2.1. System design

The KRRS mainly consists of a link with a pivot joint that simulates the knee movement. The joint is actuated by a geared DC motor through a cable-driven transmission system. An optical encoder is mounted on the motor shaft to measure the joint position. The motor, transmission system, and encoder are shown in Fig. 2, and their specifications are illustrated in Table 1.

Figure 2.

DC motor with transmission system.

The patient’s shank is attached to the link in parallel by a brace, as shown in Fig. 1(b). The length of the link can be adjusted in a range between 45 and 63 cm, with an adjustment step of 1.5 cm. This feature permits the KRRS to fit the various lengths of the patient’s shank.

2.1.1. Involuntary movements measurements mechanism

To measure potential involuntary movements of the knee, we developed a new mechanism based on the bar load cells. We used two straight bar load cells, mounted on the link sides. The brace is connected to the KRRS link via a passive joint. Once the knee moves involuntarily, this passive joint allows the brace to move and hit one of the load cells. The motion range of the passive joint is limited, and it is restricted by two high-stiffness springs at each side. Figure 3 represents (a) the schematic and (b) the CAD model of the load cells positions along with the brace, springs, and passive joint.

Figure 3.

(a) The schematic and (b) CAD model of the passive joint, brace, springs, and load cells.

2.2. Control unit

Since the proposed KRRS works in direct contact with a human, any delay in the control loop may endanger the patient’s safety. For instance, the delay in acquiring the measured interaction force will lead to mishandle this excessive force and the robot may exceed the predefined limits if there is a delay in reading the actual angular position. Such cases can cause serious harm to the patient’s leg. Thus, it is essential that the control unit should work in real time and avoid any delay in both receiving commands from the physiotherapist and acquiring data from the sensors.

Even PCs provide the ability to design control software easily using high-level programming language environments such as MATLAB and LabVIEW. However, they are not designed to achieve uninterrupted real-time control [Reference Gupta and Sharma27]. They share their resources between the control software and other system activities. Some of these activities are under user control, while some others are not. This affects the speed and repeatability of the operation of the robot and is out of user’s control. Moreover, when a PC freezes or undergoes hard drive failure, due to some of these activities, the I/O unit controlling the robot will not be often neither controllable nor monitored.

Table I.

Specification of actuating system.

To provide a real-time system, an ARM Cortex-M4-based STM32 microcontroller (model number: STM32F407ZGT6) is utilized as the control unit of KRRS. This microcontroller is programmed by C programming language, and it works on frequency 168 MHz. All control procedures of the robot are implemented by this microcontroller. On the other hand, a host PC is connected to the control unit. Using a UI on this host PC, the therapist can observe the therapy session, as well as command the control unit. Another AVR Atmega328-based board is used for safety, which controls a relay that shuts down the whole system immediately when the emergency pushbutton is pressed.

2.3. User interface

Based on C# programming language, an appropriate and user-friendly UI has been developed. This UI is available as an executable file (*.exe). Thus, it can be installed on any Windows-based PC. By this UI, a physiotherapist can perform the following functions:

• Send the main commands to the control unit: start, stop, pause, and resume.

• Select the suitable range of motion for the patient within the range of motion of the knee. Figure 4 shows the range of motion of the tibia with respect to the vertical axis according to ref. [Reference Akdoğan, Taçgın and Adli28].

• Set the exercise mode: passive or active. In both modes, the robot drives the patient’s limb on a reference trajectory. Patients’ interaction forces are not considered in the passive mode. While in the active mode, these interaction forces are measured and considered during the session.

• Monitor data of the therapy session in real time: these data are divided into two sets:

Figure 4.

Range of motion of the knee joint.

The first set of data is for evaluating the therapy progress of the patient: position and interaction torques. The other set is for evaluating the performance of the robot: tracking error, control output, and CE. Moreover, these data can be saved to an Excel file for further analysis.

4.1. The outer control loop

During the exercise session by the KRRS, the patient’s limb may move either voluntarily by the patient or involuntary because of a neurological disorder. This movement generates an interaction torque. This torque is undesirable and may cause pain or injury for the patient. To reduce the effects of these interaction torques and avoid secondary injury, we implement an admittance (reverse impedance) control strategy. The undesirable interaction torques are measured by load cells. Then, they are applied to the input of admittance control. The admittance control attains the flexibility of the robot by changing its trajectory suitably according to the measured torques.

Using a mass-damper-spring impedance model [Reference Song, Yu and Zhang30], the interaction torque can be given as follows:

(16) \begin{equation}M_{i}\ddot{\tilde{\theta }}+C_{i}\dot{\tilde{\theta }}+K_{i}\tilde{\theta }=\tau _{int}\end{equation}

(17) \begin{equation}\tilde{\theta }=\theta _{des}-\theta _{r}\end{equation}

where $M_{i}, C_{i}, K_{i}$ are the inertia, damping, and stiffness impedance terms, respectively. These terms should be chosen in a way that ensures a smooth and safe change in the reference trajectory. $\theta _{des}$ is the predefined trajectory and $\theta _{r}$ is the reference trajectory, which is the input of the inner control loop. $\tilde{\theta }$ is the robot trajectory deviation from the predefined trajectory. $\tau _{int}$ is the undesirable interaction torque. When $\tau _{int}=0$ , we have $\tilde{\theta }=0$ and $\theta _{r}=\theta _{des}$ , and when the interaction torque is exerted, the reference trajectory changes and deviates from the predefined trajectory. Substituting Eq. (17) in (16), the reference trajectory, which is proper to the exerted interaction torques, can be obtained as follows:

(18) \begin{equation}M_{i}\left(\ddot{\theta }_{des}-\ddot{\theta }_{r}\right)+C_{i}\left(\dot{\theta }_{des}-\dot{\theta }_{r}\right)+K_{i}\left(\theta _{des}-\theta _{r}\right)=\tau _{int}\end{equation}

(19) \begin{equation}\ddot{\theta }_{r}=\ddot{\theta }_{des}-M_{i}^{-1}\left\{C_{i}\left(\dot{\theta }_{des}-\dot{\theta }_{r}\right)+K_{i}\left(\theta _{des}-\theta _{r}\right)-\tau _{int}\right\}\end{equation}

4.2. The inner control loop

This loop achieves the trajectory tracking task, where the robot guides the leg to move along a reference trajectory with the highest possible accuracy. The tracking error, which is the difference between the reference angular position $(\theta _{r})$ and the robot actual angular position $(\theta )$ , and its first derivative are given as follows:

(20) \begin{equation}e=\theta _{r}-\theta\end{equation}

(21) \begin{equation}\dot{e}=\dot{\theta }_{r}-\dot{\theta }\end{equation}

A PD-RBF NN is proposed to be applied in this loop. However, before discussing the aforementioned proposed approach, two other controllers are introduced for a comparative study purpose: an ASMC, as a representation of the model-based controller, and a conventional PD controller, as a model-free controller. The results of these controllers are used in a comparative study to prove the efficiency of the PD-RBF controller. The reasons behind choosing these two controllers, PD and ASMC, are

• PD, as a variation of PID, is the most basic and simplest controller for any system and one of the most widely used controllers for robots in trajectory tracking tasks. However, its performance is subject to the nonlinearity level of the system [Reference Feng, Yin, Weng, Ma, Zhou, Jia and Zhang31]. Therefore, since our system is nonlinear, we start from PD and seek to improve its performance for our system. Then validate the improved PD (PD-RBF) by comparing its performance with the conventional PD.

• Sliding mode control (SMC) has a robust property and a complex structure, which is commonly used to control nonlinear systems with known uncertain bounds [Reference Shtessel, Edwards, Fridman and Levant32]. ASMC is an improved version of SMC to control nonlinear systems when the boundaries of the uncertainty are unknown, which is more practical [Reference Roy, Baldi and Fridman33]. Thus, since the proposed PD-RBF is an advanced control, it is reasonable to examine its performance with a controller more complex than PD, such as ASMC, to have fair comparison.

4.2.1. Adaptive sliding mode control

In rehabilitation robots, many control methodologies based on SMC have been developed [Reference Yousefi, Alipour, Tarvirdizadeh and Hadi9, Reference Babaiasl34–Reference Torabi, Sharifi and Vossoughi36]. In this research, an ASMC with PID surface has been introduced to perform trajectory tracking control.

The sliding surface can be defined as follows [Reference Eker37]:

(22) \begin{equation}s=\dot{e}+\beta _{1}e+\beta _{2}\int edt\end{equation}

where $\beta _{1}$ and $\beta _{2}$ are positive gains.

The aim of the SMC is to force the error to approach the sliding surface and then move the error along the sliding surface toward the zero.

To achieve this aim, the control law of SMC should be designed in a way that the following expression is maintained:

(23) \begin{equation}\dot{s}=0\end{equation}

This condition is necessary to ensure that the error stays on the sliding surface.

where $\dot{s}$ is the time derivative of the sliding surface and derived as follows:

(24) \begin{align}\dot{s}&=\ddot{e}+\beta _{1}\dot{e}+\beta _{2}e\\[5pt] & =\left(\ddot{\theta }_{r}-\ddot{\theta }\right)+\beta _{1}\dot{e}+\beta _{2}e \nonumber \end{align}

Substituting (15) in (24) yields:

(25) \begin{equation}\dot{s}=\left(\ddot{\theta }_{r}-M_{R}^{-1}\left\{\tau -\tau _{int}-G\left(\theta \right)-D(t)\right\}\right)+\beta _{1}\dot{e}+\beta _{2}e\end{equation}

Equation (25) has known (nominal and measurable) and unknown, but bounded, terms. Herein, we introduce a control law that consists of two terms, an equivalent and auxiliary, the former considers the known term while the latter tries to vanish the impact of the unknown term. The control law is

(26) \begin{equation}\tau =\tau _{eq}+\tau _{aux }\end{equation}

making $\dot{s}=0$ , and neglecting the unknown term $D(t)$ , the equivalent control law $\tau _{eq}$ is obtained as

(27) \begin{equation}\tau _{eq}=M_{R}\left(\ddot{\theta }_{r}+\beta _{1}\dot{e}+\beta _{2}e\right)+G_{R}\left(\theta \right)+\tau _{int}\end{equation}

The term of system uncertainties and disturbance ( $D(t)$ ) leads the tracking error to drift away from the sliding surface. In order to overcome this issue, an auxiliary control law is defined. This auxiliary control law has a switching discontinuous function which forces the tracking error to lie on the sliding surface. The auxiliary control law $\tau _{aux}$ is given as:

(28) \begin{equation}\tau _{aux}=M_{R}\left(K_{1}s+K_{2}sign\left(s\right)\right)\end{equation}

where $K_{1}$ and $K_{2}$ are positive gains. The sign function, $sign(\blacksquare )$ , is the ideal discontinuous function, which guarantees an infinitely fast switching. However, in practice, it is impossible to achieve infinitely fast switching control because of the finite time delays for the control computation and limitations of the physical actuators. This imperfect switching in practice raises the oscillation problem, known as chattering. Chattering is a harmful phenomenon because it leads to low control accuracy, may damage the physical parts of the system, and requires lots of energy for good efficiency. A common method to reduce chattering is to define a boundary layer near the sliding surface and then use a saturation function instead of the sign function. When the system is within the boundary layer, a continuous control will be achieved and the chattering will be alleviated. The saturation function $sat(\blacksquare )$ is defined as follows:

(29) \begin{equation}sat\left(s\right)=\begin{cases} \varphi & s\geq \varphi \\ -\varphi & s\leq \varphi \\ s & -\varphi \lt s\lt \varphi \end{cases}\end{equation}

where $\varphi$ is the boundary layer thickness.

Thus, $\tau _{aux}$ will be

(30) \begin{equation}\tau _{aux}=M_{R}\left(K_{1}s+K_{2}sat\left(s\right)\right)\end{equation}

Further, in order to improve the robustness of the SMC law, an adaptive term has been introduced.

Assumption 1:

The term $D(t)$ is unknown but bounded, that is, $\Lambda \gt \left| D(t)\right|$ , where $\Lambda$ is unknown positive number. Thus,

$\hat{\Lambda}$ is the estimated value of $\Lambda$ and obtained from the adaptation law:

(31) \begin{equation}\dot{\hat{\Lambda }}=\kappa \left| s\right|\end{equation}

where $\kappa$ is a positive number. $\hat{\Lambda }$ is added to $\tau _{aux}$ as follows:

(32) \begin{equation}\tau _{aux}=K_{1}s+K_{2}sat\left(s\right)+\hat{\Lambda }sat\left(s\right)\end{equation}

Finally, by substituting (27) and (32) in (26), the ASMC law, $\tau _{ASMC}$ , is given as

(33) \begin{equation}\tau _{ASMC}=M_{R}\left(\ddot{\theta }_{r}+\beta _{1}\dot{e}+\beta _{2}e\right)+G_{R}\left(\theta \right)+\tau _{int}+K_{1}s+K_{2}\mathrm{sat}\left(s\right)+\hat{\Lambda }sat\left(s\right)\end{equation}

Figure 6 presents the block diagram of the admittance ASMC approach.

Stability analysis

To ensure the stability of the designed controller, a candidate Lyapunov positive definite function $V$ is defined as follows:

(34) \begin{equation}V=\frac{1}{2}\kappa M_{R}s^{2}+\tilde{\Lambda }^{2}\end{equation}

where

(35) \begin{equation}\tilde{\Lambda }=\Lambda -\hat{\Lambda }\end{equation}

Assumption 2:

Assigning the initial value of $\hat{\Lambda }$ as $\hat{\Lambda }_{\textbf{0}}\ll \Lambda$ , where $\hat{\Lambda }_{\textbf{0}}$ is the value of $\hat{\Lambda }$ at t = 0, and for a relatively small value of $\kappa$ in Eq. (31), $\hat{\Lambda }$ will increase slowly. Moreover, as much as the sliding surface $s$ converges to zero, $\dot{\hat{\Lambda }}$ converges to zero. This will ensure that $\hat{\Lambda }$ will flatten quickly and will not exceed $\Lambda$ . Thus, the value of $\tilde{\Lambda }$ will be $\geq 0$ . Generally, it is guaranteed that $\tilde{\Lambda }$ is positive, always.

The derivative of Lyapunov function is given as follows:

(36) \begin{equation}\dot{V}=\kappa M_{R}s\dot{s}-2(\Lambda -\hat{\Lambda })\dot{\hat{\Lambda }}\end{equation}

Substituting (33), in (25), and, substituting the obtained formula and (31) in (36) lead to:

(37) \begin{align} \dot{V}& =\kappa s\left(-K_{1}s-K_{2}sat\left(s\right)-\hat{\Lambda }sat\left(s\right)+(D(t)\right)-2(\Lambda -\hat{\Lambda })\kappa \left| s\right| \nonumber \\ & =-\kappa K_{1}s^{2}-\kappa sK_{2}sat\left(s\right)-\kappa s\hat{\Lambda }sat\left(s\right)+\kappa s(D(t)-2(\hat{\Lambda }-\Lambda )\kappa \left| s\right| \nonumber \\ & \quad \leq -\hat{\Lambda }\kappa \left| s\right| +\left| (D(t)\right| \kappa \left| s\right| -2\left(\Lambda -\hat{\Lambda }\right)\kappa \left| s\right| +\Lambda \left| s\right| -\Lambda \left| s\right|\end{align}

Thus, the first derivative of Lyapunov function $\dot{V}$ can be written as follows:

(38) \begin{equation}\dot{V}\leq -\left(\Lambda -\left| (D(t)\right| \right)\kappa \left| s\right| -\left(\Lambda -\hat{\Lambda }\right)\kappa \left| s\right|\end{equation}

Since $\left| D(t)\right| \lt \Lambda$ , according to the assumption 1, and by choosing $\hat{\Lambda }_{0}$ and $\kappa$ properly (based on assumption 2), that provide the condition $\tilde{\Lambda }\geq 0$ , $\dot{V}$ is negative definite, and according to the global stability theorem of Lyapunov, the stability of the system is guaranteed.

4.2.2. PD controller:

PID controller and its other types, PD and PI, are model-free controller where they do not require any knowledge about the dynamic system. These controllers are widely used for controlling both linear and nonlinear systems due to their simple structure, robustness, ease of implementation, low cost, and high performance [Reference Feng, Yin, Weng, Ma, Zhou, Jia and Zhang31, Reference Kara and Mary38, Reference Sharkawy and Koustoumpardis39]. The PD equation used in this research is described as follows:

(39) \begin{equation}\tau _{pd}=K_{p}e+K_{d}\dot{e}\end{equation}

The total torque to be applied on the robot is

(40) \begin{equation}\tau =\tau _{pd}+\tau _{int}\end{equation}

where $K_{p}$ and $K_{d}$ are the positive proportional and derivative gains, respectively. $\tau _{pd }$ is the PD controller output. Figure 7 shows the block diagram of PD controller.

Figure 7.

Block diagram of the PD controller.

Figure 8.

The architecture of the RBF neural network.

Figure 9.

Block diagram of the PD-RBF controller.

4.2.3. PD augmented by RBF

Since the patient’s limb is connected directly to the robot in parallel and the physical characteristics of the leg differ from one patient to another, the nonlinear characteristics of the robot vary dramatically for each patient. This change in the characteristics makes obtaining the accurate dynamic model difficult and affects the performance of the tracking control due to the high nonlinearity. To overcome this problem and provide the KRRS with highly precise tracking performance, we propose a PD controller augmented by an RBF NN block as feedforward loop. The aim of this feedforward loop is to compensate for the nonlinearity terms in the system and improve the tracking accuracy of the model-free controller (PD). RBF is widely implemented to approximate the nonlinearity due to its simplicity and fast convergence [Reference Nguyen, Lin, Su and Tran40, Reference Li and Tian41]. Figure 8 shows the architecture of the proposed RBF NN, which consists of an input layer, hidden layer, and output layer. Gaussian functions are used as an RBF in the hidden layer. Then, the output of RBF, $\tau _{rbf }$ , is obtained as follows:

(41) \begin{equation}h_{i}=\exp \left(-\frac{\left\| X-c_{i}\right\| ^{2}}{2b_{i}^{2}}\right)\end{equation}

where $i=1,\ldots,4$ is the number of the neuron and $h_{i}$ , $c_{i}$ , $b_{i}$ are the Gaussian function, center value, and the width values of the neuron $i$ , respectively. $X$ is the reference trajectory $\theta _{r}$ .

And the output of the RBF NN is given as follows:

(42) \begin{equation}\tau _{rbf}=\boldsymbol{{w}}^{\mathrm{T}}\boldsymbol{{h}}\boldsymbol{{sat}}\left(\tau _{pd}\right)\end{equation}

where $\boldsymbol{{w}}\in \mathcal{R}^{4\times 1}$ is the weights vector.

Defining the weights adaptive law as follows:

(43) \begin{equation}\dot{\boldsymbol{{w}}}=\frac{1}{2}\textbf{P}\,\boldsymbol{{h}}\left| \tau _{pd}\right|\end{equation}

where $\textbf{P}\in \mathcal{R}^{4\times 4}$ is a positive diagonal matrix.

Thus, the proposed control output, shown in Fig. 9, is given as follows:

(44) \begin{equation}\tau =\tau _{rbf}+\tau _{pd}+\tau _{int}\end{equation}

Stability analysis

To analyze the stability of the proposed control strategy, the following assumptions are considered, without any lake of generality:

Assumptions 3:

Define a new term includes the nonlinear and unknown terms of Eq. (15), this new term is unknown but bounded.

(45) \begin{equation}\Pi(\theta,t)=G\left(\theta \right)+D(t)\end{equation}

The ideal approximation of the NN output ${\boldsymbol{{w}}^{*}}^{\mathrm{T}}\boldsymbol{{h}}$ , that is, ${\boldsymbol{{w}}^{*}}^{\mathrm{T}}\boldsymbol{{h}}\gt \left| \Pi (\theta,t)\right|$ where $\boldsymbol{{w}}^{\boldsymbol{*}}$ denotes the best approximation of the NN weights.

Assumptions 4:

$K_{p}$ and $K_{d}$ are chosen to ensure that the following inequalities are true:

(46) \begin{equation}2K_{p}K_{d}\gt M_{R}K_{d}^{-1}K_{p}^{2}\end{equation}

(47) \begin{equation}K_{d}^{2}\gt M_{R}K_{p}\end{equation}

Rewriting the dynamic Eq. (15) as function of error and substituting Eqs. (45), (44), (42), and (39) in it as follows:

(48) \begin{equation}M_{R}\ddot{e}+K_{d}\dot{e}+K_{p}\mathrm{e}+\boldsymbol{{w}}^{\mathrm{T}}\boldsymbol{{h}}\mathrm{sat}\left(\tau _{pd}\right)-\Pi \left(t\right)=0\end{equation}

Defining the Lyapunov candidate as follows:

(49) \begin{equation}V=\frac{1}{2}M_{R}K_{d}^{-1}\left(K_{p}e+K_{d}\dot{e}\right)^{2}+\frac{1}{2}\left(2K_{p}K_{d}-M_{R}K_{d}^{-1}K_{p}^{2}\right)e^{2}+2\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\tilde{\boldsymbol{{w}}}\end{equation}

where

(50) \begin{equation}\tilde{\boldsymbol{{w}}}=\boldsymbol{{w}}^{\boldsymbol{*}}-\boldsymbol{{w}}\end{equation}

Assumption 5:

Assigning the initial value of $\boldsymbol{{w}}$ as $\boldsymbol{{w}}_{\textbf{0}}\ll \boldsymbol{{w}}^{\boldsymbol{*}}$ , where $\boldsymbol{{w}}_{\textbf{0}}$ is the value of $\boldsymbol{{w}}$ at t = 0, and for a relatively small values of $\textbf{P }$ in Eq. (43), $\boldsymbol{{w}}$ will increase slowly. Moreover, as much as the error converges to zero, $\dot{\boldsymbol{{w}}}$ converges to zero. This will ensure that $\boldsymbol{{w}}$ will be flatten quickly and will not exceed $\boldsymbol{{w}}^{\boldsymbol{*}}$ . Thus, the value of $\tilde{\boldsymbol{{w}}}$ will be $\geq \textbf{0}$ . Generally, it is guaranteed that $\tilde{\boldsymbol{{w}}}$ is positive, always.

Differentiating (49), we obtain the following relation for $\dot{V}$ .

(51) \begin{align}\dot{V}&=\left(K_{p}e+K_{d}\dot{e}\right)\left(M_{R}K_{d}^{-1}K_{p}\dot{e}+M_{R}\ddot{e}\right)+\left(2K_{p}K_{d}-M_{R}K_{d}^{-1}K_{p}^{2}\right)e\dot{e}-4\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\dot{\boldsymbol{{w}}}\\[5pt] & = M_{R}K_{d}^{-1}K_{p}^{2}e\dot{e}+M_{R}K_{p}e\ddot{e}+M_{R}K_{p}\dot{e}^{2}+M_{R}K_{d}\dot{e}\ddot{e}+\left(2K_{p}K_{d}-M_{R}K_{d}^{-1}K_{p}^{2}\right)e\dot{e}-2\tilde{\boldsymbol{{w}}}\textbf{P}\boldsymbol{{h}}\left| \tau _{pd}\right| \nonumber \end{align}

Substituting (48) and (43) in (51):

(52) \begin{align}\dot{V}&=M_{R}K_{d}^{-1}K_{p}^{2}e\dot{e}+\left(K_{p}e+K_{d}\dot{e}\right)\left(-K_{p}e-K_{d}\dot{e}-\boldsymbol{{w}}^{\mathrm{T}}\boldsymbol{{h}}\mathrm{sat}\left(\tau _{pd}\right)+\Pi \left(t\right)\right)+M_{R}K_{p}\dot{e}^{2} \\ &\quad +\left(2K_{p}K_{d}-M_{R}K_{d}^{-1}K_{p}^{2}\right)e\dot{e}-2\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\textbf{P }\boldsymbol{{h}}\left| \tau _{pd}\right| \nonumber \end{align}

Then, the derivative of the Lyapunov function can be written as follows:

(53) \begin{align} \dot{V}& =-K_{p}^{2}e^{2}-\left(K_{d}^{2}-M_{R}K_{p}\right)\dot{e}^{2}-\boldsymbol{{w}}^{\mathrm{T}}\boldsymbol{{h}}\mathrm{sat}\left(\tau _{pd}\right)\tau _{pd}+\Pi \left(t\right)\tau _{pd}-2\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\textbf{P }\boldsymbol{{h}}\left| \tau _{pd}\right| \nonumber \\[5pt] & =-K_{p}^{2}e^{2}-\left(K_{d}^{2}-M_{R}K_{p}\right)\dot{e}^{2}-\boldsymbol{{w}}^{\mathrm{T}}\boldsymbol{{h}}\left| \tau _{pd}\right| +\Pi \left(t\right)\tau _{pd}-2\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\textbf{P }\boldsymbol{{h}}\left| \tau _{pd}\right| +{\boldsymbol{{w}}^{\boldsymbol{*}}}^{\mathrm{T}}\textbf{P}\boldsymbol{{h}}\left| \tau _{pd}\right| -{\boldsymbol{{w}}^{\boldsymbol{*}}}^{\mathrm{T}}\textbf{P}\boldsymbol{{h}}\left| \tau _{pd}\right| \nonumber \\[5pt] & =-K_{p}^{2}e^{2}-\left(K_{d}^{2}-M_{R}K_{p}\right)\dot{e}^{2}-\left({\boldsymbol{{w}}^{\boldsymbol{*}}}^{\mathrm{T}}\textbf{P}\boldsymbol{{h}}-\left| \Pi \left(t\right)\right| \right)\left| \tau _{pd}\right| -\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\textbf{P }\boldsymbol{{h}}\left| \tau _{pd}\right|\end{align}

(54) \begin{equation}\dot{V}\leq -\left(K_{d}^{2}-M_{R}K_{p}\right)\dot{e}^{2}-\left({\boldsymbol{{w}}^{\boldsymbol{*}}}^{\mathrm{T}}\textbf{P}\boldsymbol{{h}}-\left| \Pi \left(t\right)\right| \right)\left| \tau _{pd}\right| -\tilde{\boldsymbol{{w}}}^{\mathrm{T}}\textbf{P }\boldsymbol{{h}}\left| \tau _{pd}\right|\end{equation}

Considering assumptions 3, 4, and 5 stated earlier, it can be seen that $\dot{V}$ is negative definite, and the system is stable.

5.1. Passive mode

In this mode, the admittance control loop is disabled and the interaction torques between the KRRS and the leg are not considered. The robot follows the predefined trajectory without any changes. Figure 11 represents the predefined trajectory. First, the robot is tested, for each controller, without any load (the leg is not connected to the robot). Figure 12 shows the tracking error for the controllers in the passive no-load mode.

Then, the patient’s leg is connected, and the KRRS moves it along the predefined trajectory. Figure 13 shows a leg connected to the robot. Figure 14 represents the tracking error for the controllers in the passive with-load mode.

Figure 13.

The robot with a connected leg.

Figure 14.

Tracking error in the passive with-load mode.

Performance indices, IAE and CE, results of each controller in this mode are reported in Table 6.

Table VI.

IAE and CE of the controllers in the passive mode.

Figure 15.

Trajectory changes in the presence of the interaction torque $\tau _{int}$ and tracking performance of ASMC controller.

Figure 16.

Trajectory changes in the presence of the interaction torque $\tau _{int}$ and tracking performance of PD controller.

Figure 17.

Trajectory changes in the presence of the interaction torque $\tau _{int}$ and tracking performance of PD-RBF controller.

Figure 18.

Tracking error of the ASMC, PD, and PD-RBF controllers.

5.2. Active mode

The admittance control loop is enabled in this mode, and the exerted interaction forces are measured by load cells and filtered to consider only forces of the sudden undesirable movements. Then these forces are fed back into the admittance equation, where the predefined trajectory is modified accordingly to reduce the effects of these interaction forces and ensure safe and smooth movements. Figures 15, 16, and 17 show the tracking performance of the ASMC, PD, and PD-RBF, respectively. These figures also represent how the trajectory is modified when interaction torque is applied. The tracking error of these controllers is illustrated in Fig. 18.

The CE of the robot in the active mode is subject to the interaction torques. Thus, to have a reasonable comparison between the controllers in their CE, the applied interaction torques on each of them should be similar. Since such a condition cannot be realized in practical experiment, a new ratio between the control output and the applied interaction torque is defined as follows:

(60) \begin{equation}\eta =\frac{\int \left| \tau \right| dt}{\int \left| \tau _{int}\right| dt}\end{equation}

$\eta$ ratio represents the CE with respect to the interaction torques exerted by the patient.

Table 7 demonstrates the IAE and the ratio $\eta$ for each controller in the active mode.