Multicriteria Optimization of Lower Limb Exoskeleton Mechanism

Typical leg exoskeletons employ open-loop kinematic chains with motors placed directly on movable joints; while this design offers flexibility, it leads to increased costs and heightened control complexity due to the high number of degrees of freedom. The use of heavy servo-motors to handle torque in active joints results in complex and bulky designs, as highlighted in existing literature. In this study, we introduced a novel synthesis method with analytical solutions provided for synthesizing lower-limb exoskeleton. Additionally, we have incorporated multicriteria optimization by six designing criteria. As a result, we offer several mechanisms, comprising only six links, well-suited to the human anatomical structure, exhibit superior trajectory accuracy, efficient force transmission, satisfactory step height, and having internal transfer segment of the foot.


Introduction
Exoskeleton robots have found broad application for augmenting power and aiding in rehabilitation [1,2,3,4,5,6,11].Power augmentation is important for tasks involving heavy load transportation with limited muscle strength, while robotassisted technologies employing upper and lower limb exoskeletons are used for rehabilitating individuals who have experienced a loss of mobility in their joints and muscles.Studying human walking apparatus and motion diagrams representing the leg movement were useful in various fields including the development of human prosthetics, human mimicking robots, and advancements in research areas such as biomimetics, military combat, cinematography, toys, and terrestrial and extraterrestrial exploration [11,24].For a comprehensive overview of bipedal walking robots and exoskeletons, refer to [1,2,3,4].
Various design and control architectures of exoskeletons were summarized in the relevant references [5,3].In recent years, various design schemes of lower limb exoskeletons aimed at achieving compact devices and meeting specific optimality criteria have been proposed [25,12,30].Commonly, the kinematic scheme of the leg exoskeleton is based on an Fig. 1: Prof. J.Shigley's Foot-Path Diagram system provided in electronic system, which enables to use the prototype for various load handling capacities.An eightbar leg mechanism dimensional synthesis is presented in [18] as well.As compared to the prototype, the advantage of the proposed geometric model consists of a greater step height that helps the robotic structure to overcome larger obstacles.
The Klann linkage is another single DOF walking mechanism that is patented and is widely used on multi-legged robots [19,20,21,22].The capabilities of standard non-reconfigurable quadruped Klann legs can be significantly extended applying the method proposed in [19].Reconfigurable legged robots based on 1 DOF Klann mechanism are highly desired because they are effective on rough and irregular terrains and they provide mobility in such terrain with simple control schemes.However, both Theo Jansen's solution and Klann mechanism inspired leg designs are very cumbersome, especially they cannot be used as lower limb exoskeleton since it does not fit the size limit and overall dimension restrictions.Other related works include [29,31,35,36,37,38].
Tsuge [23] proposed a kinematic synthesis method developed to achieve a mechanical system that guides a natural ankle trajectory for human walking gait.The author analyzed existing Watt I and Stephenson III six-bar linkage synthesis methods and applied the developed additional linkage synthesis procedures for treadmill training mechanism design.A new six-bar linkage system was proposed to support natural movement of the human lower limb.However the accuracy of generated straight-line segment of foot path is poor and relative horizontal velocity of the foot is not constant.
In [28] presented 6 configurations of an 8-bar leg mechanism, with three fixed pivots that make it strong and stable, validated on experimental prototype.The paper emphasizes that this mechanism offers the largest stride-to-size ratio, allowing for the construction of a compact and lightweight walking mechanism with low inertia.Consequently, it is well-suited for speed walking.In this study we synthesized a mechanism that is even more compact, and comprising only six links.We introduced a novel synthesis method, and as the result of multicriteria optimization, we achieved a compact solution matching human anatomy, with high accuracy of trajectory generation, optimized force transfer, minimized chassis height, and with internal transfer of foot.
The desired foot trajectory consists of two segments (Fig. 1): 1. straight-line segment A − B with stride L, corresponding to the support phase of step cycle (when the foot P is on the ground); 2. swing phase segment B −C − A with a step height h (the leg transfer phase).This paper is organized as follows: Section 2 introduces the structure of the lower-limb exoskeleton mechanism and our proposed synthesis method.Section 3 presents analytical solutions of the synthesis problem.Multicriteria optimization conditions are defined in Section 4. Obtained design solutions are discussed in Section 5, and the final design is presented in Section 6.And Section 7 provides conclusions.

Lower Limb Exoskeleton Mechanism Structure and Synthesis Problem Formulation
Human walking can be analyzed in three planes: the sagittal plane, coronal plane, and transverse plane [6,7].Among these, the sagittal plane motion is predominant.The designed leg exoskeleton aids in hip and knee flexion/extension movements while the wearer stays in place.This practice focuses on motions within the sagittal plane.Consequently, a planar linkage with angular outputs can effectively facilitate these movements.
The kinematic scheme of six-bar Stephenson III type lower-limb exoskeleton mechanism is illustrated in Fig. 2a with the input link AB and the foot P mounted on coupler EF.Rotation of the crank AB is described by ϕ i , Here, ϕ 0 is an initial angular position of the crank AB with respect to the horizontal axis.∆Φ is the maximum rotation angle, ∆Φ > π, in order to supply the overlap of support phases of alternating two legs.Kinematic analysis of the mechanism is provided in Appendix A.
The rotation of the crank GF is represented by the angle θ i , i = 1, N. ⃗ r P [x P , y P ] is a local radius vector, rigidly associated with the moving coordinate system Exy; and ⃗ R P i [ξ P i , η P i ] and ⃗ R E i [ξ E i , η E i ] signify the absolute positions of the foot center P i and the point E respectively.Then the relationship for ⃗ R P i can be expressed as: Given the desired absolute coordinates of the foot P * i (Fig. 2b): with stride L and desired start position P * 1 (ξ 0 , η 0 ), the synthesis condition is stated as follows: Considering the constraint equations Eq. ( 2) one can write in scalar form as follows: Given the mechanism link lengths except (x P , y P ), synthesis task consists in determining the parameters x P , y P , ξ 0 , η 0 , L that satisfy these constraints approximately.
Then the least square approximation problem is formulated as follows: 3 Analytical Solution of the Synthesis Problem 3.1 4-parameter synthesis Unknown variables are: x 1 := x P , x 2 := y P , x 3 := ξ 0 , x 4 := η 0 .The conditions ∂S ∂x j = 0, j = 1, 4 lead to the following equations involving four unknowns x 1 , x 2 , x 3 , x 4 : Namely, Let us introduce notations: Then we come to the system of linear equations expressed as follows: The only solution of this system in case The necessary conditions also serve as sufficient conditions for attaining the minimum of the function S: the matrix , since the determinants of all corner minors are non-negative:

5-parameter synthesis
Let's explore a scenario where L = x 5 is an additional variable subject to optimization.Referring to Eq. ( 12), from the condition ∂S ∂⃗ x = ⃗ 0 we get where where where where The last equation is derived from Eq. ( 6), j = 5: Finally, taking into account and we obtain the following system of linear equations where and The Hessian H S = d 2 S d⃗ x 2 , and it is non-negative definite (for the proof refer to Appendix B).
, Therefore, the last condition supplies the minimum of the function S, if det H S ̸ = 0.For analytical solution in this case we refer to Eq. (12a) -(12d): x assuming k 2 + m 2 ̸ = N 2 ; and x 5 can be found directly by substituting Eq. ( 31) -(34) into Eq.(22).In order to choose the acceptable solutions from a trial table, which contains great amount of data, a certain design criteria will be used.The trajectory of the foot center P, called a step cycle, consists of two phases: "support phase" and "transfer phase" (swing).During the support phase the foot center P should trace horizontal straight-line.The main criteria of synthesis is the accuracy of straight line generation during support phase, that has to be minimized: At the same time we deal with a complicated synthesis task since the following additional criteria should be taken into account.
The chassis height H = −x 4 has to be minimized or (which is the same condition) η 0 = −H to be maximized: The worse transference angle has to be maximized where L BC , L CD , L FG , L EF are lengths of corresponding links, |BD|, |EG| are distances between the centers of corresponding joints.
Anatomy matching: hip to shin relation L FP /L FG (the ratio of the thigh to the lower leg) has to be around one (c 4 = 1) Te penalty function c 5 for external transfer of the foot center P has to be minimized.

Global Search and Multi-Criteria Optimization of Exoskeleton Link Dimensions
Let us keep notation ξ, η for absolute coordinates of joints as it was in the previous sections.n = 13 variable parameters of the exoskeleton mechanism are varied within a given search area by using so called random LP-τ sequences, evenly distributed in n-dimensional parallelepiped [32,33,34].Global search carried out specifying the following limits on variable parameters For each set of n = 13 random variables analytical solutions for 5 design parameters x 1 , x 2 , ..., x 5 are determined applying formulae Eq. ( 31) - (34), and Eq. ( 22), and the trial table is obtained by calculating design criteria values.Analyzing the obtained trial table, 25 preliminary solutions are selected as specified in Table 1 and shown in Fig. 7 (7).The criteria values (c 1 , ..., c 5 ) for the obtained solutions are found to be vary within the following limits: The parameter values for solution on Fig. 7a are shown in Tables 11-13 (Appendix C) Global Search within the Narrowed Search Area.Analyzing the results, we study functionality of the mechanism.The most elusive was meeting criteria c 5 (achieving internal swing with trajectory turned inward).Thus, a design criterion c 6 was introduced to increase the step height h i = η P i − η 0 defined as that has to be maximized.(Note that we have not used h = max h i as a design criterion c 6 , since it does not reflect the foot trajectory that goes below the limit η = η 0 as demonstrated in Fig. 7 (Appendix C).Meanwhile, when using the sum these trajectories will have negative sign which decreases the sum).Arranging and cutting the obtained trial table, eliminating solutions with unacceptable criteria values, we are looking for the compromise solutions that meet all designing criteria, so we clarify new boundaries for design variables.Then we carry out new search of the design parameters within the new search area and obtain new trial table.After several repetition of this sequence of actions we came to the following search area specified by the boundaries: As the result we obtained a number of solutions presented on Table 2, having internal transfer segment of the foot, so that in the swing phase the foot trajectory is turned inward.Part of these solutions are plotted on Fig. 3.The solutions in Table 2 are arranged by criterion c 6 in descending order.Despite the step height (height of the foot transference) not being very high, we obtained the desired solutions with high accuracy (about 1 percent from step stride) and fine transmission angle (from 32 • to 55 • ) (Tables 3 -5).One can observe that solutions 29884 (Fig. 3a) and 26230 (Fig. 3c) exhibit a low value of criterion c 4 , leading to the displacement of the knee joint E to an undesirable lower position that does not conform to human anatomy.Solution 31076 (Fig. 3b) possesses an acceptable c 4 value, but the straight-line segment height H = −c 2 = −η 0 is excessive.Same shortcomings take place in solutions 27664, 1832, 7592, and 17728, thus the relevant figures have not been plotted.Ultimately, our analysis identified solution 7146 (Fig. 3d) as the most suitable choice.Although it exhibits a suboptimal swing height, it has excellent accuracy, transmission angle, and a satisfactory knee joint position E. As demonstrated in Fig. 3, the lower the c 6 value, the lower the foot swing height (h).Thus, the rest of the solutions are not shown.In order to improve the swing height the local search around the solution 7146 is carried out.The selected solutions presented on Table 6, arranged in descending order of the criterion c 6 .
Because of the substantial number of solutions, we included only few images on Fig. 4, Table 7 explains why we have selected these specific images.The remaining solutions are not depicted, since one can observe, that step height decreases.The design parameters for the solution 8398 are presented on Tables 8 -10.The typical schemes of leg exoskeletons are based on open-loop kinematic chain with the motors mounted directly on the moveable joints.While the design choice offer greater flexibility and ease of design, their large number of DOF contributes to increased costs and complexities in control.Using heavy servo-motors to meet significant torques aroused in active joints leads to complicated and cumbersome design.Existing literature emphasizes bulkiness and substantial weight of this kind of devices.
Another approach involves the utilization of 1-DOF mechanisms.However, many of them characterized by a substantial number of linkages, often reaching eight or more.For example, the best schemes of such design are selected in figure below.As you can see in Fig. 5, even the mechanism (b), which is often considered the best and published in the MMT International Journal as one of the respectable designs [28], is overly bulky with a large number of links also.If employed as an lower-limb exoskeleton mechanism, its geometry would not align with the human anatomy.In the figure also shown that the mechanism we designed (a) is well-suited to the anatomical parameters of humans.Note, that the numbers on the axes are relative values (not in meters), when L -the step length -is equal to 1.
In this study, we introduced a novel synthesis method with analytical solutions provided for synthesizing lower-limb exoskeleton.Additionally, we have incorporated multicriteria optimization by six designing criteria.As a result, we offer several mechanisms, comprising only six links, well-suited to the human anatomical structure, exhibit superior trajectory accuracy, efficient force transmission, satisfactory step height, and having internal transfer segment of the foot.
Thus, the Hessian matrix H S will be Therefore, the matrix H S is non-negative definite, det H S = 0 is the singularity of the synthesis problem.
Appendix C

Fig. 3 :
Fig. 3: Visualization of the mechanisms with swing phase trajectories turned inward

Fig. 4 :
Fig. 4: The results by the local search with improved swing height

Table 2 :
The trial table fragment: the best solutions by criterion c 6

Table 6 :
The results of the local search around the solution 7146: the best solutions by criterion c 6

Table 7 :
Comments to the images

Table 11 :
Random search design parameters p 1 , ..., p 7 for the mechanism on Fig.7a

Table 12 :
Random search design parameter values p 8 , . .., p 13 for the mechanism on the Fig.7a