In recent years, a large variety of mechanisms for the rehabilitation of the upper limbs have been created by researchers. The scope of this work covers kinematic and kinetostatic analysis, inertial optimization and the use of passive control elements. The objective of this work is to reduce the cost of the mechanism by minimizing the required input torque for generating a movement consistent with the Minimal Jerk Model.

The position of the center of gravity of each link and the addition of linear translational springs, as passive control elements, are used in optimizing the design of the mechanism. First, I would like to express my deepest gratitude to my supervisor Konstantinos Kostas for his help and mentorship. Secondly, I would like to express my appreciation to my co-supervisor Evagoras Xydas for his feedback and help in the implementation of this thesis.

This work was completed thanks to their advice and dedicated hours of discussion. Finally, I would like to thank my parents and friends, who have supported me all along.

## Introduction

### Aim and objectives

One of the main objectives of this work is to design a mechanism with low engine power requirements, which will obviously reduce the overall cost. We specifically focus on reducing the required input torque on the drive link and this can be achieved by carefully selecting the drive link, optimizing the position of the center of gravity of each link and using linear translation springs to support the motor itself and/or to replace. . Obviously, at the same time we must be able to maintain the correct trajectory and/or velocity and acceleration profiles and therefore a full kinematic and kinetostatic analysis is required.

Methodologies and techniques

Thesis structure

## Literature Review

### Four bar linkage mechanism

A four-bar linkage (see Figure 2.1) is a flexible mechanism that is applied to transfer motion or provide mechanical advantage. The four-bar linkage consists of four links: the input link (crank), the output link (rocker), the ground link, and the connecting link. A point in the connecting link is called a tie point, while in robotics this is considered an end effect.

When the input link rotates, the coupling point follows a trajectory called the coupling curve. Four rod mechanisms are classified into two main groups based on the rotational capacity of their links. In 1883, Grashof presented a simple rule that determines the rotatability of the links: consider a four-bar link where 𝑙1 is the shortest link, 𝑙4 is the longest link, and the lengths of the remaining two links are 𝑙2 and 𝑙3.

In addition to this categorization, there are many types of four bar mechanisms such as Hoeken's, Watt's, Roberts's, Chebyshev's and Peaucellier's. Hoeken's four-bar mechanism (see Figure 2.2) is used because of its ability to generate both straight line and curved motion.

### Minimum Jerk Model

To find the position expressions for curved motion, it was assumed that in order to transfer the hand from start position to end position within defined time, it should pass through third specified positions 𝑥1 and 𝑦1 at unspecified time 𝑡1. Fitt's law states that the time required to move the hand to the specified target is a function of Experiment calculating the total movement time done in [17] and it is known that the total time is equal to 0.5 seconds.

Moreover, coupling point conforming to MJM must maintain a straight-line motion with allowable fluctuations. These differences can be measured by three methods: constant stimuli method, limit method and adjustment method. The JND is defined as the difference between the two comparison stimuli that was evaluated larger in 75% (or 25%) cases.

The limit method takes the same approach, but the comparison stimulus starts at much lower values than the standard and gradually increases. In the adaptation method, the comparison stimulus can be varied until it is equal to the standard.

## Mechanism’s design & Analysis

*Kinematic analysis**Kinetostatic analysis**Inertial optimization**Spring forces analysis*

The Hoeken four bar coupling can give a suitable straightness of the coupler point and is snap-on, meaning the drive motor can be used. Kinematic analysis of the mechanism performed in this section to determine the angles, positions, velocities and accelerations of the links. First, the mechanism model with known dimensions was assembled in the GIM software (see Figure 3.3) to obtain the initial, intermediate and final positions of the coupling point.

Coupling curve must match MJM, so specific angles 𝜃2, 𝜃3 and 𝜃4 are calculated for the known coupling point position. After that, velocity analysis must be done to define angular velocities of each link. 𝑉𝑃𝑥 is the linear velocity of coupling point, which is derived by differentiating equation 2.2 with respect to time and corresponds to MJM.

To calculate angular accelerations of each link the same method as in velocity analysis is used, apart from differentiating twice with respect to time. Free body diagrams (see Figure 3.5) of each link are drawn to identify all forces and torques acting on it. External force can be calculated by applying imaginary spring between patient's hand and attachment point (see Figure 3.6).

One way to reduce the required torque input, found in the force analysis section, is to change the position of the linkage's center of gravity (CG). The position vectors were changed in the following range: 𝑅𝐶𝐺 ∈ [0.1 ∗ 𝐿𝑛; 0.9 ∗ 𝐿𝑛], where 𝐿𝑛 is the length of the input and output links. Linear springs can be used as passive control elements to generate the required torque according to the MJM model.

First, it is necessary to find the required input torque 𝑇4 acting on the output link taking into account that the external 𝑇2 is equal to zero. An accelerating spring and a decelerating spring can be used to move the mechanism under torque 𝑇4. Then the deceleration spring decelerates the mechanism until it comes to the rest position and parallel to the output link.

To calculate spring force acting on output link, only the derivation of accelerating spring is considered, since calculation for the second spring is equal. Notations in Figure 3.8 are as follows: 𝐹 is fixed point of spring to ground, 𝐶 is a contact point of spring with output link, 𝐿𝑠1 is a distance from pivot point to fixed point, 𝐿𝑠2 is a distance from pivot point to contact point, 𝛽 is angle between fixed point and X abscissa, 𝛾 is angle between spring force and output link, 𝐿0 is a free length of spring, 𝐿𝑠 is the length of spring at a certain angular position 𝜃4 of output link, 𝜀 is the direction angle of spring force.

## Implementation & Results

### Curved motion

The analysis of position, velocity, acceleration and force for the curved motion is performed the same as for the rectilinear motion, but with the exception that the MJM has different equations. According to the MJM of curved motion, the hand must pass through the time point 𝑡1. In the case of curved movement, the coupler and output links rotate in the opposite direction compared to straight-line movement.

For the curved movement, it is assumed that only horizontal deviation of reduced user path from MJM profile is taken into account. The reduced user position and MJM position for curved motion on the horizontal plane is shown in Figure 4.14. Required torque on the input and output link for curved movement is depicted in figures 4.15 and 4.16.

The same approach as for straight-line motion is used to optimize the torque generated by the springs for curved motion. In case of curved movement, the output link rotates clockwise, so the acceleration spring will be located on the left side of the output link. Due to dimensional limitations, the distance from the fixed coupling of the output link to the fixed end of the acceleration spring is limited to between 0.03 m and 0.08 m.

The distance from fixed connection of output link to connection point with accelerating spring is limited between 0.03 m and 0.10 m. According to Figure 4.17, torque generated by accelerating spring does not ideally match the required torque due to dimensional constraints. As in straight line motion, most of the torque generated by spring and the required torque on the drive motor is reduced.

The above results showed that auxiliary springs can be used to reduce the required torque for the drive motor.

## Conclusion

It is required to design adjustable mechanism that generates different profiles, which can improve the rehabilitation procedure. When springs are used instead of actuators, it is important to determine how to set the mechanism to the "home position" after performing a motion cycle and how to connect the springs to the links. According to the results, there is a sharp displacement of the torque when the accelerating spring stops acting on the mechanism and the mechanism "meets" the decelerating spring.

Further research will investigate the installation of springs so that an accelerating part of the movement smoothly "switches" to a decelerating part. Also, kinetostatic analysis performed in this work should be validated by solving forward dynamic problem. In conclusion, the results of this thesis represent that proposed methods can be used as a basis in the development of cost-effective mechanisms that can be used in upper limb rehabilitation.

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