1. Introduction

2. Human Lower Extremity Anatomy

3. Human Lower Extremity Kinematic and Dynamic Modeling

Figure 2. Human lower extremity rehabilitation exoskeleton robot.

Figure 2. Human lower extremity rehabilitation exoskeleton robot.

3.1. Kinematic and Dynamic Modeling

The kinematic model was developed using the modified Denavit-Hartenberg (DH) parameters method. A separate link frame was assigned to each degree of freedom. Figure 3 illustrates the link frame assignments according to the modified DH parameters. The modified D-H parameters for each joint are given in Table 2.

The general form of the transformation matrix is presented in Equation (22):

$${}_{\mathrm{I}}{}^{\mathrm{i}-1}\mathrm{T}=\left[\begin{array}{cccc}\mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\theta }}_{\mathrm{i}}& -\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\theta }}_{\mathrm{i}}& 0& {\mathsf{\alpha }}_{\mathrm{i}-1}\\ \mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\theta }}_{\mathrm{i}}\mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\alpha }}_{\mathrm{i}-1}& \mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\theta }}_{\mathrm{i}}\mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\alpha }}_{\mathrm{i}-1}& -\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\alpha }}_{\mathrm{i}-1}& -\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\alpha }}_{\mathrm{i}-1}{\mathrm{d}}_{\mathrm{i}}\\ \mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\theta }}_{\mathrm{i}}\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\alpha }}_{\mathrm{i}-1}& \mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\theta }}_{\mathrm{i}}\mathrm{s}\mathrm{i}\mathrm{n}{\mathsf{\alpha }}_{\mathrm{i}-1}& \mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\alpha }}_{\mathrm{i}-1}& \mathrm{c}\mathrm{o}\mathrm{s}{\mathsf{\alpha }}_{\mathrm{i}-1}{\mathrm{d}}_{\mathrm{i}}\\ 0& 0& 0& 1\end{array}\right]$$

(22)

By substituting the modified DH parameter values from Table 3 into Equation. (22), the resulting transformation matrices, Equation (23) to Equation (29), were obtained.

$${}^{0}T_1=\left[\begin{array}{cccc}\mathrm{c}\mathrm{o}\mathrm{s}(\begin{array}{c}{\theta }_1)\end{array}& -sin\left({\theta }_1\right)& 0& 0\\ sin\left({\theta }_1\right)& cos\left({\theta }_1\right)& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(23)

$${}^{1}T_2=\left[\begin{array}{cccc}sin\left({\theta }_2\right)& cos\left({\theta }_2\right)& 0& 0\\ 0& 0& 0& 0\\ cos\left({\theta }_2\right)& -sin\left({\theta }_2\right)& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(24)

$${}^{2}T_3=\left[\begin{array}{cccc}cos\left({\theta }_3\right)& -sin\left({\theta }_3\right)& 0& 0\\ 0& 0& 0& -l_1\\ -sin\left({\theta }_3\right)& -cos\left({\theta }_3\right)& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

(25)

$${}^{3}T_4=\left[\begin{array}{cccc}cos\left({\theta }_4\right)& -sin\left({\theta }_4\right)& 0& 0\\ 0& 0& -1& 0\\ sin\left({\theta }_4\right)& cos\left({\theta }_4\right)& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(26)

$${}^{4}T_5=\left[\begin{array}{cccc}cos\left({\theta }_5\right)& -sin\left({\theta }_5\right)& 0& 0\\ 0& 0& 1& -l_2\\ -sin\left({\theta }_5\right)& -cos\left({\theta }_5\right)& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(27)

$${}^{5}T_6=\left[\begin{array}{cccc}\mathit{sin}\left({\theta }_6\right)& \mathit{cos}\left({\theta }_6\right)& 0& 0\\ 0& 0& -1& 0\\ -\mathit{cos}\left({\theta }_6\right)& \mathit{sin}\left({\theta }_6\right)& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(28)

$${}^{6}T_7=\left[\begin{array}{cccc}cos\left({\theta }_7\right)& -sin\left({\theta }_7\right)& 0& a_1\\ 0& 0& 1& 0\\ -sin\left({\theta }_7\right)& -cos\left({\theta }_7\right)& 0& 0\\ 0& 0& 0& 1\end{array}\right]$$

(29)

The homogeneous transformation matrix determines the position and orientation of the end frame relative to the base frame. The homogeneous transformation matrix was constructed by multiplying the individual transformation matrices from Equation (23) to Equation (29). For the lower extremity, the homogeneous transformation matrix represents the position and orientation of the foot relative to the hip joint.

$${\displaystyle \genfrac{}{}{0pt}{}{0}{7}}T=\left[{\displaystyle \genfrac{}{}{0pt}{}{0}{1}}T{\displaystyle \genfrac{}{}{0pt}{}{1}{2}}T{\displaystyle \genfrac{}{}{0pt}{}{2}{3}}T{\displaystyle \genfrac{}{}{0pt}{}{3}{4}}T{\displaystyle \genfrac{}{}{0pt}{}{4}{5}}T{\displaystyle \genfrac{}{}{0pt}{}{5}{6}}T{\displaystyle \genfrac{}{}{0pt}{}{6}{7}}T\right]$$

(30)

Lagrange’s method is a widely used mathematical approach for deriving dynamic equations of motion. It involves differentiating energy terms with respect to the system variables and time. The dynamic model of the human lower extremity was developed using Lagrange’s method [38]. The kinetic energy of a link at any given moment can be calculated using Equation (31).

$$k_i=\left[{\displaystyle \frac{1}{2}}{m}_i{.v}_{ci}^T.v_{ci}+{{\displaystyle \frac{1}{2}}}^i{\omega }_i^T{.}^{ci}{I}_i.{\omega }_i\right]$$

(31)

In Equation (31), $m_i$ represents the mass of the link, while $v_{ci}$ and${\omega }_{ci}$ denote the linear and angular velocities of the link at its center of mass. $I_i$ is the moment of inertia of the link about its center of mass. The total kinetic energy of the model can be calculated using Equation (32).

$$k=\sum _{i=1}^n{k}_i$$

(32)

In Equation, (32), $n$ represents the total number of degrees of freedom, $n=7$ for the developed human lower extremity dynamic model.

The potential energy of any link can be calculated using Equation (33),

$$u_i=-m_i{.}^0{g}^T{.}^0{P}_{ci}+u_{ref}$$

(33)

$$u=\sum _{i=1}^n{u}_i$$

(34)

In Equation (33)

, $m_i$ represents the mass of the link, $g$ is the gravitational acceleration, and${}_{0}P_{ci}$ denotes the position of the link’s center of mass relative to the ground reference. Finally, by using the Lagrange energy method, the required joint torque can be determined as follows:

$${\tau }_i=\left[{\displaystyle \frac{d}{dt}}{\displaystyle \frac{\delta k}{\delta {\dot{\theta }}_i}}-{\displaystyle \frac{\delta k}{\delta {\theta }_i}}+{\displaystyle \frac{\delta u}{\delta {\theta }_i}}\right]$$

(35)

The dynamic equations of motion for the robot are expressed in Equation (36),

$${\tau }_{Joint}=\left[M\left(\theta \right)\ddot{\theta }+V\left(\theta ,\dot{\theta }\right)+G\left(\theta \right)\right]$$

(36)

In Equation (36), $M\left(\theta \right)$ is the mass matrix, a $\left(7x7\right)$ symmetric positive definite matrix, $V(\theta ,\dot{\theta })$ is $\left(7x1\right)$ matrix represents the Coriolis and the centripetal terms, and $G\left(\theta \right)$ is $\left(7x1\right)$ matrix representing the gravitational term. ${\tau }_{Joint}$, $\left(7x1\right)$ matrix, represents the joint torque requirements. The robot’s dynamic equation of motion can be rewrite as:

$$\ddot{\theta }=\left[M{\left(\theta \right)}^{-1}\left({\tau }_{Joint}-V\left(\theta ,\dot{\theta }\right)-G\left(\theta \right)\right)\right]$$

(37)

Since $M\left(\theta \right)$ is a positive definite matrix, ${M\left(\theta \right)}^{-1}$ always exists. Figure 4 shows a schematic diagram of the ideal/model robot dynamics without counting the joint frictions.

Friction between two mating parts with relative motion is unavoidable. After considering joint friction torques, the robot dynamical equations become,

$${\tau }_{Joint}=\left[M\left(\theta \right)\ddot{\theta }+V\left(\theta ,\dot{\theta }\right)+G\left(\theta \right)+{\tau }_{friction}\right]$$

(38)

where,

$${\tau }_{friction}=\left[F\left(\dot{\theta }\right)\right]$$

(39)

Equation (38) can also be written as,

$$\ddot{\theta }=\left[M{\left(\theta \right)}^{-1}\left({\tau }_{Joint}-V\left(\theta ,\dot{\theta }\right)-G\left(\theta \right)-F\left(\dot{\theta }\right)\right)\right]$$

(40)

Figure 5 presents the schematic diagram of the robot dynamics with friction effects.

Figure 5. Internal architecture of the physical Robot model.

Figure 5. Internal architecture of the physical Robot model.

4. Realistic Model Reference Computed Torque Control

4.3. Realistic Model Reference Computed Torque Controller

In order to make the Model Reference Computed Torque Controller more efficient and realistic multiple changes were made in the Realistic Model Reference Computed Torque Controller.

Equation (36) shows that the total torque required to operate the robot is used for accelerating the robot’s links and payloads (M matrix), compensating for gravitational effects (G matrix), and counteracting Coriolis and centrifugal effects (V matrix). Often joints friction are considered as the disturbances. It has been observed that the magnitude of Coriolis and centrifugal forces is significantly smaller compared to the torque needed for acceleration and gravity compensation.

Figure 10. Comparison between the total torque, torque required for link acceleration (M), to overcome the gravity (G), Coriolis and centrifugal forces (V).

Figure 10. Comparison between the total torque, torque required for link acceleration (M), to overcome the gravity (G), Coriolis and centrifugal forces (V).

Calculating the Coriolis and centrifugal terms at every sampling period for the robot’s specific position and velocity demands more computational power than executing the mass and gravity matrices (M and G matrices).

To optimize the realistic model reference computed torque controller, we excluded the Coriolis and centrifugal terms from the dynamic model, treating them as disturbances. Additionally, we divided the control algorithm into two loops: slower loop for joint torque estimation and faster loop contains the PID controller for minimizing the tracking error between the reference trajectory and the plant output. The slower loop run at (100Hz), while the faster loop run at 1kHz. Running 2 loops at two different rates reduces the computational load and energy consumption for real time computing. Predicting torque at slower speed cause some inaccuracies. The PID controller was used to remove the errors and improve the robustness.The developed realistic model reference computed torque controller showed excellent trajectory tracking performance, maintaining small tracking errors even in the presence of disturbances. By excluding the Coriolis and centrifugal force terms $V\left(\theta ,\dot{\theta }\right)$, it achieves significantly higher energy efficiency compared to traditional computed torque control and model reference computed torque control methods. Figure 11 shows the schematic diagram of the realistic model reference computed toque controller.

Stability Analysis of the Proposed Realistic Model Reference Computed Torque Controller:

Ensuring controller stability is a very fundamental requirement of a control system. The following section will prove the combined system stability by computing the stability of the individual loops.

Loop1:

The dynamic model of the robot can be represented by Equation (47),

$$\left[M\left(\theta \right)\ddot{\theta }+V\left(\theta ,\dot{\theta }\right)+G\left(\theta \right)\right]={\tau }_{Joint}$$

(47)

The control torque for the model is calculated by using Equation (48),

$${\tau }_M=M\left(\theta \right)[{\ddot{\theta }}_d+K_v({\dot{\theta }}_d-\dot{\theta })+K_P({\theta }_d-\theta \left)\right]$$

(48)

By comparing Equations (47) and (48),

$$(\ddot{{\theta }_d}-\ddot{\theta })+K_V(\dot{{\theta }_d}-\dot{\theta })+K_P({\theta }_d-\theta )=0$$

(49)

The robot tracking error is defined as $E=({\theta }_d-\theta )$. Using Equation (49), the error dynamics can be expressed as:

$$\ddot{E}+K_V\dot{E}+K_{P}E=0$$

(50)

By applying the Laplace transform on Equation (50), the closed-loop characteristic polynomial can be written as,

$${\mathsf{\Delta }}_{\mathrm{C}}\left(S\right)=|S^{2}I+K_{V}S+K_P|$$

(51)

$$\mathrm{where},K_P=diag\left\{K_{Pi}\right\}$$

(52)

$$K_V=diag\left\{K_{Vi}\right\}$$

(53)

According to the Routh-Hurwitz stability criteria in Table 4, Equation (51) is asymptotically stable as long as the gain matrices $(K_P,K_V)$ are positive definite.

Loop 2:

Calculates the deviation of the plant output from the model $(e,\dot{e})$ and forces the plant to compensate for the tracking errors in a controlled way. Based on the developed RMRCTC scheme, the plant input torque $\left({\tau }_P\right)$ can be calculated as,

$${\tau }_P={\tau }_M+K_I\int \left({\theta }_M-{\theta }_P\right)+K_P\left({\theta }_M-{\theta }_P\right)+K_V\left(\dot{{\theta }_M}-\dot{{\theta }_P}\right)+{\tau }_f$$

(54)

In developing this controller, it is assumed that the model and the plant share the same dynamical structure, differing only in their parameters.

The tracking error $(E,\dot{E})$ between the model and the plant is defined as,

$$E=({\theta }_M-{\theta }_P)$$

(55)

$$\dot{E}=({\dot{\theta }}_M-{\dot{\theta }}_P)$$

(56)

$${\tau }_P={\tau }_M+K_V\dot{E}+K_I\int E+K_{P}E+{\tau }_f$$

(57)

$${\tau }_P\left(S\right)={\tau }_M\left(S\right)+{\displaystyle \frac{K_{I}E\left(S\right)}{S}}+K_{P}E\left(S\right)+K_{V}SE\left(S\right)+{\tau }_f\left(S\right)$$

(58)

$${\tau }_P\left(S\right)-{\tau }_M\left(S\right)-{\tau }_f\left(S\right)={\displaystyle \frac{\left({S^{2}K}_V+S{K}_P+K_I\right)E\left(S\right)}{S}}$$

(59)

Using Equation (59), the error dynamics can be expressed as:

$${\displaystyle \frac{E\left(S\right)}{{\tau }_P\left(S\right)-{\tau }_M\left(S\right)-{\tau }_f\left(S\right)}}={\displaystyle \frac{S}{S^2{K}_V+S{K}_P+K_I}}$$

(60)

Equation (60) represents the transfer function of the tracking error

Table 5 shows that the control system will be asymptotically stable if the gain matrices $K_P,K_V,K_I$ are positive definite. Here, $K_P,K_I,K_V$ are the $\left(7x7\right)$diagonal matrices.

$$K_P=diag\left\{K_P\right\}$$

(61)

$$K_I=diag\left\{K_I\right\}$$

(62)

$$K_V=diag\left\{K_V\right\}$$

(63)

An analysis of the stability of Loop1 and Loop 2, quickly leads to the conclusion that the proposed control system is asymptotically stable as long as the gain matrices are positive definite.

5. Simulation Results and Discussion

6. Controller Performance Verification

Conclusions

References

1. K. Kyoungchul and J. Doyoung, “Design and control of an exoskeleton for the elderly and patients,” IEEE/ASME Transactions on Mechatronics, vol. 11, pp. 428-432, 2006.
2. K. Kong, H. Moon, B. Hwang, D. Jeon, and M. Tomizuka, “Impedance Compensation of SUBAR for Back-Drivable Force-Mode Actuation,” IEEE Transactions on Robotics, vol. 25, pp. 512-521, 2009. [CrossRef]
3. Colombo, Gery, Joerg, Matthias, Schreier, Reinhard, et al., “Treadmill training of paraplegic patients using a robotic orthosis.,” Journal of Rehabilitation Research & Development vol. 37, pp. 693-700, 2000.
4. M. Bernhardt, M. Frey, G. Colombo, and R. Riener, “Hybrid force-position control yields cooperative behaviour of the rehabilitation robot LOKOMAT,” in 9th International Conference on Rehabilitation Robotics, 2005. ICORR 2005., 2005, pp. 536-539.
5. J. F. Veneman, R. Kruidhof, E. E. G. Hekman, R. Ekkelenkamp, E. H. F. V. Asseldonk, and H. v. d. Kooij, “Design and Evaluation of the LOPES Exoskeleton Robot for Interactive Gait Rehabilitation,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 15, pp. 379-386, 2007. [CrossRef]
6. J. F. Veneman, R. Ekkelenkamp, R. Kruidhof, F. C. T. v. d. Helm, and H. v. d. Kooij, “Design of a series elastic- and Bowden cable-based actuation system for use as torque-actuator in exoskeleton-type training,” in 9th International Conference on Rehabilitation Robotics, 2005. ICORR 2005., 2005, pp. 496-499.
7. S. K. Banala, S. K. Agrawal, and J. P. Scholz, “Active Leg Exoskeleton (ALEX) for Gait Rehabilitation of Motor-Impaired Patients,” in 2007 IEEE 10th International Conference on Rehabilitation Robotics, 2007, pp. 401-407.
8. S. K. Agrawal, S. K. Banala, A. Fattah, V. Sangwan, V. Krishnamoorthy, J. P. Scholz, et al., “Assessment of Motion of a Swing Leg and Gait Rehabilitation With a Gravity Balancing Exoskeleton,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 15, pp. 410-420, 2007. [CrossRef]
9. S. H. Kim, S. K. Banala, E. A. Brackbill, S. K. Agrawal, V. Krishnamoorthy, and J. P. Scholz, “Robot-assisted modifications of gait in healthy individuals,” Experimental Brain Research, vol. 202, pp. 809-824, 2010/05/01 2010.
10. P. Stegall, K. N. Winfree, and S. K. Agrawal, “Degrees-of-freedom of a robotic exoskeleton and human adaptation to new gait templates,” in 2012 IEEE International Conference on Robotics and Automation, 2012, pp. 4986-4991.
11. H. Kawamoto and Y. Sankai, “Power assist method based on phase sequence driven by interaction between human and robot suit,” in RO-MAN 2004. 13th IEEE International Workshop on Robot and Human Interactive Communication (IEEE Catalog No.04TH8759), 2004, pp. 491-496.
12. G. M. Kenta Suzuki, H. Kawamoto, Y. Hasegawa, and Y. Sankai, “Intention-Based Walking Support for Paraplegia Patients with Robot Suit HAL.pdf,” 2005. [CrossRef]
13. H. Kawamoto, T. Hayashi, T. Sakurai, K. Eguchi, and Y. Sankai, “Development of single leg version of HAL for hemiplegia,” in 2009 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, 2009, pp. 5038-5043.
14. G. Zeilig, H. Weingarden, M. Zwecker, I. Dudkiewicz, A. Bloch, and A. Esquenazi, “Safety and tolerance of the ReWalk exoskeleton suit for ambulation by people with complete spinal cord injury: a pilot study,” J Spinal Cord Med, vol. 35, pp. 96-101, Mar 2012.
15. A. Esquenazi, M. Talaty, A. Packel, and M. Saulino, “The ReWalk Powered Exoskeleton to Restore Ambulatory Function to Individuals with Thoracic-Level Motor-Complete Spinal Cord Injury,” American Journal of Physical Medicine & Rehabilitation, vol. 91, 2012. [CrossRef]
16. D. Shi, W. Zhang, W. Zhang, and X. Ding, “A Review on Lower Limb Rehabilitation Exoskeleton Robots,” Chinese Journal of Mechanical Engineering, vol. 32, p. 74, 2019/08/30 2019.
17. E. Strickland. (2011). Good-bye, Wheelchair, Hello Exoskeleton. Available: http://spectrum.ieee.org/biomedical/bionics/goodbye-wheelchair-hello-exoskeleton.
18. R. J. Farris, H. A. Quintero, S. A. Murray, K. H. Ha, C. Hartigan, and M. Goldfarb, “A Preliminary Assessment of Legged Mobility Provided by a Lower Limb Exoskeleton for Persons With Paraplegia,” IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 22, pp. 482-490, 2014. [CrossRef]
19. D. Sanz-Merodio, M. Cestari, J. C. Arevalo, and E. Garcia, “Control Motion Approach of a Lower Limb Orthosis to Reduce Energy Consumption,” International Journal of Advanced Robotic Systems, vol. 9, p. 232, 2012/12/06 2012. [CrossRef]
20. P. D. Neuhaus, J. H. Noorden, T. J. Craig, T. Torres, J. Kirschbaum, and J. E. Pratt, “Design and evaluation of Mina: A robotic orthosis for paraplegics,” in 2011 IEEE International Conference on Rehabilitation Robotics, 2011, pp. 1-8.
21. L. Wang, S. Wang, E. H. F. v. Asseldonk, and H. v. d. Kooij, “Actively controlled lateral gait assistance in a lower limb exoskeleton,” in 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2013, pp. 965-970.
22. J.-H. Kim, J. W. Han, D. Y. Kim, and Y. S. Baek, “Design of a Walking Assistance Lower Limb Exoskeleton for Paraplegic Patients and Hardware Validation Using CoP,” International Journal of Advanced Robotic Systems, vol. 10, p. 113, 2013/02/01 2013. [CrossRef]
23. K. Hian Kai, J. H. Noorden, M. Missel, T. Craig, J. E. Pratt, and P. D. Neuhaus, “Development of the IHMC Mobility Assist Exoskeleton,” in 2009 IEEE International Conference on Robotics and Automation, 2009, pp. 2556-2562.
24. N. L. Tagliamonte, F. Sergi, G. Carpino, D. Accoto, and E. Guglielmelli, “Human-robot interaction tests on a novel robot for gait assistance,” in 2013 IEEE 13th International Conference on Rehabilitation Robotics (ICORR), 2013, pp. 1-6.
25. Y. Mori, J. Okada, and K. Takayama, “Development of a standing style transfer system “ABLE” for disabled lower limbs,” IEEE/ASME Transactions on Mechatronics, vol. 11, pp. 372-380, 2006. [CrossRef]
26. T. Yoshimitsu and K. Yamamoto, “Development of a power assist suit for nursing work,” in SICE 2004 Annual Conference, 2004, pp. 577-580 vol. 1.
27. K. Yamamoto, M. Ishii, H. Noborisaka, and K. Hyodo, “Stand alone wearable power assisting suit - sensing and control systems,” in RO-MAN 2004. 13th IEEE International Workshop on Robot and Human Interactive Communication (IEEE Catalog No.04TH8759), 2004, pp. 661-666.
28. A. B. Zoss, H. Kazerooni, and A. Chu, “Biomechanical design of the Berkeley lower extremity exoskeleton (BLEEX),” IEEE/ASME Transactions on Mechatronics, vol. 11, pp. 128-138, 2006. [CrossRef]
29. H. Kazerooni, R. Steger, and L. Huang, “Hybrid Control of the Berkeley Lower Extremity Exoskeleton (BLEEX),” The International Journal of Robotics Research, vol. 25, pp. 561-573, 2006.
30. B. Chen, C.-H. Zhong, X. Zhao, H. Ma, X. Guan, X. Li, et al., “A wearable exoskeleton suit for motion assistance to paralysed patients,” Journal of Orthopaedic Translation, vol. 11, pp. 7-18, 2017/10/01/ 2017. [CrossRef]
31. B. Chen, C. Zhong, X. Zhao, H. Ma, L. Qin, and W. Liao, “Reference Joint Trajectories Generation of CUHK-EXO Exoskeleton for System Balance in Walking Assistance,” IEEE Access, vol. 7, pp. 33809-33821, 2019. [CrossRef]
32. S. Hyon, J. Morimoto, T. Matsubara, T. Noda, and M. Kawato, “XoR: Hybrid drive exoskeleton robot that can balance,” in 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems, 2011, pp. 3975-3981.
33. D. Nguyen-Tuong and J. Peters, Learning Robot Dynamics for Computed Torque Control Using Local Gaussian Processes Regression, 2008.
34. T. Kurz. (2015). Normal Ranges of Joint Motion. Available: http://web.mit.edu/tkd/stretch/stretching_8.html.
35. A. A. o. O. Surgeons., Joint Motion: Methods of Measuring and Recording: Chicago : American Academy of Orthopaedic Surgeons, 1965.
36. F. P. Kendall, E. K. McCreary, P. G. Provance, M. M. Rodgers, and W. A. Romani, Muscles: Testing and Function, with Posture and Pain, 5th ed.: LWW, 2005.
37. G. Nikolova and Y. Toshev, “Comparison of two approaches for calculation of the geometric and inertial characteristics of the human body of the Bulgarian population,” Acta of Bioengineering and Biomechanics, vol. 10, pp. 3-8, 2008.
38. J. J. Craig, Introduction to robotics : mechanics and control: Pearson/Prentice Hall, Upper Saddle River, N.J., 2005.
39. S. Liu and G. S. Chen, “Dynamics and Control of Robotic Manipulators with Contact and Friction,” 2018.
40. E. Pennestrì, V. Rossi, P. Salvini, and P. P. Valentini, “Review and comparison of dry friction force models,” Nonlinear Dynamics, vol. 83, pp. 1785-1801, 2016/03/01 2016. [CrossRef]
41. J.-L. Ha, R.-F. Fung, C.-F. Han, and J.-R. Chang, “Effects of Frictional Models on the Dynamic Response of the Impact Drive Mechanism,” Journal of Vibration and Acoustics, vol. 128, pp. 88-96, 2005. [CrossRef]
42. S. K. Hasan and A. K. Dhingra, “Development of a model reference computed torque controller for a human lower extremity exoskeleton robot,” Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 235, pp. 1615-1637, 2021/10/01 2021. [CrossRef]