SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial Life
Vol. 9(2); September (2022), pp. 128–135
ISSN (Online):2352-6386; ISSN (Print):2405-9021
https://alife-robotics.org/jrnal.html

Research Article

A MBD-based Knee Link Model to Analyze Singular Postures toFind Necessary Restrictions Toward Personalized Knee Support
Exoskeleton
Shintaro Kasai1, Pancho Dachkinov1, Kohei Tanaka1, Hiroaki Wagatsuma1,2
Kyushu Institute of Technology, 4 Hibikino, Wakamatsu-ku, Kitakyushu, 808-0196, JapanRIKEN CBS, Japan

1
2

ARTIC LE INFO
Article History
Received 25 November 2021
Accepted 24 March 2022

Keywords
Knee support
Exoskeleton-type assistive device
Multi-body dynamics (MBD)
Joint dysfunction

ABSTRACT

Assistive devices have been studied in the senses of the universal design and a possible
simplification for mass production, and such a concept can be extended to the personalization byusing advanced 3D printing technologies, which allows to provide suitable exoskeleton-typedevices to fit for the subject’s body to achieve symptom relief. A systematic analytical mechanics
is highly important for the investigation of proposed integrative systems and multibody dynamics
and its computer simulation enhance the capability to find potential problems in design, forexample risky postures for the subject. We introduced multibody dynamics to analyze a knee
support exoskeleton by using a linkage system newly designed as a simplification of the device.
In the computer experiment, a singular posture was successfully found as the limitation of thedevice to move safely for subjects having joint dysfunctions.

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

or the prevention of serious motor dysfunctions. The
devices have developed in recent years by using motorizedmechanisms and spring-damper parts [2,3]. If joints in thehuman body were modeled as a simple rotational joint tobe able to replace by rotational actuators, pendulum
models can be considered to enhance walking functions innormal and fast gaits [2,4]. On the other hand, it is still
unclear to define best parameters to realize a dynamic
adaptation to motion kinetics in actual walking
environments. Then model-based kinematic and kinetic
analyses are highly important for the clarification of
specifications such as joint torque and knee stiffness [4,5].
The systematic analysis can extend future capabilities of
exoskeleton-type assistive devices and contribute theassurance of quality [6,7,8].
In the present study, we studied a systematic theoreticalanalysis based on the multibody dynamics (MBD)

1. Introduction
In the ambulatory function, knee joints bear the bodyweight and then it is vulnerable especially in elderly
persons. Problems in joints, or knee injuries, impact on
their abilities in daily life [1]. Therefore, a biomechanicalanalysis to compare normal and injured conditions in theknee joint function is necessary for finding potential
problems and possible supports by assistive devices. In
other words, the introduction of theoretical andcomputerized design methods for knee assistive devices
has a large potential for parameter optimizations of thedevice’s specification for sake of making a successful
rehabilitation program. Assistive devices to attach thehuman body known as exoskeletons have been provided in
the stage of the convalescence stage of the rehabilitation

Corresponding author’s E-mail: kasai.shintaro660@mail.kyutech.jp, dachkinov.pancho-nikolaev608@mail.kyutech.jp,
tanaka.kohei886@mail.kyutech.jp, waga@brain.kyutech.ac.jp https://www.kyutech.ac.jp/128[9,10,11,12,13] applied to the proposed knee mechanismin the aim of motion kinetics analysis toward the knee joint
support. The systematic analytical method based on MBD
clarifies joint torque and knee stiffness in the target motion,
which helps in the investigation of supportive mechanisms
for a load reduction of the joint. The model-based analysis
provides detail inspection for timing and position analyses
to be able to improve support devices.
2. Methodology
2.1. Knee link modelFig. 2. A prototype exoskeleton-type knee joint support
device [15].

Human knee joints are organized internally with two
ligaments to form a four-bar linkage mechanism as
discussed by Shenoy et al. [14] (Fig. 1). It is called a rollback mechanism because the center of the upper rotational
point (black circle in Fig. 1) shifts back during the bending
motion when the knee straight posture changes to knee
bent postures. The mechanism cannot be replaced by asimple rotary joint. Such a complex biomechanism is noteasily reproduced and supported in the form of theexoskeleton-type device, while welfare device expert
designers reproduce a suitable device based on theirknowledge and experiences. Fig. 2 is a proto-type of theknee support device to attempt to reproduce the roll-backmechanism by the combination of rotational joints andtranslational movements [15]. However, the effectivenessof the design need to be clarified not only in the posture
analysis but also the force estimation at each rotationalparts in the device. This is important for further theoreticaloptimization to pursue what is the best structure.

2.2. MBD for knee link model
In consideration of linkage motions with angler change inindividual joints, a constant angular velocity can be given
to the joint p3 (Fig. 1 (b)) to mimic the knee joint extension,
which was analyzed in the multibody dynamics (MBD)[9,10,11,12,13]. By using MBD, the numerical analysis
provides the posture and force analyses by describing the
motion equation in an algebraic way to represent thestructural system with various components. The
differential equations are described in the form of thematrix with given constraints, and the actual solution isderived from the numerical integration method in thecomputer experiment. In MBD, the differential algebraic
equation is reconstructed based on generalized coordinates,
which are provided from transformations of local
coordinates in individual components. As the forward
dynamics analysis, the differential algebraic equation asshown in Eq. (1) is initially formed to represent the target
system. The meanings of the characters in Eq. (1) are as
Table 1.
Individual positions of bodies of the system and velocities,
acceleration and other factors are obtained in the analysis.
(1)Fig. 1. A schematic illustration of the biomechanics of theknee joint [14]. Upper and lower bodies in the joint arebound with the posterior cruciate ligament (PCL,
represented by the gray bar) and the anterior cruciate
ligament (ACL, the black bar).

129

Table 1. The planning and control components.
MΦ𝑞𝑞

Mass matrix
Jacobian matrix differentiatedfrom constraint equation in
generalized coordinates𝑞𝑞̈λGeneralized acceleration matrix
Lagrange multiplier

𝑄𝑄 𝐴𝐴
γ

Generalized force
Acceleration equation

Fig. 3 shows the angle and the mass-center coordinates ofeach link and Table 2 shows the lengths of each link. The
following equations express the generalized coordinate
matrix and the generalized velocity matrix for each masscenter.

Fig. 3. The definition of generalized coordinates.

(2)(3)
The generalized coordinate matrix and the generalized
velocity matrix were derived as follows.
(4)(5)
Fig. 4. The definition of kinematic constraints of rotary joint.
Table 2: Parameters in the link model
Link nameSides

Length
[mm]

𝐿𝐿1

O-r1

82

r1-r2

47

𝐿𝐿3

r2-r4

37

r3-d

190

𝐿𝐿5

r3-r4

14

r1-r3

80

𝐿𝐿2
𝐿𝐿4
𝐿𝐿6

For describing the connections and movements of eachlink in the knee link model, the definitions of theconstraints of each link and joint were needed. The
kinematic constraints of rotary joints in the general multibody systems were defined as shown in Fig. 4 andrepresented as follows.

(6)

In the expanded form, the kinematic constraints in thesystem finally represented as follows.

130

(13)

(7)

(14)
The link 1 is fixed vertically, and then 𝑥𝑥1 , 𝑦𝑦1 and 𝜃𝜃1 isprovided according to the absolute constraints as follows.
In absolute constraints, the angle constraints of 𝜙𝜙12 (L1 L2) and 𝜙𝜙54 (L5 – L4) are also defined.
(8)

(15)

When a constant angular velocity is given in the joint r3,
the driving constraints is represented as follows.
In MBD, the Jacobian matrix is used for the differential
algebraic equation of the motion equation shown in Eq. (1).
The generalized acceleration matrix can numerically becalculated for the detail analysis.

(9)
The kinematic, absolute and driving constraints are
combined in the matrix as follows.

2.3. Spring-Damper Components
As the benefit of the MBD-based analysis, any link can be
replaced to the spring-damper component and thecomponent bridges between two points as shown in Fig. 5.
In this analysis, the spring-damper component was used
for the force control system provided by themusculoskeletal system of the human knee joint.

(10)
Finally, the Jacobian matrix is calculated as follows.
(11)

(12)

131

experiment. Fig. 6 showed the temporal sequence of the
behavior of the system in a cycle of 𝑇𝑇 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 = 50.

Fig. 5. The definition of the spring-damper (motor)
component attached to two points. c, k and fa respectively
denote damping constant, spring constant, and force
provided by the implemented actuator.

3. Results
Kinematic and kinetic (dynamics) analysis can be done in
the framework of MBD, which are systematically andnumerically computed with the differential algebraic
equation as described in the previous section. In the knee
rigid link model, positions and angles of individual links
were derived in the form of generalized coordinates in the
numerical analyses by using the Newton-Raphson method
and the numerical integration of the Runge-Kutta Gill’s
method [16] in MATLAB implementations.
The constraint equations such as Eq. (7) can flexibly be
modified depending on parts in the system, forces andtorques to realize a target motion can be estimatedsystematically and environmental conditions can be
included in the analysis. Therefore, the MBD-based
analysis has advantages in 1) the extendibility of parts inthe system, 2) driving force estimation by inverse
dynamics, 3) variability of external forces.

Fig. 6. The MBD-based kinematic analysis of the system.

When a constant angular velocity was given to 𝜃𝜃4 towardbending direction with the knee link extended, the knee
link model reproduced the bending motions of knee joint.
By the systematic and theoretical analysis, angles, angularvelocities and angular accelerations of individual links
were clearly analyzed. Giving constant angular velocity to
r3, the angle 𝜃𝜃4 was decreasing from 𝜃𝜃4 = 90°. However,when the link angle reaches a certain angle, the rotation of
the joint is locked, and then the knee movement isprevented. Interestingly, the result indicated that this knee
link model has a limitation of the range of the joint
extension in the form of a singular posture (Fig. 7). It
happened when link L3 and L5 were aligned in a samestraight line. It realized that the rotation of the link islocked at this particular angle and suggest that it isnecessary function to prevent the breakage of the knee
joint. The original device was designed to embed its
limitation with respect to the freedom of the knee joint. In
this sense, the result proved the importance of MBD-based
singular posture analysis.
The result indicates the limitation of the range of thestretching knee motion, which suggests a necessary knee
joint support to prevent the knee joint dysfunction byattaching the assistive device.

3.1. The movement of the knee link model
In the first place, the movement analysis was demonstrated
to test the range of the motion. In the kinematic analysis,
the relative angular constraints were given as ∅12 = 𝜋𝜋⁄4
and ∅54 = 𝜋𝜋⁄4 and the driving constraint was given as therepetitive vertical movement of the point d, which was
defined as follows.
𝑑𝑑𝑦𝑦 = 0.5(𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 )∙ sin(2𝜋𝜋⁄𝑇𝑇 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 ∙ 𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 + 1) − 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚(16)

where 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 3, 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 2, 𝑇𝑇 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 = 2000 ∙ Δ𝑡𝑡 and Δ𝑡𝑡 =0.025 and 𝑡𝑡 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 denotes the time step of the computer132

Fig. 8. The MBD-based kinetic (dynamics) analysis of thesystem without necessary constraints to keep the posture.

Then, a spring-damper mechanism was added to connect
between points d and r1 as shown in Fig. 9. Parameters
were the same as before except 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 = 0.8 , 𝑝𝑝𝑚𝑚𝑚𝑚𝑚𝑚 =0.4388 for the minimization of the effect from the driving
force to observe the effect by the spring-damper system.
The external force is generated by the spring damper
system as follows.
1 0
𝑑𝑑14
0 1
𝑄𝑄1𝑒𝑒 = 𝑓𝑓𝑠𝑠 �𝑇𝑇 �|𝑑𝑑14 |
𝑙𝑙1 ⁄2
�𝐴𝐴1 𝑉𝑉 �
��
0
(17)
1 0
𝑑𝑑14
0 1
𝑄𝑄4𝑒𝑒 = −𝑓𝑓𝑠𝑠 �𝑇𝑇 �|𝑑𝑑14 |
−𝑙𝑙4 ⁄2
���𝐴𝐴4 𝑉𝑉 �
0

Fig. 7. The singular point posture of the knee model (left). A
schematic illustration of the attachment position of thehuman body (right).

3.2. The joint load dynamics with spring-damper
components
In the second place, the force analysis was demonstrated
in the MBD-based kinetic analysis. As illustrated in Fig. 4(right), the center of mass of the human body is kept in thestraight line to keep the standing posture. It reflects to thekinematic constraint to be the same x position of points a(O) and d. Due to the difference from the kinematic
analysis to examine possible postures of the linkage, thekinetic analysis based on the set of equations of motion
involves mass of all links. As demonstrated in the MBDbased kinetic analysis in Fig. 8, during the driving
constraint as well as the previous section by Eq. (16)(different parameters were 𝑇𝑇 𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢 = 1000 ∙ Δ𝑡𝑡 and Δ𝑡𝑡 =0.0025 for the observation in the early stage to falling
down precisely), the behavior was different from thekinetic analysis and it showed a falling down motion due
to its weight. As the dynamics to be physical action of thesystem without any external force to keep posture, which
was clearly demonstrated an actual behavior in the world.

where

𝑑𝑑𝑥𝑥
�,
𝑑𝑑𝑦𝑦
(18)
0 −1
𝑉𝑉 = �
�.
1 0Therefore, the final form of the external force of Eq. (1) isgiven as the summation as follows.
𝑑𝑑14 = �0
𝑄𝑄1𝐴𝐴 = �−𝑚𝑚1 𝑔𝑔� + 𝑄𝑄1𝑒𝑒00
𝑄𝑄4𝐴𝐴 = �−𝑚𝑚4 𝑔𝑔� + 𝑄𝑄4𝑒𝑒
0(19)

In addition, the parameter of the spring-damper system(Fig. 5) is given as follows.
𝑓𝑓𝑠𝑠 = −𝑘𝑘(𝑙𝑙 − 𝑙𝑙0 ) − 𝑐𝑐𝑙𝑙 ̇ + 𝑓𝑓𝑎𝑎
where 𝑘𝑘 = 1, 𝑙𝑙0 = 1, 𝑐𝑐 = 0.5 and 𝑓𝑓𝑎𝑎 = −10.
133

(20)

3.3. The results of the kinetic analysis
After the implementation of the spring-damper system, thetemporal evolution in the MBD-based kinetic analysis wasdemonstrated. Fig. 9 showed the initial stage as ashrinkage behavior and it was recovered according to thespring-damper system (Fig. 10).
Fig. 10. The recovery stage to push the system up according
to the spring-damper-motor system, which corresponds to
the knee extension.

4. Conclusion
By using the theoretical background of the MBD analysis
and implementation techniques, the knee joint linkage
model was examined in the computer experiment. It
provided not only the kinematic analysis to find thesingular posture, but also force analysis in the kinetic
analysis based on the differential algebraic equations in
multibody system to represent equations of motion andinternal and external constraints. The analysis with thespring-damper system is beneficial for the estimation of
the parameter values to realize the target motion. In thepresent study, we assumed the extension and flexion of theknee joint in the form of the spring-damper system and thefitness of the assistive device will be analyzed in thefurther extended analysis in the MBD-based analysis. It
indicates that the MBD-based analysis is beneficial for thereverse engineering in the process of the support device
design to estimate the ideal load reduction at specific knee
bending and extension movements. In the further analysis,
detail analyses are necessary such as a comparative
analysis of models that combines the knee model andvarious assistive device mechanisms to develop the knee
assistive devices.

Fig. 9. The initial stage to go down the system due to its
weight, which corresponds to the knee flexion.

This result indicates that the MBD-based analysis effective
to specify which parts are necessary and what parameters
is crucial to achieve the target behavior. In addition, if theforce dynamics from the human body and environmental
factors can be considered in the combination of force
controls, the parts need to be repriced depending to Fig. 7
shows the torque generated in each joint in the bending
motion of the link obtained by inverse dynamic analysis.
From Fig.7, it is possible to understand the magnitude andtiming of the torque generated in each joint. Thus, theproposed linkage model and MBD-based analysis
contributed to find a risk in persons with knee joint
dysfunction and this timing analysis is important toefficiently
supportjoint
movement.

Acknowledgements
This work was supported in part by JSPS KAKENHI(16H01616,
17H06383),
ProjectonRegional
Revitalization Through Advanced Robotics (Kyushu
Institute of Technology/Kitakyushu city, Japan) andKitakyushu Foundation for the Advancement of Industry,
Science and Technology (FAIS). The authors gratefully
acknowledge ARIZONO orthopedic supplies Co., Ltd. forassistance with the formulation of exoskeleton-typeassistive devices in their products.

134References
1. L. Zhang, G. Liu, B. Han, Z. Wang, Y. Yan, J. Ma and P.
Wei, “Knee Joint Biomechanics in Physiological Conditionsand How Pathologies Can Affect It: A Systematic Review,”
Applied Bionics and Biomechanics, 2020, Article Number
7451683.
2. L. Zhang, Y. Liu, R. Wang, C. Smith and E. GutierrezFarewik, “Modeling and Simulation of a Human Knee
Exoskeleton’s Assistive Strategies and Interaction,”
Frontiers in Neurorobotics, 2021, 15 620928.
3. W. Wu, J. Fong, V. Crocher, P. Lee, D. Oetomo, Y. Tan and
D. Ackland, “Modulation of shoulder muscle and joint
function using a powered upper-limb exoskeleton,” Journalof Biomechanics, 2018, Apr 27;72:7-16.
4. J. Vantilt, K. Tanghe, M. Afschrift, A. Bruijnes, K. Junius,
J. Geeroms, E. Artbelien, F. Groote, D. Lefeber, I. Jonkers
and J. Schutter, “Model-based control for exoskeletons withseries elastic actuators evaluated on sit-to-stand movements,”
Journal of Neuroengineering and Rehabilitation, 2019, Jun
3;16(1):65.
5. S. Schrade, Y. Nager, A. Wu, R. Gassert and A. Ijspeert,“Bio-inspired control of joint torque and knee stiffness in arobotic lower limb exoskeleton using a central patterngenerator,” IEEE, 2017, Jul;2017:1387-1394.
6. R. McGrath, M. Ziegler, M. Pires-Faernandes, B. Knarr, J.
Higginson and F. Sergi, “The effect of stride length on lower
extremity joint kinematics at various gait speeds,” PLoS
One, 2019, Feb 22;14(2):e0200862.
7. K. Gui, H. Liu and D. Zhang, “A generalizes framework toachieve coordinated admittance control for multi-joint
lower limb robotic exoskeleton,” IEEE, 2017, Jul;2017:228233.
8. K. Mankala, S. Banala and S. Agrawal, “Novel swing-assist
um-motorized exoskeletons for gait training,” Journal of
Neuroengineering and Rehabilitation, 2009, Jul 3;6:24.
9. J. A. C. Ambrósio, Impact of Rigid and Flexible Multibody
Systems: Deformation Description and Contact Models, in
Virtual Nonlinear Multibody Systems. NATO ASI Series(Series II: Mathematics, Physics and Chemistry), eds. W.
Schiehlen and M. Valášek, Springer, Dordrecht, vol 103,
2003, pp. 57-81.
10. P. E. Nikravesh, Planar Multibody Dynamics: Formulation,
Programming with MATLAB, and Applications, 2nd edn.,
CRC Press, Boca Raton, 2018.
11. K. Komoda and H. Wagatsuma, Energy-efficacy
comparisons and multibody dynamics analyses of legged
robots with different closed-loop mechanisms, Multibody
System Dynamics 40, 2017, pp. 123–153.
12. D. Batbaatar and H. Wagatsuma, A Proposal of theKinematic Model of the Horse Leg Musculoskeletal System
by Using Closed Linkages, Proceedings of the 2019 IEEE
International Conference on Robotics and Biomimetics(ROBIO), Dali, China, 2019, pp. 869–874.
13. J. Baumgarte, Stabilization of constraints and integrals of
motion in dynamical systems, Computer Methods in
Applied Mechanics and Engineering 1(1), 1972, pp. 1–16.

135

14. Arizono orthopedic supplies Co., Ltd.: Products,
http://arizono.co.jp/products/products_links/products_2/
15. R. Shenoy, P.S. Pastides, D. Nathwani, (iii) Biomechanicsof the knee and TKR, Orthopaedics and Trauma, 27(6),2013, pp. 364-371.
16. Wolfram Research, Inc., Runge-Kutta Gill’s method,
https://mathworld.wolfram.com/GillsMethod.html

Authors Introduction
Mr. Shintaro Kasai

He received his Bachelor’s degree inEngineering in 2021 from the Facultyof Engineering, Kyushu Institute oftechnology in Japan. He is currently amaster student in Kyushu Institute ofTechnology, Japan

Mr. Pancho Dachkinov
He received his Master’s degree inEngineering in 2018 from theTechnical University of Sofia,
Bulgaria. He worked in the Instituteof Robotics – Bulgarian Academy of
Sciences 2016-2019. He is currently adoctoral student in Kyushu Instituteof Technology, Japan
Mr. Kohei Tanaka

He is currently a fourth grade
undergraduate student in DepartmentofMechanicalandControlEngineering, Faculty of Engineering,
Kyushu Institute of Technology,
Japan. He is interested in 3D printing
and compliant mechanisms.

Dr. Hiroaki Wagatsuma
He received his M.S., and Ph.D.
degrees
from
Tokyo
Denki
University, Japan, in 1997 and 2005,
respectively. In 2009, he joined
Kyushu Institute of Technology,
where he is currently an Associate
Professor of the Department of
Human Intelli- gence Systems. His
research interests include non- linear
dynamics and robotics. He is a member of IEEE.