123 × 10−5	7.3216 × 10−5

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