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Conference Paper

Kinematic and Dynamic Analysis of Lower Limb Movement: 
Towards the Design of a Wearable Rehabilitation Assistant 
Device 

Filippos Margaritis1,*, Konstantinos Mitsopoulos1, Kostas Nizamis2, Alkinoos Athanasiou1 and Panagiotis D. Bamidis1 

1 Medical Physics and Digital Innovation Laboratory, School of Medicine, Aristotle University of Thessaloniki, Greece. 
2 Systems Engineering and Multidisciplinary Design, University of Twente, Netherlands.

* Corresponding Author Email: filimarg@ece.auth.gr 

ABSTRACT

This study outlines a comprehensive approach to the kinematic and dynamic analysis of lower limb movement, with the express 
purpose of designing an efficient wearable rehabilitation assistant device for the lower body. The approach begins by conducting 
a kinematic analysis of the lower limbs, presenting the degrees of freedom and each joint’s range of motion. A kinematic model 
is designed by deciding on a kinematic chain configuration and calculating the Denavit Hartenberg (DH) parameters. Next, dif-
ferential kinematic analysis is employed to calculate the velocity of the limbs, generated by the corresponding muscle groups 
during different types of movements. This can provide significant insights into the design of a device that can accurately track 
and assist these movements. Furthermore, a dynamic analysis is performed to calculate joint moments and forces. This analysis 
provides insights into the forces that the joints experience during movement. When combined with electromyography (EMG) 
data, it allows for a more holistic description of muscle activity and a more accurate estimation of individual muscle forces and 
joint loads. The research also lays out a plan for the wearable device’s implementation. Based on OpenSenseRT1 an open-source 
software and hardware project, that utilized the OpenSim2 API, real-time inverse kinematics of a movement can be calculated 
using data from inertial measurement units (IMUs). This data is then used to compute the error in a person’s movement during 
lower limb rehabilitation exercises. This error, along with the error derived from real-time dynamic analysis and EMG data, can 
be integrated to improve the control accuracy of the wearable device. 

Keywords—Lower limb kinematic analysis, Lower limb dynamic analysis, OpenSim, OpenSenseRT system, IMU inverse 
kinematics, Real-time inverse kinematics, Real-time motion analysis, Wearable rehabilitation assistant device.

Copyright © 2024. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY): Creative Commons - 
Attribution 4.0 International - CC BY 4.0. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright 
owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduc-
tion is permitted which does not comply with these terms.

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63 J Global Clinical Engineering Vol.6 Special Issue 6: 2024

INTRODUCTION

The human body is a complex system, comprising of 
various interconnected parts that function in harmony 
to enable mobility. The lower limbs, particularly, play 
a crucial role in locomotion and maintaining balance. 
Understanding the movement of these limbs, especially 
in scenarios such as injury or disease, is vital to develop-
ing effective rehabilitation strategies. This study delves 
into this subject, presenting a comprehensive approach 
to analyzing the kinematics and dynamics of lower limb 
movement and designing an implementation plan for the 
wearable device. 

KINEMATIC ANALYSIS  

To calculate the kinematic model of the lower body, first, 
the Degrees of Freedom (DoF) and the Range of Motion 
(RoM) of each joint were found. The selection of DoF for 
each joint was influenced by the project’s current focus on 
individuals with tetraplegia or paraplegia, thus excluding 
the need to maintain a standing balance. The lower limb 
can be modeled as a sequence of rigid links connected by 
one universal rotary joint representing the hip and two 
revolute joints representing the knee, and ankle joints. 
The DoF of a joint defines the number of independent 
movements it can make. In our model the hip has three 
DoF allowing flexion—extension, adduction—abduction, 
and the internal—external rotation of the joint, the knee 
has one DoF allowing flexion/extension motions, and the 
ankle also has one DoF allowing dorsiflexion and plantar 
flexion. For the sake of simplicity, without loss of general-
ity, the hip joint is equivalently modeled as three closely 
placed revolute joints instead of a spherical joint.3

AThe Range of Motion (RoM) of a joint, on the other 
hand, refers to the total amount of movement that can 
occur at a joint in each of its possible planes of move-
ment. From Range of Joint Motion Evaluation Chart4 
and Kinesiology: Scientific Basis of Human Motion (B&B 
PHYSICAL EDUCATION)5, the RoM of the lower limb joints 
is shown in Table 1.

The kinematic analysis is performed using the Rigid 
Body Segment Model Approach, assuming that the bones 
are completely rigid while they may have some flexibility. 
Each body segment is linked to the next by a joint, allowing 

specific degrees of freedom. This forms a chain of rigid 
bodies, also known as a kinematic chain.  

The kinematic chain consists of local reference frames 
for each joint’s motion, which helps us identify the posi-
tion and orientation of each body segment. In Figure 1, 
these local frames are then expressed relative to a fixed 
global reference frame, Frame {0}, located at the pelvis’s 
center between the hip joints. Our model incorporates 
seven frames, labeled {0} to {6}. Frame {0} serves as a 
stable global reference positioned at the pelvis center. 
Frames {1}, {2}, and {3} are associated with the hip joint’s 
three movements. Frame {4} corresponds to the knee 
joint’s flexion and extension, while Frame {5} is linked 
to dorsiflexion and plantar flexion. Frame {6}, finally, is 
the end-effector frame, marking the kinematic chain’s 
terminal point.

TABLE 1. Range of Motion (RoM) of lower limb joints.

Joint Movement Degree

Hip Flexion/Extension 100°/30°

Hip Abduction/Addaction 40°/20°

Hip Internal/External Rotation 40°/50°

Knee Flexion/Extension 150°/0°

Ankle Dorsiflexion/Plantar Flexion 20°/40°

FIGURE 1. Representation of the lower limb’s kinematic chain: 
Featuring the frames of joint movements, θ angle variables, and 
the coordinate systems for each frame, in accordance with the 
Denavit-Hartenberg convention.

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The design of the kinematic chain necessitates the 
definition of the rotation axes for each frame and the 
direction of motion for each segment, while also taking 
into consideration any anatomical constraints or joint 
limitations that affect the range of motion at each joint. 
This procedure is guided by the principles of the Denavit 
Hartenberg (DH) convention, which provides a system-
atic method for representing the kinematic equations of 
a manipulator. This convention is particularly useful in 
the context of serial manipulators, where a matrix is used 
to represent the pose (position and orientation) of one 
body relative to another. Applying the DH parameters to 
Figure 1, we can produce the Table 2:

Variable angles φ change based on the position of the 
lower limbs and are constrained by the range of motion at 
each joint. In particular, the angle φ1 is within the interval 
[−50°, 40°], φ2 in [−20°, 40°], φ3 in [−30°, 100°], φ4 in [0°, 
150°] and φ5 in [−40°, 20°].  

The DH parameters are used to find the homoge-
neous transformation matrix and solve the forward and 
inverse kinematics problems. Transformation matrices 
and coordinate systems are utilized to find any position 
and orientation for any frame relative to the base frame. 
In particular, the homogeneous transformation matrices 
allow us to combine 3 × 3 rotation matrices and 3 × 1 
displacement vectors into a single 4 × 4 matrix, adding an 
additional row: [0 0 0 1]. The general form of the Transfor-
mation Matrix (T), according to Introduction to robotics: 
mechanics and control6, that defines frame {i} relative 
to the frame {i−1}, in accord with the DH convention, is:

By chaining together the transformation matrices of 
each joint starting from the base of the robot, we can ob-
tain the overall transformation matrix of the lower limb. 

This matrix can then be used to calculate the position 
and orientation of the end-effector for a given set of joint 
angles (forward kinematics) or to determine the joint 
angles required to achieve a desired end-effector position 
and orientation (inverse kinematics). 
From the transformation matrix calculated previously, 
we obtain:  

where R is the rotation matrix and P is the position vec-
tor of frame 5 with respect to the reference base frame 
0.7 The position vector P provides the position of the 
desired frame, and the rotation matrix R provides the 
orientation, those two are the solutions to the forward 
kinematics problem. 

On the other hand, solving the inverse kinematics prob-
lem involves determining the joint angles. A frequently 
used method for this involves multiplying each side of 
the transformation equation (2) by the corresponding 
inverse transformation matrix, depending on the frame’s 
angle we are aiming to solve for.8 However, the solution 
to the inverse kinematics problem is not pertinent to this 
study, and we will not delve into it further.

DIFFERENTIAL KINEMATIC ANALYSIS 
Differential kinematics focuses on the relationship 

between the joint velocities and the corresponding end-
effector’s linear and angular velocity. It provides a way to 

TABLE 2. Denavit Hartenberg (DH) parameters of the lower limb model.

Joint i αi-1 ai-1 di θi

Pelvis 0 - - d0 0° (x1‖x2)

Hip 1 α0 0 0 φ1 + 90°

Hip 2 −90° 0 0 φ2 + 90°

Hip 3 +90° 0 0 φ3 + 90°

Knee 4 0° (z3‖z4) l1 0 φ4

Ankle 5 0° (z4‖z5) l2 0 φ5

End-effector 6 0° (z5‖z6) l3 - -

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65 J Global Clinical Engineering Vol.6 Special Issue 6: 2024

analyze how changes in joint velocities affect the motion 
of the end-effector. To facilitate this analysis, we introduce 
a matrix quantity known as the Jacobian, which maps 
velocities in joint space to velocities in Cartesian space. 
There are two types of Jacobian matrices: Geometric and 
Analytical. The Geometric Jacobian is based on the pose 
matrix of the lower body end-effector. On the other hand, 
the Analytical Jacobian is based on a minimal parametrized 
form for representing the position and the orientation of 
the end-effector frame. In our case, Geometric Jacobian 
is more suitable since the pose matrix is available, rather 
than the minimal representation form that would be 
required for an analytical approach. As such, when we 
refer to the Jacobian, we will be referring specifically to 
the Geometric Jacobian.7,9

The rotation matrix, denoted as R, and the position vec-
tor, P, are dependent solely on the variables of the joints. 
In our specific scenario, where the joints are revolute, 
these variables correspond to the angles θ of the model 
and are represented as q, where qi = θi. The relationship 
between these joint variables, the Jacobian matrix, and 
the velocity of the end-effector, can be expressed as fol-
lows7 (on page 107):

where q and   are given by:

The matrix   is a 6 × 5 matrix where 5 is the number 
of links.

The angular velocity of the end-effector can be ex-
pressed relative to the rotation matrix R as follows: 

Through the computations described7 (on page 108 
and 111) it is retrieved that the angular velocities’ Jaco-
bian Jω is being expressed, for every i−th revolute joint, 
as follows:

The total lower half of the Jacobian is thus given as:
On the other hand, the linear velocity of the end-effector 
is just the derivative of the position vector  and by the 
chain rule for differentiation:

Again, following the computation described7 (on page 
110) is retrieved that the linear velocities’ Jacobian Ju is 

being expressed, for every i−th revolute joint, as it follows:

The total upper half of the Jacobian is thus given as:
Combining the upper and lower halves of the Jacobian, 

we can deduce that the Jacobian of the lower limb model 
is of the form:

where the i-th column is given by:

The above procedure works not only for computing 
the velocity of the end-effector but also for computing 
the velocity of any frame on the model.

DYNAMIC ANALYSIS 
While the kinematic equations outline the motion of 

the robot without considering the forces and moments 
causing the motion, the dynamic equations explicitly 
describe the relationship between force and motion. The 
dynamic equations of motion can be calculated using 
Newtonian, Lagrangian, or Hamiltonian mechanics. In 
our case, the Lagrangian approach was selected mainly 
because is based on the system’s kinetic and dynamic 
energy, rather than forces. This can simplify the analysis 
of complex systems and avoid the need for complex force 
equations. Assuring that the constraint forces satisfy 

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J Global Clinical Engineering Vol.6 Special Issue 6: 2024 66

the principle of virtual work we can introduce the Euler 
Lagrange equations of motion:
where L = K − P is the Lagrangian function, K is the kinetic 
energy, P is the dynamic energy, and forces τ represent 
the generalized forces’ function.
As shown in Robot modeling and control7 (on page 205), 
the kinetic energy of the manipulator can be computed 

using the calculated Jacobian matrices. The form of it 

equals:
Similarly, can be expressed as:
where D(q) is a symmetric positive definite matrix that 
is called an inertia matrix.

Assuming that the mass of every link is concentrated 
at its center, the potential energy of the i−th link of the 
lower body can be computed as follows:
where the vector g represents the direction of gravity 

in the inertial frame and the vector rci denotes the coor-
dinates of the center of mass of the i-th link. The total 
potential energy of our model is given by the sum:
Having the kinetic energy in the quadratic form of the 

vector q̇ and assuming that the potential energy of every 
link of the model is independent of q̇ the Euler-Lagrange 
equations of motion can be specialized as:
where k = 1, ... ,5.

IMPLEMENTATION PLAN 
The goal of the wearable device is to aid in rehabilita-

tion exercises by utilizing electrostimulations. To control 
the electrostimulator (EMS), it is necessary to compute 
the error of the person's motion. This is achieved through 
the analysis of joints’ angle errors provided by kinematic 
analysis and the forces and torque errors computed by 
dynamic analysis. Additionally to these errors, EMG data 
can be utilized by analyzing them using the CEINMS 
software.10 The real-time kinematic analysis will be 
implemented based on OpenSenseRT (as shown in Fig-
ure 2), an open-source software and hardware project 
that utilizes the IMU inverse kinematics algorithm from 
OpenSim.

ACKNOWLEDGEMENTS
This research project was supported by the Hellenic 

Foundation for Research and Innovation (H.F.R.I.) https://
www.elidek.gr under the “2nd Call for H.F.R.I. Research 
Projects to support Faculty Members & Researchers” 
(Project Number: 4391).

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FIGURE 2. Block diagram of the project’s implementation.

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