Design of a Hybrid Adaptive Controller for Series Elastic Actuators of Robots

The integration of a Series Elastic Actuator (SEA) represents a revolutionary advancement in the fields of robotics and mechatronics. By incorporating an elastic element in series with the motor, SEAs exhibit unparalleled attributes such as compliance, force sensing, and energy storage capabilities, surpassing the limitations of conventional rigid actuators. This distinctive design empowers the actuator to absorb, store, and judiciously release mechanical energy, enabling precise and adaptable movements.

The impetus behind SEAs lies in the pursuit of creating robotic systems that emulate human interaction and prioritize safety. By endowing robots with compliance through elastic elements, SEAs enhance responsiveness to external forces and dynamic environmental fluctuations, proving invaluable in physical engagement domains like collaborative robots (cobots), exoskeletons, prosthetics, rehabilitation devices, and diverse applications in healthcare and entertainment [1].

However, the adoption of SEAs introduces a set of complex challenges, including intricate design, energy inefficiency due to elastic effects, intricate control, and achieving stability, non-linear behavior, and demanding modeling. These intricacies necessitate innovative engineering solutions, novel control paradigms, and astute trade-off comprehension tailored to specific applications [2].

A hallmark feature of SEAs is their force-sensing ability, derived from the design's elasticity. The actuator, in real-time, discerns and quantifies external forces and torques. While this real-time force feedback proves invaluable in precision-critical applications, such as enabling robots to execute delicate tasks or adapt motions based on detected interactions, it also prompts the need for advanced control strategies [3].

In the realm of SEA control algorithms, two predominant categories are robust and adaptive controllers. Robust control aims to forge systems capable of withstanding uncertainties and disturbances without necessitating real-time adjustments, ensuring stability even in worst-case scenarios. In contrast, adaptive control continuously adapts settings based on real-time estimations of system dynamics, accommodating changes and minimizing disparities between desired and actual outputs [4].

Notably, these controllers, including PID Controllers, Sliding Mode Controllers (SMC), impedance controllers, SFC, MRAC, and Fuzzy Logic Controllers (FLC), may leverage either robust or adaptive control algorithms and may be used individually or combined to form hybrid controllers [5-10].

Within this context, a PID controller, widely utilized in diverse industries, calculates an output signal based on the discrepancy between the desired setpoint and the present process value, aiming for stable and precise process control. However, PID control may struggle with systems manifesting complex or highly nonlinear dynamics, compromising accuracy and stability [11-19].

SMC is a robust technique crafting a 'sliding surface' that mirrors a desired state trajectory, offering swift convergence and resilience to uncertainties. Yet, its susceptibility to high-frequency chattering requires consideration, potentially impacting stability and performance [20-30].

Impedance control, vital in robotics and mechatronics, regulates interactions between a robot and its environment by governing the dynamic interplay between applied forces or torques and ensuing displacements or velocities at the robot's end effector. While impedance control facilitates adaptable responses; discrepancies in system parameters or uncertainties can undermine its efficacy, impacting overall stability and performance [31-37].

State Feedback Control (SFC), governing dynamic systems based on internal state variables, finds application in diverse domains, achieving performance and stability by precise control. However, measurement errors or noise in accurate state measurements may compromise performance [38-47].

Model Reference Adaptive Control (MRAC), altering system control parameters in real-time to mirror a desired reference model, proves valuable in navigating systems with uncertain or time-varying dynamics. Yet, effective MRAC relies on accurate modeling of system dynamics, crucial for control performance [48-54].

FLC, mimicking human decision-making through linguistic variables and fuzzy sets, handles imprecise and uncertain data. FLC is advantageous in complex, nonlinear, or uncertain systems but requires expertise and iterative refinement [55-62].

The paper introduces an approach that integrates MRAC and SFC to effectively address challenges posed by SEAs. The anticipated benefit of this integration stems from the complementary strengths of MRAC and SFC. MRAC, known for its adaptability and robustness, compensates for uncertainties, while SFC, with its precision, contributes to accurate control based on internal state variables. This synergistic approach aims to enhance overall system performance, overcoming challenges such as intricate design, energy inefficiency, and control stability associated with SEAs. The combination leverages each approach's strengths to provide a comprehensive solution tailored to the specific challenges posed by SEAs. The hybrid technique is designed to overcome system uncertainties, reduce load disturbances, and improve the overall performance and stability of the system. Additionally, a lumped parameter model serves as the reference model, facilitating performance adjustment and the computation of necessary gains for the designed SFC. The calculation of an adaptation law, guided by Lyapunov's theory, incorporates terms aimed at enhancing system performance and mitigating parameter uncertainties and load disturbances.

In the pursuit of optimizing the performance of a SEA, the implementation of a model reference approach stands out as a key technique. By utilizing this method, the detection of errors in the actuator's performance becomes more accurate and precise. To further enhance the actuator's capabilities, a controller grounded in state feedback theory is thoughtfully crafted. This controller, with its adept ability to calculate appropriate gains, effectively counters system uncertainties and disturbances while impeccably tracking the desired performance trajectory. One of the remarkable aspects of this design is the utilization of Lyapunov's theory, which serves as the guiding principle for continuously updating and adjusting the aforementioned gains. This process ensures a robust and adaptive actuation system that excels in meeting the ever-evolving challenges posed by real-world applications. Figure 2 shows the functional block diagram of the proposed controller.

2.png

Figure 2. Functional block diagram of the proposed controller

3.1 Reference model

Recalling Eq. (1) and Eq. (3) and rewriting them in simplifying them using the $D$-operator, i.e., $D=d / d t$, then the result is

$\theta_m(t)=\frac{1}{J_m}\left[\frac{u(t)+\left(B_j D+K_j\right) \theta_l(t)}{D^2+\frac{B_j}{J_m} D+\frac{K_j+K_b}{J_m}}\right]$     (12)

and

$\theta_l(t)=\frac{-\tau(t)+\left(B_j D+K_j\right) \theta_m(t)}{J_l D^2+B_j D+K_j}$     (13)    

Assuming that both Eq. (12) and Eq. (13) have the standard form of the second-order system. This means that they can be rewritten as

$\theta_{m r}(t)=\frac{\omega_{n 1}^2 u(t)}{D^2+2 \zeta_1 \omega_{n 1} D+\omega_{n 1}^2}$             (14)

$\theta_{l r}(t)=\frac{\omega_{n 2}^2 \theta_{m r}(t)}{D^2+2 \zeta_2 \omega_{n 2} D+\omega_{n 2}^2}$         (15)

where, $\theta_{m r}(t)$ and $\theta_{l r}(t)$ are the angular positions obtained by the reference model of the motor and load sides, respectively. Furthermore, the parameters $\omega_{n 1}, \omega_{n 2}, \zeta_1$, and $\zeta_2$ are the natural frequency of the motor side, the natural frequency of the load side, the damping ratio of the motor side, and the damping ratio of the load side, respectively. Again, if it is assumed that Assuming, $x_{r 1}(t)=\theta_{m r}(t), x_{r 2}(t)=$ $\dot{\theta}_{m r}(t), x_{r 3}(t)=\theta_{l r}(t)$, and $x_{r 4}(t)=\dot{\theta}_{l r}(t)$, the reference model can be expressed in the following state space form

$\dot{x}_r(t)=A_r x_r(t)+B_r r(t)$      (16)

and the output has this form

$y_r(t)=C_r x_r(t)$        (17)

then the system matrices of the reference model can be expressed as

$A_r=\left[\begin{array}{cccc}0 & 1 & 0 & 0 \\ -\omega_{n 1}^2 & -2 \zeta \omega_{n 1} & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \omega_{n 2}^2 & 0 & -\omega_{n 2}^2 & -2 \zeta \omega_n\end{array}\right]$           (18a)

$B_r=\left[\begin{array}{llll}0 & \omega_{n 1}^2 & 0 & 0\end{array}\right]^T$    (18b)   

and

$C_r=C$             (18c)

where, $x_r(t)$ denotes the reference model state vector, and $\dot{x}_r(t)$ is its derivative over time. The output vector of the reference model is represented by $y_r(t)$. Additionally, $A_r$ is the state matrix of the reference model, $B_r$ is the input matrix for the refence input $r(t)$, and $C_r$ is the output matrix of the reference model. In the current study, $r(t)$ is a $1 \times 4$ column vector that serves as the representation of the desired reference input to be tracked.

A reference model serves as a guide for adjusting the performance of an actual system. This model represents the desired performance that the system should achieve. The designer customizes the performance of the reference model based on their preferences. The settling time of a system, $t_s$, can be calculated using the Eq. (19):

$t_s=\frac{4}{2 \zeta \omega_n}$          (19)

To clarify, the goal is to have a settling time of 1 second for both sides of the system. Additionally, the damping ratios are set to 1 for both sides. This choice implies that the natural frequency $\omega_{n 1}$ is equal to $\omega_{n 2}$, and both are set to 4. In simpler terms, the reference model's settling time formula and parameter values are used to fine-tune the system's performance. Substituting these values in Eqs. (18a) and (18b) gives

$A_r=\left[\begin{array}{cccc}0 & 1 & 0 & 0 \\ -16 & -8 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 16 & 0 & -16 & -8\end{array}\right]$            (20a)

and

$B_r=\left[\begin{array}{llll}0 & 16 & 0 & 0\end{array}\right]^T$         (20b) 

Eq. (16), Eq. (18c), Eq. (20a) and Eq. (20b) offer a comprehensive framework for characterizing the optimal response of SEAs.

3.2 Adaptive control law

The primary objective of the controller is to ensure that the behavior of the actual SEA follows the behavior of the reference model in order to achieve the desired outcome. Therefore, the error, $e(t)$,can be expressed as

$e(t)=x(t)-x_r(t)$      (21)

Thus, the error rate of change, $\dot{e}(t)$, can be written as

$\dot{e}(t)=\dot{x}(t)-\dot{x}_r(t)$        (22)

Substituting Eq. (9) and Eq. (16) into Eq. (22) gives

$\dot{e}(t)=A x(t)+B u(t)-A_r x_r(t)-B_r r(t)$           (23) 

In this case, the disturbance torque is assumed to be zero, and the system takes on the conventional state space form. Expressing the control law can be achieved through the application of state feedback control theory, as illustrated below:

$u(t)=k_r r(t)-k_x x(t)$                (24)

where, $k_r$ and $k_x$, are 1×4 vectors represent the gain of the reference input, and the feedback gain of the state. In fact, the integration of state feedback control theory with the reference model inherently addresses the impact of disturbances on the system. This integration compels the system to dynamically follow the reference input $r(t)$, contingent upon the disparities between the states of the actual system and those of the reference model.

Substituting Eq. (23) into Eq. (24) results in the following expression:

$\dot{e}(t)=\left(A-B k_x\right) x(t)+\left(B k_r-B_r\right) r(t)-A_r x_r(t)$       (25)

The controller's goal is to ensure that state $x(t)$ tends to be identical to state $x_r(t)$, and hence the error is zero. This provides

$A_r=A-B k_x$    (26)

and

$B_r=B k_r$      (27)

In real-life applications, the gains $k_x$ and $k_r$ cannot be calculated because the system parameters represented by the matrix $A$ and $B$ cannot be identified properly or they change over time due to friction, wear, deformation, etc. To overcome such a problem, the control law is reformulated to be

$u(t)=\hat{k}_r(t) r(t)-\hat{k}_x(t) x(t)$         (28)

where, $\hat{k}_r(t)$ and $\hat{k}_x(t)$ are the estimations of $k_r$ and $k_x$ and have the same dimensions, respectively. Now, substituting Eq. (28) into Eq. (23), which gives

$\begin{gathered}\dot{e}(t)=\left(A-B \hat{k}_x(t)\right) x(t)+\left(B \hat{k}_r(t)-B_r\right) r(t) -A_r x_r(t)\end{gathered}$       (29)

It can be noted that the estimation gains $\hat{k}_r(t)$ and $\hat{k}_x(t)$ are varying with time, and they can be expressed as

$\tilde{k}_r(t)=k_r-\hat{k}_r(t)$      (30a)

and

$\tilde{k}_x(t)=k_x-\hat{k}_x(t)$         (30b)

where, $\tilde{k}_r(t)$ and $\tilde{k}_x(t)$ are the estimation errors of $k_r$ and $k_x$, respectively. Substituting Eqs. (30a) and (30b) into Eq. (29) gives

$\begin{gathered}\dot{e}(t)=\left(A-B k_x-B \tilde{k}_x(t)\right) x(t) \\ +\left(B k_r-B \tilde{k}_r(t)-B_r\right) r(t)-A_r x_r(t)\end{gathered}$    (31)

Further, substituting Eq. (26) and Eq. (27) into Eq. (31), which gives

$\dot{e}(t)=A_r e(t)-B \tilde{k}_x(t) x(t)-B \tilde{k}_r(t) r(t)$         (32)

Now, apply Lyapunov stability analysis to the following candidate function:

$V(t)=\frac{1}{2} e^2(t)+\frac{1}{2} \tilde{k}_r^2(t)+\frac{1}{2} \tilde{k}_x^2(t)$       (33)

The time derivative of the candidate function can be written as

$\dot{V}(t)=e(t) \dot{e}(t)+\tilde{k}_r(t) \tilde{k}_r(t)+\tilde{k}_x(t) \tilde{k}_x(t)$        (34)

The time derivatives of $\tilde{k}_r(t)$ and $\tilde{k}_x(t)$ are

$\dot{\tilde{k}}_x(t)=-\dot{\hat{k}}_x(t)$         (35a)

and

$\dot{\tilde{k}}_r(t)=-\dot{\hat{k}}_r(t)$      (35b)

Substituting Eq. (35a), Eq. (35b), and Eq. (32) into Eq. (34), which leads to

$\dot{V}(t)=A_r e^2(t)-B x(t) e(t) \tilde{k}_x(t)$$-B r(t) e(t) \tilde{k}_r(t)-\dot{\hat{k}}_x(t) \tilde{k}_x(t)$$-\dot{\hat{k}}_r(t) \tilde{k}_r(t)$      (36)

In order to ensure that $\dot{V}(t)$ goes to zero at time rans to infinity, then the following relationships should be provided

$\dot{\hat{k}}_r(t)=-B r(t) e(t)$      (37a)

and

$\dot {\hat{k}}_x(t)=-B x(t) e(t)$        (37b)

As previously mentioned, since the system parameters are considered unknown, the vector B can be represented as follows:

$B=b\left[\begin{array}{llll}0 & 1 & 0 & 0\end{array}\right]^T$          (38)

Here, b is a constant utilized to fine-tune controller performance and tailor it based on the designer's preferences.

The proposed controller exhibits versatile applicability across SEAs and analogous systems. The incorporation of state feedback control integrated with a reference model establishes a flexible framework that can seamlessly adapt to diverse systems sharing similar dynamics. The fundamental principles and methodologies inherent in our model are inherently generalizable, streamlining the application process to different SEAs or systems that demonstrate akin characteristics. This inherent adaptability underscores the robustness and broad utility of our controller in addressing a spectrum of practical scenarios.

This paper highlights the revolutionary impact of SEAs in robotics and mechatronics, emphasizing their unique attributes such as compliance, force sensing, and energy storage. While SEAs address critical needs in human-robot interaction and safety, they also pose challenges like intricate design and control complexities. The paper proposes an innovative integration of MRAC and SFC to address SEA challenges. This hybrid approach aims to leverage the adaptability of MRAC and the precision of SFC, offering a comprehensive solution to enhance overall system performance and overcome challenges associated with SEAs, such as energy inefficiency and control stability. The use of a lumped parameter model and Lyapunov's theory guides the adaptation law, optimizing system performance and mitigating uncertainties and disturbances.

In conclusion, this study delves into a meticulous exploration of the performance and adaptability of the proposed controller within a dynamic system framework. The investigation encompasses various facets, each shedding light on the controller's robustness and efficacy. Through a systematic analysis of tuning gain b, the impact on the system's response was unveiled, revealing distinct settling times for varying b values. Notably, the reference model consistently demonstrated exceptional performance, even when settling times varied between the motor and load sides. This underscores the controller's potential to adeptly capture system behavior.

Moreover, the inquiry encompasses parameter uncertainties, deliberately chosen at a 25% variation in parameter values. These uncertainties were selected deliberately to be sufficiently substantial, allowing for a robust assessment of the controller's performance under challenging conditions. Impressively, despite parameter fluctuations, the angular positions remained consistent, reaffirming the controller's adaptability in maintaining desired behaviors. The proportional adjustment of control torque in response to parameter variations exemplified the controller's dynamic nature, robustly mitigating the effects of uncertain parameters. The results demonstrate that a 25% increase in parameters corresponds to a 25% increase in control torque, highlighting a proportional reaction to parameter changes. Conversely, a 25% reduction in parameters results in a corresponding 25% reduction in control torque.

The controller's prowess is further validated under load disturbances, showcasing its adaptability to external influences. Whether subjected to unit-step or sinusoidal disturbances, the controller exhibits commendable responses, adeptly mitigating surges in control torque and maintaining desired angular positions. These findings underscore the controller's versatility in managing diverse operational scenarios. Additionally, the results indicate that the application of a unit step disturbance leads to a 49.4% increase in control torque. This significant rise emphasizes the controller's ability to dynamically adapt and exercise expanded control authority in response to system changes. The effect of the sinusoidal disturbance on control torque is also notable, with a detectable 25.1% increase.

In summary, the proposed controller proves to be a robust and adaptive control strategy, effectively addressing uncertainties in parameters and dynamic disturbances. Its ability to uphold system stability, ensure precise control, and achieve desired behaviors across diverse challenges positions it as a valuable asset for optimizing system performance. Through a comprehensive exploration, this study underscores the controller's potential for real-world applications in dynamic systems, contributing to the evolution of control methodologies and enhancing operational outcomes. Looking ahead, future extensions of this study may involve implementing optimizing algorithms to determine the tuning parameter b. Additionally, consideration of non-linearities associated with SEAs could further broaden the scope of this research.