Neuroadaptive Control of a Continuum Robot for Finger Rehabilitation

Abstract

This study has designed an easy-to-wear parallel continuum robot-based hand rehabilitation system that supports and enhances the finger’s flexion, extension, abduction, and adduction movements. The primary novelty of the proposed system lies in its ability to focus therapeutic exercises on a single joint, a feature not commonly found in existing rehabilitation robots. A kinematic model of the system was developed, and to perform both kinematic and dynamic analyses, a multibody model was constructed in the MATLAB Simulink environment. Joint angle control was implemented using a nominal controller, and to account for individual uncertainties in joint dynamics, a neuroadaptive controller was integrated with the nominal controller. This approach aims for the neural network architecture to learn these uncertainties during control iterations and incorporate them into the control, resulting in a robust controller. Thus, a model reference control approach was proposed for active and passive rehabilitation processes. The system model was tested in a simulation environment, and then all tests were repeated in the physical system. The simulation and real system results include the real system’s open-loop responses, nominal controller responses for each joint, responses, and the results for active, passive, and assistive control modes.

1. Introduction

Hand dysfunction has a significant negative impact, making daily life challenging. It tremendously influences life, independence, and daily activities and causes many restrictions on movement. Losing the functions of the hand makes it difficult to deal with daily activities. Achieving a certain freedom of movement might be possible after challenging progress [1,2]. Intensive repetitive rehabilitation can help to regain the function of the hand after stroke and other neurological disabilities [3]. On the other hand, some autoimmune diseases might cause hand dysfunctions, like rheumatoid arthritis (RA) [4]. RA does not cause loss of movement but difficulty in joint movement. Conventional hand rehabilitation is based on in-person sessions with professionals. These time-consuming therapies are not easily reachable by all patients. Considering the importance of starting the rehabilitation process for neurological disorders from the first day of hit, it seems that technological solutions need to be suggested for this area [2].

Recently, robot-based hand training devices have been developed, and they are becoming part of the repetitive rehabilitation process. Besides clinical-based robotic rehabilitation, researchers also commonly suggest wearable and more flexible rehabilitation robots because of their advantages. Wearable robot-assisted repetitive therapies have benefits, such as moving the process to the patient’s house. Physicians can efficiently manage the process. They can also collect a lot of data from robotic rehabilitation solutions [5].

Finger rehabilitation robots utilize various actuators designed to facilitate specific movements and support rehabilitation processes for patients recovering from injuries or conditions affecting hand function. Among the most prominent actuator types are soft pneumatic actuators, which have gained significant attention due to their inherent compliance and ability to mimic the natural movements of the human hand. Soft pneumatic actuators, such as the McKibben pneumatic artificial muscles [6], are widely recognized for their effectiveness in rehabilitation robotics.

These actuators consist of an inflatable bladder surrounded by braided mesh, allowing for bending and flexing motions that closely resemble human finger movements [7,8]. The design of these actuators enables them to provide a gentle and adaptable force, which is crucial for rehabilitation settings where patient comfort and safety are paramount [9]. Research has shown that soft actuators can enhance the interaction between the robotic device and the user, promoting a more natural rehabilitation experience [10].

Another approach involves fabric-based soft actuators integrated into wearable exoskeletons. These devices often incorporate inflatable chambers that allow for controlled bending and extension of the fingers, facilitating rehabilitation exercise [11]. The combination of soft materials and inflatable structures not only enhances the flexibility of the actuators but also reduces the overall weight of the rehabilitation device, making it more comfortable for prolonged use [12]. Studies indicate that such designs can effectively assist patients in regaining motor control and strength in their fingers following strokes or other debilitating conditions [13].

Remarkable improvements in wearable hand robots have been made in recent years. This field has received contributions from both academic and commercial researchers worldwide. Wearable hand rehabilitation robots from the market and the literature may be rehabilitative, assistive, or haptic. Rehabilitative devices manipulate the hands of patients with a particular trajectory, whereas assistive wearable robotics manipulate the hands to support the grasping of objects. Haptic devices provide feedback to the patient to improve the smoothness of their movement [14].

Wearable hand rehabilitation robotics can be classified based on their structure and degree of freedom (DoF). They may be designed as exoskeleton- or end effector-based [15]. Alternatively, depending on the independently manipulated point, the designs could be 1,2, or 3 DoF designs [16,17,18]. While it appears to support more complex movements with a high DoF, it also increases the complexity of controlling the robot [14]. Manipulation can also be achieved through rigid linkage mechanisms powered by actuators [19] or tendon/cable-driven [20]. Some soft robotic solutions use pneumatic sources to actuate their joints [21].

Unlike classical serial or parallel manipulators, continuum robotics (CRs) is a new research area. CRs can be designed to achieve many degrees of freedom [22]. CRs are fabricated of elastic materials instead of rigid links connected by discrete joints [23] and composed of a series of individually bending segments. CRs may be tendon-actuated, a concentric tube, magnetically actuated, or soft robot-based [24,25,26].

The flexible structure of CRs and their ability to move over a wide range has made them useful in the healthcare field. There are designs specifically for moving endoscopic devices through vessels or body cavities [27]. Studies, particularly in fields such as surgical robots, are rapidly expanding [28].

Many proposed hand rehabilitation robots only address flexion and extension movements of the hand [29]. Furthermore, rigid linkage-based models are especially challenging for patients with arthritis or post-stroke conditions [30]. Some hand rehabilitation systems allow joints to be controlled individually using 2–3 DoF mechanisms. However, achieving this with fewer actuators seems difficult. Furthermore, the need for separate actuators and additional mechanisms for abduction and adduction movements increases the physical size of hand rehabilitation systems [31].

This study proposes a newly designed parallel continuum robot-based rehabilitation robot to be used for both active and passive rehabilitation tasks. Because of the CRs’ flexibility, the Continuum Rehabilitation Robot (CRR) can assist with abduction and adduction movements without additional actuators or mechanisms in the system. The CRR is designed and implemented in Matlab/Simulink environments as a multibody sim-mechanics model as well as an experimental prototype using 3D printing.

This article is organized as follows. Section 2 introduces the methodology employed in designing the CRR and the mathematical model of the lumped backbone and kinematic model of the CRR. Section 3 presents the results of our simulation and experimental studies, including rehabilitation scenarios and control experiments. Section 4 discusses the findings in the context of the experiments and potential improvements.

2. Materials and Methods

2.1. Concept Design

A series of three continuum robots like actuators are configured in a parallel arrangement to rehabilitate finger joints. Each actuator consists of three segments interconnected using hook and loop tape (HOLT) to form a triangular shape, as illustrated in Figure 1b. The use of HOLT facilitates the ease of application and adjustment for the patient. Each continuum robot as an actuator features a flexible additional backbone situated at its core. These backbones are not fixed to the rings. Their length between rings can increase or decrease. These backbones are just used to keep the rings aligned. The robot positioned at the apex of the triangular template is designed to assist with extension movements, while the other two continuum robots are responsible for assisting with flexion movements.

As seen in Figure 1b, the whole system is considered as one CR system. Here, the finger takes the role of the stationary backbone of the CR. The perspective and dorsal views of the system can be seen in Figure 1a,c, respectively.

As shown in Figure 2, a finger consists of four distinct bones in its anatomy: the metacarpal, proximal phalanx, middle phalanx, and distal phalanx. The metacarpal and proximal phalanx are joined by the metacarpophalangeal (MCP) joint, the proximal and middle phalanges by the proximal interphalangeal (PIP) joint, and the middle and distal phalanges by the distal interphalangeal (DIP) joint.

As shown in Figure 3 in the MCP joint, extension and flexion movements occur around the $y_p$ axis in clockwise and counterclockwise directions, respectively, while adduction and abduction occur around the $z_p$ axis in clockwise and counterclockwise directions, respectively. In the PIP joint, the finger bends through flexion and extends through extension, both occurring around the $y_m$ axis in a clockwise and counterclockwise direction, respectively. At the DIP joint, extension, which opens the finger, occurs around the $y_d$ axis in a clockwise direction, while flexion, which closes the finger, occurs in the opposite direction.

As illustrated in Figure 1, each continuum robot in the system is specifically connected to one of the phalanges: distal, middle, and proximal. Each phalanx is accommodated by its own continuum robot segment. The system’s flexibility significantly enhances its wearability. Thanks to the HOLT fastening mechanism, the system can be applied even if the patient’s fingers are not fully extended; the flexible backbones and HOLT connections allow for effective fitting and adjustment. Figure 4 shows the prototype of the CRR.

As shown in Figure 4c, tendons are driven by servos with pulley systems. A mechanism is used to pull the tendon on one side and release it on the other, enabling flexion and extension movements for the DIP and PIP joints. Here,${\tau }_{de}$ and ${\tau }_{me}$ represent the extensor tensions, while ${\tau }_{df} \mathrm{and} {\tau }_{mf}$ represent the flexor tensions for the DIP and PIP joints, respectively.

Two servos are employed for the MCP joint to facilitate abduction and adduction movements when needed. In this case, ${\tau }_{pe}$ is extensor tension and ${\tau }_{p{f}_b} \mathrm{and} {\tau }_{p{f}_d}$ are the flexor tensions for MCP joint. If tension is applied only to ${\tau }_{p{f}_b}$, abduction occurs. Conversely, if ${\tau }_{p{f}_b}$ is released and ${\tau }_{p{f}_d}$ is applied, adduction occurs.

2.2. Kinematic Relations of the Finger

The equations of motion for the kinematics of a finger with the tendon tensions produced by continuum robots can be derived using the Euler–Lagrange method. The position equations for each phalanx are given in Equations (1)–(4). As shown in Figure 5, $x_i$ and $y_i$ represent the joint positions, $m_i$ denotes the mass of each phalanx, $l_i$ is the length of each phalanx, ${\theta }_i$ indicates the angle formed by each phalanx with respect to the norm, and ${\alpha }_i$ indicates the angle of each joint between phalanges. $r_i$ represents the distance of the tendons from the axis, and $T_{xy}$ denotes the tension generated by each tendon. The phalanx lengths for the 3D-printed hand model were measured using an electronic vernier. For patient-specific applications, image processing tools will be used to measure lengths and angles.

$$x_1=l_1\cos \left({\theta }_1\right), y1=l_1\sin \left({\theta }_1\right)$$

(1)

$$x_2=l_1\cos \left({\theta }_1\right)+l_2\cos \left({\theta }_1+{\theta }_2\right), y1=l_1\sin \left({\theta }_1\right)+l_2\sin \left({\theta }_1+{\theta }_2\right)$$

(2)

$$x_3=l_1\cos \left({\theta }_1\right)+l_2\cos \left({\theta }_1+{\theta }_2\right)+l_3\cos \left({\theta }_1+{\theta }_2+{\theta }_3\right),$$

(3)

$$y_3=l_1\sin \left({\theta }_1\right)+l_2\sin \left({\theta }_1+{\theta }_2\right)+l_3\sin \left({\theta }_1+{\theta }_2+{\theta }_3\right)$$

(4)

Using the equations of motion, the kinetic energies can be written for each $i_{th}$ joint as $T_i={\displaystyle \frac{1}{2}}{m}_i\left({\left({\displaystyle \frac{d}{dt}}{x}_i\right)}^2+{\left({\displaystyle \frac{d}{dt}}{y}_i\right)}^2\right).$ The equations are shown in Equations (5)–(7).

$$T_1={\displaystyle \frac{1}{2}}{m}_1{l}_1^2{\dot{\theta }}_1^2$$

(5)

$$T_2={\displaystyle \frac{1}{2}}{m}_2{\left(m_2{l}_2\sin \left({\theta }_1+{\theta }_2\right)\left({\dot{\theta }}_1+{\dot{\theta }}_2\right)+l_1\sin \left({\theta }_1\right){\dot{\theta }}_1\right)}^2+{\displaystyle \frac{1}{2}}{m}_2{\left(l_2\cos \left({\theta }_1+{\theta }_2\right)\left({\dot{\theta }}_1+{\dot{\theta }}_2\right)+l_1\cos \left({\theta }_1\right){\dot{\theta }}_1\right)}^2$$

(6)

$$\begin{array}{l}{T}_3={\displaystyle \frac{1}{2}}{m}_3[{\left(l_{3}cos\left(\theta 1+\theta 2+\theta 3\right)\left({\dot{\theta }}_1+{\dot{\theta }}_2+{\dot{\theta }}_3\right)+l_{2}cos\left(\theta 1+\theta 2\right)\left({\dot{\theta }}_1+{\dot{\theta }}_2\right)+l_{1}cos\left(\theta 1\right){\dot{\theta }}_1\right)}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\left(l_{3}sin\left({\theta }_1+{\theta }_2+{\theta }_3\right)\left({\dot{\theta }}_1+{\dot{\theta }}_2+{\dot{\theta }}_3\right)+l_{2}sin\left({\theta }_1+{\theta }_2\right)\left({\dot{\theta }}_1+{\dot{\theta }}_2\right)+l_{1}sin\left({\theta }_1\right){\dot{\theta }}_1\right)}^2]\end{array}$$

(7)

The potential energies can also be written as follows:

$$V_1=g\times l_1\times m_1\times sin\left({\theta }_1\right(t\left)\right)$$

(8)

$$V_2=g\times m_2\times \left(l_{2}sin\right({\theta }_1+{\theta }_2)+l_{1}sin({\theta }_1\left)\right)$$

(9)

$$V_3=g\times m_3\times \left(l_{3}sin\right({\theta }_1+{\theta }_2+{\theta }_3)+l_{2}sin({\theta }_1+{\theta }_2)+l_{1}sin({\theta }_1\left)\right)$$

(10)

With potential and kinetic energy equations, the Euler–Lagrange equation can be written as follows. And as shown in Equation (12), the motion equations for each $i_{th}$ joint can be calculated with the Euler–Lagrange method [32].

$$L={(L}_1+L_2+L_3)-(V_1+V_2+V_3),$$

(11)

$${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial L}{\partial \dot{{\theta }_i}}}\right)-{\displaystyle \frac{\partial L}{\partial {\theta }_i}}=F_d+F_i,F_d=b_i\dot{{\theta }_i}$$

(12)

Here, $b_i$ denotes the damping coefficient with the unit of N.s/rad for each $i_{th}$ joint. $F_d$ is a frictional force in terms of angular velocity. $F_i$ is the net force applied to the joint via tensions generated by tendons.

The torques generated by each driven tendon affect each joint differently, depending on the kinematic parameters of the joints. Net force equations can be seen Equations (13)–(15). Here, ${\gamma }_i$ is the tendon pulling angle. As shown in force equations when the DIP joint is ran by just flexor and extensor tendon tensions, the PIP joint is affected by the tendons connected to the distal phalange and the MCP joint is affected by the tendons connected to the middle and distal phalanges.

$$\begin{array}{l}{F}_1=sin\left({\gamma }_1\right)\times \left(T_{11}-T_{12}\right)+\sin \left(\mathit{PIP}\right)\times (\sin \left(DIP\right)F_3+F_2)-sin\left({\theta }_2-{\theta }_1\right)\times cos\left({\gamma }_2\right)\times \left(T_{21}+T_{22}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+cos\left({\gamma }_3\right)\times cos\left({\theta }_3-{\theta }_2\right)\times (T_{31}+T_{32})\end{array}$$

(13)

$$F_2=sin\left({\gamma }_2\right)\times \left(T_{21}-T_{22}\right)\times \sin \left(DIP\right)F_3-sin\left({\theta }_3-{\theta }_2\right)\times cos\left({\gamma }_3\right)\times (T_{31}+T_{32})$$

(14)

$$F_3=sin\left({\gamma }_3\right)\times (T_{31}-T_{32})$$

(15)

2.3. Kinematic Analysis for Continuum Robots as an Actuator

To mathematically describe the continuum rehabilitation robotics (CRRs) structure, each side backbone of the robots is parameterized with a lumped backbone parameterization method depending on the piecewise constant curvature explained by [33,34]. For the appropriate geometry, backbones connected to the rings above the finger are left free to lengthen and shorten between rings.

A discrete model represents the pose of the backbone. The finite backbone parameters are denoted by $X_{\mu }=[\kappa ,\varphi ,\theta ]$. Here, $\kappa ={\displaystyle \frac{1}{r}}$ is the curvature of the backbone, $\varphi$ is the angle between the bending plane and reference axis, and $\theta$ is the curve angle, as shown in Figure 6a. Here, the reference frame is just above the metacarpal bones and is described as $R_0$. Also, $s_{pm}$ is the length of the arc of the backbone between the proximal and middle phalanges. $x_p,y_p,z_p$ is the frame of the backbone fixed on the proximal phalange.

As seen in Figure 6b, ${l_m}_1,l_{p2}$ are the lengths between the connection point of the rings and the PIP joint. And these two lengths are equal (${l_m}_1=p_2$). Also, the distance between tendons is described with $r_{te}$ for the extension tendon and $r_{tf}$ for the flexion tendon. And this distance is supposed to be equal for appropriate geometrical kinematics. If we assume that the rings connected to the phalange as perpendicular, we can calculate the angle of curve $\theta$ by using the desired joint angle with basically $\theta =pi-PI{P}_d$. And we know the description of $\theta =\kappa s$. And we know that $r_k={\tan }^{-1}\left({\displaystyle \frac{PIP}{2}}\right)l_{m1}$. This means that we know the curve radius. The position of any point on backbones can be described as in Equation (16).

$$p\left(s\right)= \left[\begin{array}{c}{\displaystyle \frac{cos\left(\varphi \right)}{\kappa }}\left(1-\cos \left(\kappa s\right)\right)\\ {\displaystyle \frac{sin\left(\varphi \right)}{\kappa }}\left(1-\cos \left(\kappa s\right)\right)\\ {\displaystyle \frac{1}{\kappa }}\sin \left(\kappa s\right)\end{array}\right]$$

(16)

Here, $p\left(s\right)$ is any exact point on the backbone far from the reference frame with s. Two sequential rotations around the z and y axes provide the coordinate frame of any point on the arc, given in relation to the frame connected to the section’s base ($R_z\left(\varphi \right)R_y\left(\kappa s\right))$. The resulting transformation matrix is given as

$$T\left(s\right)=\left[\begin{array}{ccc}\cos \hspace{0.17em}\left(\varphi \right)& -\sin \hspace{0.17em}\left(\varphi \right)& 0\\ \sin \hspace{0.17em}\left(\varphi \right)& \cos \hspace{0.17em}\left(\varphi \right)& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos \hspace{0.17em}\left(\kappa s\right)& 0& \sin \hspace{0.17em}\left(\kappa s\right)\\ 0& 1& 0\\ -\sin \hspace{0.17em}\left(\kappa s\right)& 0& \cos \hspace{0.17em}\left(\kappa s\right)\end{array}\right]$$

(17)

As shown in Figure 7, different geometries are formed for flexion and extension. Actuators that create tension in the tendons need to generate a $\Delta l$, as specified in Equation (18).

$$\Delta l_{f_{pm}}=l_{f_{pm}}-\theta \left(r_k-r_{tf}\right), \Delta l_{e_{pm}}=l_{e_{pm}}-\theta \left(r_k+r_{te}cos\varphi \right)$$

(18)

Here, $\Delta l_{e_{pm}}$ and $\Delta l_{f_{pm}}$ are for extension and flexion, respectively. If we consider that the abduction and adduction motion angle of $\varphi$ becomes different from 0, for flexion, where the finger as a backbone is positioned above, the value of $l$ should be $l_f=r_k-r_{tf}·cos\varphi$. Conversely, for extension, where the finger as a backbone exerts force below its center, the value should be $l_e=r_k+r_{te}·cos\varphi$.

Figure 7. Kinematic relations for flexion and extension models.

Figure 7. Kinematic relations for flexion and extension models.

When the length of the backbone between two nodes ($s_{ij}$) is equal to the length of the tendon ($l_{ij}$),$\kappa s$ or ${\theta }_{ij}$ becomes $\pi$ radians. As shown in Figure 7, separate tendons are used for each phalanx. Tendons activating the distal, middle, and proximal phalanges can be activated individually. We can control ${\theta }_{ij}$ by pulling the appropriate tendon with a tension of ${\tau }_{ie}$ for extension and ${\tau }_{if}$ for flexion. Here, i represents the phalanx rings connected on the right-hand side, and j represents the left-hand side. For this geometry, any joint can be locked by applying ${\tau }_{ie}$ or ${\tau }_{if}$ to create stall tension on the opposite side. For example, to generate the flexion solely at the PIP joint without affecting other joints, we can apply a small amount of ${\tau }_{pe}$ and pull ${\tau }_{mf}$ with the amount of tension.

When we apply ${\tau }_{de}$ without any lock tension to other tendons, the angle of ${\theta }_{0d}$ can be observed between the base and distal phalanges. At the same time, ${\theta }_{md}$ and ${\theta }_{pm}$ can be observed between the middle and distal and proximal and distal, respectively. Here, assuming that all rings fixed phalanges too tightly, the angles between rings and phalanges are ${\displaystyle \frac{\pi }{2}}$ rad degrees and the angles of joints are $\pi -{\mathsf{\theta }}_{\mathrm{m}\mathrm{d}}$. Therefore, the joint angles can be calculated with the given Equation (19).

$${\left[DIP,PIP,MCP\right]}^T={\left[\pi -{\theta }_{md}, \pi -{\theta }_{pm}, \pi -{\theta }_{0m}\right]}^T$$

(19)

Angles of the $\varphi$ can be calculated with the tendon lengths and given by [35]. This angle is described as between the bending plane and the reference axis. Here, we use 2 tendon systems and reference axes.

$$\varphi ={\tan }^{-1}\left({\displaystyle \frac{3\left(l_{t3}-l_{t2}\right)}{\sqrt{3}\left(l_{t2}+l_{t3}-2l{t}_1\right)}}\right)$$

(20)

$$\kappa ={\displaystyle \frac{\sqrt{l_{t1}^2+l_{t1}^2+l_{t1}^2-l_{t1}{l}_{t2}-l_{t1}{l}_{t3}-l_{t2}{l}_{t3}}}{r_d\left(l_{t1}+l_{t2}+l{t}_3\right)}}$$

(21)

2.4. Force Relations of Tendons

Tendons apply a force $F_s$ depicted in Equation (22).

$$F_s=-{\tau }_{ix}{\displaystyle \frac{\left(p_s^{\prime }\left(l\right)\right)}{‖p_s^{\prime }\left(l\right)‖}}$$

(22)

Here, ${p\prime }_s$ is the derivative of the position vector of $p\left(s\right)$ with respect to the backbone length of s. ${\tau }_{ix}$ denotes the tension of the tendon of i for x movement [36]. This force produces a torque of $T_x$ at the joints of the fingers. Here, $x$ denotes MCP, PIP, or DIP.

$$T_X=p\left(l\right)F_s$$

(23)

2.5. Multibody Model and Simulation

After designing all parts in CAD software and mating all parts with each other properly, the model can be imported to MATLAB, and thus the Simscape model has been created, as shown in Figure 8. This model can be simulated under physical conditions. Stiffness and other internal mechanical parameters can be changed and have different perspective environments.

The internal mechanical parameters for the DIP, PIP, and MCP joints have been determined from previous studies that have analyzed human hand mechanisms. All mechanical parameters are shown in

Table 1. Index finger mechanical parameters [37].

Movement	$\mathit{K}$  $*$ $(\mathbf{N}\mathbf{m}/\mathbf{r}\mathbf{a}\mathbf{d})$	$\mathit{B}$  $*$ $(\mathbf{N}\mathbf{m}\mathbf{s}/\mathbf{r}\mathbf{a}\mathbf{d})$
MCP flexion	$0.0482$	$0.0169$
PIP flexion	$0.0169$	$3.65\times {10}^{-4}$
DIP flexion	$0.002814$	$8.23\times {10}^{-5}$
Abduction	$0.098$	$3.7\times {10}^{-3}$
Phalanges	MoI *	Damping Ratio
Proximal	$3.45\times {10}^{-5}$	$0.562$
Middle	$6.33\times {10}^{-6}$	$0.665$
Distal	$1.37\times {10}^{-6}$	$0.687$
Abduction	$7.52\times {10}^{-5}$	$0.574$

* MoI: Moment of Inertia ($\mathrm{K}\mathrm{g}{\mathrm{m}}^2/\mathrm{r}\mathrm{a}\mathrm{d})$, K: Spring constant, B: damping coefficient.

.

In the sim-mechanics model, the backbone is modeled using the lumped approach, as described below. Here, with this model, we can apply a pulling force to the tendons and measure angular displacements from the joints. Also, external force can be applied to any point of the finger, and the consequences can be analyzed easily. The joint’s position and velocity can be collected for the control algorithm.

2.6. Control Algorithm

The control implementation applied force (F) in the z-direction to each continuum disk group, enabling the movement of individual phalanges. Nominal controllers are devised to regulate the reference angles for each phalanx. Due to the simulation’s cascading structure, the system operated at a slow pace, necessitating the determination of references and initial simulation runs. Angular positions and input forces could be observed during simulation, albeit not in real time.

In Figure 9, the schematic of the proposed control algorithm is shown. Here, system dynamics are estimated by an RBF-based neural network online, and the obtained dynamics are used as an adaptive controller to regulate the error between the reference model and the state of the system. Additionally, there is a nominal controller ($u_n$) in the system.

We assume that the system is unstructured, mainly because there are uncertainties in various dynamic factors. This assumption leads us to create the following equation to describe the system.

$$\dot{x}=\varsigma \left(x\right)+u$$

(24)

Here, the state vector of our system is denoted by x∈$R^n$, where $\varsigma$ represents uncertain and unmodeled dynamics in the system. The input to the system is denoted by u. Unmodeled dynamics and all uncertainties in the system can be expressed as shown in Equation (25).

$$\varsigma \left(x\right)=W^T \theta \left(x\right)+\epsilon \left(x\right)$$

(25)

Here, $\theta \left(x\right)$ represents radial basis functions (RBFs) that model unknown, unmodeled, and uncertain aspects within the system, while $W$ specifies the weights of each RBF, and $\epsilon \left(x\right)$ represents the residual error.

$$\dot{x}=W^T \theta \left(x\right)+\epsilon \left(x\right)+u$$

(26)

The signal u can be expressed as in Equation (27) to control the system. The controller consists of two parts here: the nominal controller is represented by $-\alpha (x-c)$ and the neuroadaptive controller is represented by $-{\widehat{W}}^T\theta \left(x\right).$

$$u=-\alpha \left(x-c\right)-{\widehat{W}}^T\theta \left(x\right)$$

(27)

Here, $\widehat{W}$ is the estimated weight matrix for system uncertainties and unmodeled dynamics. If the system is controlled by referencing a first-order model as given in Equation (28), an error occurs as in Equation (29).

$${\dot{x}}_r=-\alpha (x_r-c)$$

(28)

$$e=x-x_r$$

(29)

Thus, the change in the error will be as given in Equation (30).

$$e^{.}=-\alpha e-{\tilde{W}}^T \theta (x)+\epsilon (x)$$

(30)

$$\tilde{W}=\widehat{W}-W$$

(31)

Here, $\tilde{W}$ represents the difference between the estimated weights and the actual weights. To regulate the error dynamic equation, a Lyapunov candidate equation can be written as follows.

$$V={\displaystyle \frac{1}{2}}{e}^2+{\displaystyle \frac{1}{2\gamma }} {\tilde{W}}^T \tilde{W}$$

(32)

The Lyapunov candidate equation is quadratic and positive definite. When the time derivative of the energy equation is taken, it will be as shown in Equation (33).

$$\dot{V}=e\left[-\alpha e-{\tilde{W}}^T \theta \left(x\right)+\epsilon \left(x\right)\right]+{\displaystyle \frac{1}{\gamma }}{\tilde{W}}^T \dot{\tilde{W}}$$

(33)

Here, if it is assumed that the uncertainties in the system are bound by a limited $W$, then $\dot{\tilde{W}}$ can be replaced with $\widehat{W}$. Consequently, if a dynamic is written for $\dot{\widehat{W}}$ as in Equation (34) instead of $\dot{\tilde{W}},$ the Lyapunov candidate equation will be negatively semi-definite, as seen in Equation (35). This indicates that the error tends to zero as time approaches infinity.

$$\dot{\widehat{W} }=\gamma \theta (x)e$$

(34)

$$\dot{V}=-\alpha e^2+\epsilon \left(x\right)$$

(35)

Here, γ is the learning rate, and e is the error. Applying a (sigma) σ modification to the estimation model written in Equation (36) can help avoid local minimum in the solution space.

$$\dot{\widehat{W} }=\gamma (\theta (x)(x-x_r)-\sigma \widehat{W})$$

(36)

2.7. Control Action for Assistive and Active Rehabilitation

The external force applied to the rehabilitation robot by a patient or physical therapist can be measured by adding to the system force sensor. The sim-mechanics multi-dynamic simulation model can already be measured on any point of a finger.

Using measured external force, a reference model can be created to easily modify its parameters to manage the rehabilitation process adaptively. Let us say that there is a mathematical model for simulating the effect of the external force. Suppose that the virtual mechanical system dynamics are as follows.

$$m_d{\ddot{x}}_d+b_d{\dot{x}}_d+k_d{x}_d=F_{ex}$$

(37)

From this equation, $x_d$ can be obtained because of the reference model of the external force. Using this desired state $x_d$, it is possible to create a control rule for active, assistive, or resistive rehabilitation processes. For a passive rehabilitation, $x_d$ can be used to create a deviation from the predefined trajectory $x_r$ of the system.

$$x_r^{\prime }=x_r-x_d,{\dot{x}}_r^{\prime }={\dot{x}}_r-{\dot{x}}_d$$

(38)

And this new reference can be used for manipulating the hands of patients. For this mode, patients do not have any ability to move. Robots take over all action from the patient. It can be modified by changing parameters $(m_d,b_d,k_d$) to an aggressive or smooth rehabilitation session. On the other hand, for active rehabilitation, the desired state variable $x_d$ can be used to create a deviation from the actual position $x_1$ of the actuator by

$$x_r^{\prime }=x_1-x_d$$

(39)

Here, $x_r^{\prime }$ can be used for the reference signal for the control algorithm. We can easily change the dynamics of the reference model $(m_d,b_d,k_d$) to change the dynamics of the response of the system. For this approach, patients do have a motion on their hand and a robot running with the external force applied by the patient.

3. Results and Discussion

3.1. Results of the Kinematics

Sim-Mechanic Multibody Model

Individual experiments are used in this section to show an open-loop response of the multibody dynamic system. Figure 10 depicts an experiment in which tension is applied to the proximal tendons as both flexors and extensors. For this experiment, −40 N tension is applied to the proximal flexor tendons for the first five seconds of the simulation and then −40 N tension is applied to the proximal extensor while releasing tension on the flexor tendon for the second part of the simulation. The first and second graphics show 0 to 0.5-radian flexion movement caused by this action on the MCP joint.

Figure 11 shows the second experiment with abduction and adduction movements for the index finger. As shown in Figure 11, for the first five seconds, −40 N tension was applied to the proximal abduction tendon and −40 N tension to the proximal adduction tendon, respectively. This tension combination resulted in abduction and adduction movements.

The MCP joint is also affected when forces are applied to the middle flexor and extensor tendons. Figure 12 depicts this combination of the PIP flexion and extension movement. When tension is applied to the middle tendons, the PIP joint moves to approximately 0.3 rad, and the MCP joint moves to approximately 0.5 rad.

Figure 13 shows the same scenario for the DIP joint and distal tendons. When tension is applied to the distal flexor and extensor, the PIP and MCP joints move as well.

Tensions can be applied to the other tendons for extension when tension is applied for flexion at the PIP and DIP joints. Figure 14 shows the experiment with DIP flexion and extension alone. When −10 N tension is applied to the distal flexor for the first 5 s, the middle and proximal tendons receive 20 N tension. This allows for individual DIP flexion and extension without moving the other joints. Figure 15 depicts the experiment for both PIP flexion and extension alone.

3.2. Control Results

Control experiments are conducted for each joint angle using the proposed control algorithm. In the experiment shown in Figure 16, a reference is given to the controller for the MCP joint. As can be seen in the graph, the MCP joint successfully followed this reference. The process was carried out using the tension applied to the tendon connected to the proximal phalanx, which affects the MCP joint. As shown in Figure 16, a tension of around −60 N is applied to the proximal tendon.

A similar experiment is conducted for the DIP joint, as shown in Figure 17. In this scenario, the influence of the distal tendon, which drives the DIP joint, on the other joints can be seen in the graphs. While a trajectory reference was given for the DIP, the other joints were given a reference of zero. The controller applied extension tensions in the middle and proximal tendons for the MCP and PIP joints to keep the other joints stable, as shown in Figure 17.

Figure 18 shows the results of a similar experiment for the PIP joint. In this scenario, the controller generated the desired extension and flexion tensions. As seen in Figure 18, a flexion is applied to the tendon connected to the middle phalanx, while an extension tension is applied to the tendon connected to the proximal phalanx.

In the experiment shown in Figure 19, similar trajectory targets are provided for all three joints. The controllers produced tensions for these trajectories. The deviation in the MCP joint might be noticeable here, indicating that all tendons influence the flexion and extension of the MCP joint.

3.3. Assistive and Passive Rehabilitation Scenarios

The results of the planned experiment for the active rehabilitation scenario are shown in Figure 20. As seen in Figure 20, a negative force was applied to the fingertip at the 10th second. The system is expected to respond adaptively to this force. As explained in the Section 2, the deviation of the trajectory calculated using external force ($F_{ex}$) and a virtual mass-spring-damper system was added to the system’s reference. In Figure 20, a distorted reference can also be seen beside the reference in the graph titled PIP. This value shows the deviation produced by the mechanical system influenced by the external force.

As shown in Figure 20, the negative tension was applied at the 10th second, meaning that the finger is forced into flexion either by an external source or the patient’s own tendons, causing a deviation in the reference. The controller then followed this new trajectory, as seen in the graph titled PIP, showing adaptive behavior to the external force. When the direction of the force changed at the 30th second, the system adapted to the demand for extension, recreating the trajectory with a deviation in that direction. This experiment is performed with $m=0.5, k=3,\mathrm{a}\mathrm{n}\mathrm{d} b=5.$ The response behavior can be easily changed by adjusting the mechanical parameters.

Figure 21 shows the results of the passive rehabilitation experiment. In passive rehabilitation, the system is passive while the patient is active. The system completely follows the patient’s movements and tracks the trajectory generated by the force the patient’s hand applies to the system. In this experiment, the sinusoidal force profile was applied to the fingertip. This profile can be interpreted as the patient moving their finger into flexion or extension using their own tendons.

This scenario functions like a regulator. The reference is always the current position, and the system tries to maintain itself at that location. When an external force is applied, a trajectory is created by the deviation force, which becomes the controller’s reference trajectory. In Figure 21, the distorted reference can be seen in the PIP joint. The controller successfully followed this trajectory. Since the reference is always the actual position, the system will stop and no longer apply force when the patient stops applying force.

Table 2. Root mean square of errors for each joint in all simulation experiments.

Simulation Exp.	$\mathit{M}\mathit{C}\mathit{P}$	$\mathit{P}\mathit{I}\mathit{P}$	$\mathit{D}\mathit{I}\mathit{P}$
MCP trajectory	$0.029757$	$1.4137\times {10}^{-5}$	$1.1672\times {10}^{-4}$
DIP trajectory	$0.015834$	$0.10025$	$3.314\times {10}^{-3}$
PIP trajectory	$5.6519\times {10}^{-3}$	$2.82\times {10}^{-3}$	$1.4062\times {10}^{-4}$
MCP, PIP, DIP Trj	$0.03376$	$0.015434$	$2.9544\times {10}^{-3}$
Active rehab.	$0.042773$	$0.029066$	$1.7087\times {10}^{-3}$
Passive rehab.	$0.020839$	$8.6129\times {10}^{-3}$	$5.2188\times {10}^{-4}$

provides the root mean square (RMS) values of the tracking errors for the overall evaluation of the controller’s performance in the simulation experiments. As shown in the table, for the MCP trajectory simulation experiment, the PIP and DIP joints are assumed to remain stationary at 0 degrees while the MCP joint tracks the given trajectory. The same scenario applies to the DIP and PIP trajectory experiments.

In each individual experiment, the tracking errors for the DIP joint are notably low. This is because the DIP joint primarily affects only the tendons connected to the distal phalanges. In contrast, the MCP and PIP joints are influenced by other tendons due to the kinematic structure of the system, as previously discussed.

For the passive rehabilitation scenario, the manipulated joint (PIP) tracking error is lower than in the active case. This is because, in passive rehabilitation, no specific trajectory is provided to the system. Instead, the system attempts to follow the patient’s intentions by utilizing external forces.

4. Conclusions

In this study, the design and control of a finger rehabilitation system driven by a parallel continuum robot structure were investigated. Finger extension and flexion were achieved using continuum robots, which functioned as actuators. The flexion and extension movements of each finger joint, as well as the finger’s abduction and adduction, were made possible through the proposed parallel continuum robot actuator. Additionally, a neuroadaptive controller was proposed for the control of this model. The kinematic model of the proposed system was developed using the Euler–Lagrange approach. Subsequently, a simulation model was created using the MATLAB multibody tool. Through this model, open-loop responses demonstrated that each joint could be individually controlled, which is particularly relevant for patients with rheumatoid arthritis. It was also shown that the proposed controller could enable each joint to follow a desired trajectory.

In this study, active and passive rehabilitation scenarios for each joint were also developed in conjunction with the control algorithm. Each joint can operate in active rehabilitation mode, where the robot is active and the patient is passive, or in passive rehabilitation mode, where the patient is active and the robot is passive. These scenarios were validated through experiments.