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**Mechanical Sciences**
An open-access journal for theoretical and applied mechanics

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**Research article**
20 Nov 2018

**Research article** | 20 Nov 2018

Kinematic analysis and optimization of a planar parallel compliant mechanism for self-alignment knee exoskeleton

^{1}Key Laboratory of Mechanism Theory and Equipment Design and Ministry of Education, Tianjin University, Tianjin, 300350 China^{2}School of Mechanical Engineering, Tianjin University, Tianjin, 300350, China^{3}Centre for Advanced Mechanisms and Robotics, Tianjin University, Tianjin, 300350, China^{4}School of Natural and Mathematical Sciences, King's College London, University of London, Strand London, WC2R 2LS, UK

^{1}Key Laboratory of Mechanism Theory and Equipment Design and Ministry of Education, Tianjin University, Tianjin, 300350 China^{2}School of Mechanical Engineering, Tianjin University, Tianjin, 300350, China^{3}Centre for Advanced Mechanisms and Robotics, Tianjin University, Tianjin, 300350, China^{4}School of Natural and Mathematical Sciences, King's College London, University of London, Strand London, WC2R 2LS, UK

**Correspondence**: Zhibin Song (songzhibin@tju.edu.cn)

**Correspondence**: Zhibin Song (songzhibin@tju.edu.cn)

Abstract

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Misalignment between the instantaneous center of rotation (ICR) of human joint and the ICR of wearable robotic exoskeleton widely exists among most of exoskeletons widely used in rehabilitation, which results in discomfort, even endangers human safety. In order to alleviate it, this study focuses on the solution of misalignment in knee joint of lower limb exoskeletons, and proposes a compliant five-bar parallel mechanism, which offers two mobility in sagittal plane, as well as the torsional springs mounted on this mechanism, have the potential to automatically adjust the ICR of output link connected to thigh with respect to the basis link connected to shank. To reach this goal, we build the stiffness model of the mechanism and optimize its variables. And the self-alignment of the compliant five-bar parallel mechanism is verified via experimental investigations.

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Niu, Y., Song, Z., and Dai, J.: Kinematic analysis and optimization of a planar parallel compliant mechanism for self-alignment knee exoskeleton, Mech. Sci., 9, 405–416, https://doi.org/10.5194/ms-9-405-2018, 2018.

1 Introduction

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Wearable robotic exoskeletons have been researched for several years, and they are widely applied in various fields ranging from rehabilitation of patients (Hesse et al., 1995) to performance assistance (Zoss et al., 2006). Most of the devices are designed to mimic the motions of users, especially the assisted rehabilitation exoskeletons, which provide physical exercises by imitating the pattern of the movement of human limbs (Wong et al., 2012).

Keeping the correspondence of robotic exoskeleton rotating axes with human joint axes is a necessary criterion. The majority of exoskeleton rotary joints are typically modeled as hinges, however, due to the limitation of the unicentric feature, these mechanisms have a drawback of causing misalignment, in other words, causing inconsistency between human limbs and exoskeleton during movement, even though their initial positions are carefully aligned. Human joints cannot be assumed as simple kinematic pairs, since human skeletal structure consists of bones, ligaments and cartilage, and this intricate structure results in complicated kinematics of joints. Human knee is one of the most typical joints where misalignment exists. As shown in Fig. 1, the anomaly geometry of tibial and femoral condyles combined with restriction of collateral tendons influence the movement of human knee joint axis in the sagittal plane. This axis is the instant center of knee rotation, which is a composite motion shown as the combination of rolling and gliding movements within a circular scope with a diameter of 20 mm (Smidt, 1973; Frankel and Nordin, 1980; Yamaguchi and Zajac, 1989). Moreover, the variability of intra/intersubjects also negatively affects the correspondence between exoskeleton axes and the human limb joints (Zanotto et al., 2015), in view of the twisting and sliding movements of the attachment area of the limb, as well as differences among different users. There is also a fact that robotic exoskeleton inevitably add weight, inertia and friction to legs (Aguirre-Ollinger et al., 2011), and once human/robot misalignment goes worse, these uncompensated mechanical impedance will severely hinder the agility of movement, or generate undesired forces or torques to the limbs and joints because of hyperstaticity of the closed-chain mechanism, leading to malaise, even injury (Zanotto et al., 2015; Aguirre-Ollinger et al., 2011).

In the past, the majority of designs aimed to handle the misalignment in the upper extremity exoskeletons (Yalcin and Patoglu, 2012; Cempini et al., 2013; Otten et al., 2015), and recently there comes out plenty of researches focusing on lower extremity exoskeletons. Sun et al. (2015) proposed a prosthetic knee joint with a geared five-bar mechanism, which can directly modify the joint trajectory via adjusting the gear ratio. Nevertheless, motions of two links of this mechanism are mutual coupling, such that the trajectory of the joint is determined in advance, and during the movements, this design cannot solve the changing misalignment in the real time. Furthermore, this design only can fine-tune its output in a narrow range, which cannot meet demands of different users. Cai et al. (2009) introduced a knee orthosis to alleviate the internal forces produced by misalignment, and it was added with additional mobility and can adjust the ICR of the exoskeleton. Due to the length-adjustable links, this design is able to self-adjust without resorting to motorized add-ons, and reversely, it brings about unconstraint passive movement, which makes it impossible to sufficiently support little weight and potentially lead to instability. Moreover, referring to a trajectory of the ICR of knee joint, Sakai et al. (2015) proposed a device with the combination of sliding channel and wire-pulley, and its rotational center could keep pace with that of human femur. However, this design determined the trajectory of its motions based on a specific motion data, which in turn make itself fail to fit other individuals. Some studies achieved self-alignment by using amounts of sensors and control system in exoskeleton (Krut et al., 2010; Zoss et al., 2016). Those solutions compensate the misalignment by observing the macroscopic deviation, and guide the position or the torque of the human joints. Even though these approaches offer an exact adjustment for each user's knee joint, measuring a mass of explicit data costs too much and the equipment is bulky and heavy. And the user will suffer from the frictional force at the attachment position and feel uncomfortable and painful, as a result, fall into a risky situation (Pons et al., 2010). In general, even though certain studies developed some approaches could attenuate the misalignment, their mechanisms have some common disadvantages, such as increasing design complexity, being unfeasible for different individuals or loading undesirable force or torque on users.

Now that the extension and flexion motions of human knee are not unicentric, for the purpose of better bionic performance, some exoskeletons switched to adopt multiaxial joints instead of traditional uniaxial joints. Many prosthetic knee technologies are designed based on the four-bar mechanisms (Hobson et al., 1974; Radcliffe, 1977, 1994; Muñoz-César et al., 2013), which offer bionic relative ICR and enhance safety and comfortability in movements. While some six-bar mechanisms (Jin et al., 2003) are developed, and these multiaxial knee joints have polycentric characteristic to make the ICR of prosthetic joint closer to that of human knee joint in walking experience. Nevertheless, the four-bar mechanism only have one mobility and cannot flexibly change its configuration to provide diverse movements in self-adjusting to human body, and as for the six-bar mechanism, it has much more various structure and needs complex controllability. By contrast, five-bar mechanism is more flexible than four-bar and more controllable than six-bar.

This research brings a parallel compliant five-bar mechanism on the carpet which can provide a self-alignment to the movements of human knee. It has two degrees of freedom to allow random movements in sagittal plane. In order to realize real-time self-alignment in extension and flexion processes, present a better bionic performance as well as reduce the load of the exoskeleton, this study suggests to introduce compliant parts at the joints of the exoskeleton mechanism. Thus, a concept of mechanical compliance is taken into account which can automatically change the kinematic variables of the mechanism to fit the limb movements, owning to the design of the mechanism with special stiffness characteristics rather than explicit control system. Due to compliant feature and absence of fixed axis, this mechanism has better bionic performance to follow the human motions, safely and automatically.

The rest of the article is organized as follows: In Sect. 2, the structural design of the planar parallel compliant five-bar mechanism is introduced. In Sect. 3, the integrated kinematic model of the proposed mechanism is built and derived. In Sect. 4, the variables of the mechanism are optimized by using GAs method. Moreover, method of evaluating the risk caused by misalignment is discussed, and the self-aligning performance of the mechanism is estimated by conducting a series of experiments in Sect. 5, followed by conclusions in Sect. 6.

2 Basic structure of knee exoskeleton

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The planar parallel compliant five-bar mechanism, assembled in the sagittal
plane of the exoskeleton, is a closed chain and substitutes for the
traditional joint. As shown in Fig. 2, the structure is designed as a
symmetric form so as to simplify the calculation process, link AE works as
the base frame and rigidly connects to the shank exoskeleton, while the thigh
exoskeleton is connected to joint *C* with a hinge. Five torsional springs
are respectively mounted on each joint. Those springs are initially installed
without internal preload torque distribution, which simplifies the process of
installment. And the whole exoskeleton is tightly attached to human limbs
with braces. To enable the mechanism to transform its configuration flexibly
and efficiently, the wearing of the exoskeleton must ensure the extension and
flexion of knee joint is within the working scope of the five-bar mechanism
even though the attachment positions are arbitrary.

During the knee extension and flexion motions, a little disconformity between
human and exoskeleton movements will generate corresponding force acting on
the thigh bar, then impose a load on the compound hinge *C*. Since the
mobility of this mechanism is two, its output (end effector) joint *C* can
move freely within the working space. When loaded, the compliant five-bar
mechanism will approach to a stable state, thus, external work will converse
into elastic energy stored in the torsional springs. And the deformation of
the mechanism can reduce the relative human/robot misaligned wobble movement,
making the exoskeleton self-align to human body. On the other hand, the point
*C* serves as the output of the mechanism as its position reflects the
corresponding configuration of the mechanism. In fact, there is no fixed
mechanical rotation center for the five-bar mechanism and this compliant
device has a potential to adjust itself to fit the polycentric motions of the
human joint axis, and simultaneously offer self-alignment to the exoskeleton
without measuring the explicit ICR of human joint.

3 Stiffness analysis of the compliant Five-Bar Mechanism

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In order to provide with suitable and effective self-alignment, motion capabilities of the mechanism are crucial issues. The movement characteristics and precision of the end effector are directly related to the stiffness of the mechanism, such that this research uses a decoupling method to obtain the stiffness matrix of the mechanism. With this method, we can figure out the mathematical problems in torque distributions of the mechanism during extension and flexion. In view of the process of squatting which involves the largest flexion range of human knee joint and induces the larger misalignment, it is a relative slow motion, and the kinematic stiffness is a static approximation and can be calculated by deriving the torque and force with respect to the infinitesimal angular changes of joints or the infinitesimal deformation.

The parallel compliant five-bar mechanism is modeled in Fig. 3.
*L*_{i} (*i*=1–5) indicates the length of each bar, *A*∼*E* respectively
indicates each joint, *θ*_{i} represents the corresponding joint angle,
and *F* is an external force applying on the joint *C*.

Presuming this mechanism is separated from point *C* into two kinematic open
chains, the left one is assumed to be the independent chain, while the right
one, including the joint *C*, is the dependent chain. The final position of
the end effector of the mechanism is related to the given angle of left
chain, and can be obtained by analyzing the forward kinematics:

$$\begin{array}{}\text{(1)}& {\mathit{u}}_{\text{C}}(x,y)=\left[\begin{array}{l}{x}_{\text{C}}\\ {y}_{\text{C}}\end{array}\right]\phantom{\rule{0.25em}{0ex}}=\left[\begin{array}{l}{L}_{\mathrm{1}}\mathrm{cos}{\mathit{\theta}}_{\mathrm{1}}+{L}_{\mathrm{2}}\mathrm{cos}({\mathit{\theta}}_{\mathrm{1}}+{\mathit{\theta}}_{\mathrm{2}})\\ {L}_{\mathrm{1}}\mathrm{sin}{\mathit{\theta}}_{\mathrm{1}}+{L}_{\mathrm{2}}\mathrm{sin}({\mathit{\theta}}_{\mathrm{1}}+{\mathit{\theta}}_{\mathrm{2}})\end{array}\right]\phantom{\rule{0.25em}{0ex}}\end{array}$$

Differentiate Eq. (1) with respect to time, the relationship is obtained:

$$\begin{array}{}\text{(2)}& {\displaystyle}& {\displaystyle}{\dot{\mathit{u}}}_{\text{C}}={\mathbf{J}}_{\text{L}}{\dot{\mathit{q}}}_{\text{L}}\text{(3)}& {\displaystyle}& {\displaystyle}{\mathbf{J}}_{\text{L}}=\left[\begin{array}{l}{\displaystyle \frac{\partial {u}_{C\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{1}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{\partial {u}_{C\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{2}}}}\\ {\displaystyle \frac{\partial {u}_{{\text{C}}_{\mathrm{2}}}}{\partial {\mathit{\theta}}_{\mathrm{1}}}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\displaystyle \frac{\partial {u}_{{\text{C}}_{\mathrm{2}}}}{\partial {\mathit{\theta}}_{\mathrm{2}}}}\end{array}\right]\end{array}$$

**J**_{L} is the forward Jacobian of left chain, and it
associates the angular velocities of joints to the velocity of the point *C*.

The mapping relationship between angular velocities of these two chains is obtained from the kinematic constraint equations, which are built on account of the closed-chain characteristics of the mechanism, as shown in Eq. (3):

$$\begin{array}{ll}{\displaystyle}{G}_{\mathrm{1}}=& {\displaystyle}{L}_{\mathrm{1}}\mathrm{cos}{\mathit{\theta}}_{\mathrm{1}}+{L}_{\mathrm{2}}\mathrm{cos}({\mathit{\theta}}_{\mathrm{1}}+{\mathit{\theta}}_{\mathrm{2}})\\ {\displaystyle}& {\displaystyle}-{L}_{\mathrm{3}}\mathrm{cos}({\mathit{\theta}}_{\mathrm{4}}+{\mathit{\theta}}_{\mathrm{5}})-{L}_{\mathrm{4}}\mathrm{cos}{\mathit{\theta}}_{\mathrm{5}}-{L}_{\mathrm{5}}=\mathrm{0}\\ \text{(4)}& {\displaystyle}{G}_{\mathrm{2}}=& {\displaystyle}{L}_{\mathrm{1}}\mathrm{sin}{\mathit{\theta}}_{\mathrm{1}}+{L}_{\mathrm{2}}\mathrm{sin}({\mathit{\theta}}_{\mathrm{1}}+{\mathit{\theta}}_{\mathrm{2}}){\displaystyle}& {\displaystyle}-{L}_{\mathrm{3}}\mathrm{sin}({\mathit{\theta}}_{\mathrm{4}}+{\mathit{\theta}}_{\mathrm{5}})-{L}_{\mathrm{4}}\mathrm{sin}{\mathit{\theta}}_{\mathrm{5}}=\mathrm{0}\\ {\displaystyle}{G}_{\mathrm{3}}=& {\displaystyle}{\mathit{\theta}}_{\mathrm{1}}+{\mathit{\theta}}_{\mathrm{2}}+{\mathit{\theta}}_{\mathrm{3}}-{\mathit{\theta}}_{\mathrm{4}}-{\mathit{\theta}}_{\mathrm{5}}-\mathrm{2}\mathit{\pi}=\mathrm{0}\end{array}$$

Time derivative of Eq. (3):

$$\begin{array}{}\text{(5)}& {\displaystyle \frac{\text{d}\mathbf{G}}{\text{d}t}}={\displaystyle \frac{\text{d}\mathbf{G}}{\text{d}\mathit{\theta}}}{\displaystyle \frac{\text{d}\mathit{\theta}}{\text{d}t}}={\displaystyle \frac{\text{d}\mathbf{G}}{\text{d}\mathit{\theta}}}\dot{\mathit{\theta}}=\mathrm{0}\end{array}$$

where **G**=(*G*_{1} *G*_{2} *G*_{3})^{T}, Let
${\dot{\mathit{q}}}_{\text{R}}={\left({\dot{\mathit{\theta}}}_{\mathrm{3}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\dot{\mathit{\theta}}}_{\mathrm{4}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\dot{\mathit{\theta}}}_{\mathrm{5}}\right)}^{\text{T}}$, and ${\dot{\mathit{q}}}_{\text{L}}={\left({\dot{\mathit{\theta}}}_{\mathrm{1}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\dot{\mathit{\theta}}}_{\mathrm{2}}\right)}^{\text{T}}$ are
separately the angular velocity vectors for the left and right chains, and
relationship between the angular motions of two chains can be obtained as
follows:

$$\begin{array}{}\text{(6)}& {\mathbf{G}}_{\text{L}}{\dot{\mathit{q}}}_{\text{L}}+{\mathbf{G}}_{\text{R}}{\dot{\mathit{q}}}_{\text{R}}=\mathrm{0}\end{array}$$

$$\begin{array}{}\text{(7)}& {\displaystyle}{\mathbf{G}}_{\text{L}}=\left[\begin{array}{ll}{\displaystyle \frac{\partial {G}_{\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{1}}}}& {\displaystyle \frac{\partial {G}_{\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{2}}}}\\ {\displaystyle \frac{\partial {G}_{\mathrm{2}}}{\partial {\mathit{\theta}}_{\mathrm{1}}}}& {\displaystyle \frac{\partial {G}_{\mathrm{2}}}{\partial {\mathit{\theta}}_{\mathrm{2}}}}\\ {\displaystyle \frac{\partial {G}_{\mathrm{3}}}{\partial {\mathit{\theta}}_{\mathrm{1}}}}& {\displaystyle \frac{\partial {G}_{\mathrm{3}}}{\partial {\mathit{\theta}}_{\mathrm{2}}}}\end{array}\right]\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\mathbf{G}}_{\text{R}}=\left[\begin{array}{lll}{\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{3}}}}& {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{4}}}}& {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{1}}}{\partial {\mathit{\theta}}_{\mathrm{5}}}}\\ {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{2}}}{\partial {\mathit{\theta}}_{\mathrm{3}}}}& {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{2}}}{\partial {\mathit{\theta}}_{\mathrm{4}}}}& {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{2}}}{\partial {\mathit{\theta}}_{\mathrm{5}}}}\\ {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{3}}}{\partial {\mathit{\theta}}_{\mathrm{3}}}}& {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{3}}}{\partial {\mathit{\theta}}_{\mathrm{4}}}}& {\displaystyle \frac{\partial {\mathrm{G}}_{\mathrm{3}}}{\partial {\mathit{\theta}}_{\mathrm{5}}}}\end{array}\right]\end{array}$$

where **G**_{L} and **G**_{R} are respectively
the corresponding first-order influence coefficient matrices for the two
chains (Huang, 1985; Hudgens and Tesar, 1991).

Let **J**_{u} express the influence matrix for the motions propagation
between the two chains, then

$$\begin{array}{}\text{(8)}& {\dot{\mathit{q}}}_{\text{R}}={\mathbf{J}}_{u}{\dot{\mathit{q}}}_{\text{L}}\end{array}$$

where ${\mathbf{J}}_{u}=-{\mathbf{G}}_{\text{R}}^{-\mathrm{1}}{\mathbf{G}}_{\text{L}}\in {\mathrm{\Re}}^{\mathrm{3}\times \mathrm{2}}$.

During the motion of the end effector, external loads acting on joint *C*
generate torques at joints for the compliant effects. Assuming no friction
applied on the mechanism, the virtual work principle is used to establish a
torque equilibrium.

$$\begin{array}{}\text{(9)}& \sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}{\mathit{V}}_{i}^{\text{T}}{\mathit{w}}_{i}+{\dot{\mathit{u}}}_{\text{C}}^{\text{T}}\mathit{F}+{\dot{\mathit{q}}}_{\text{R}}^{\text{T}}{\mathit{\tau}}_{\text{R}}+{\dot{\mathit{q}}}_{\text{L}}^{\text{T}}{\mathit{\tau}}_{\text{L}}=\mathrm{0}\end{array}$$

where ${\mathit{V}}_{i}=({\dot{x}}_{{c}_{i}}$, ${\dot{y}}_{{c}_{i}}$,
*ω*_{i})^{T} $(i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4})$ indicates the three dimensional velocity
of link *i*, as described in more detail in the section *D*; ${\mathit{w}}_{i}={\left\{\mathrm{0},-{m}_{i}g,\mathrm{0}\right\}}^{\text{T}}$ represents the vector of each
link's gravity; ${\dot{\mathit{u}}}_{\text{C}}$ is the velocity vector of joint
*C*; $\mathit{F}={\left[{f}_{x}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{f}_{y}\right]}^{\text{T}}$ indicates the
external force applied at joint *C*; ${\dot{\mathit{q}}}_{\text{R}}={\left({\dot{\mathit{\theta}}}_{\mathrm{3}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\dot{\mathit{\theta}}}_{\mathrm{4}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\dot{\mathit{\theta}}}_{\mathrm{5}}\right)}^{\text{T}}$ and ${\dot{\mathit{q}}}_{\text{L}}={\left({\dot{\mathit{\theta}}}_{\mathrm{1}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{\dot{\mathit{\theta}}}_{\mathrm{2}}\right)}^{\text{T}}$ respectively express the angular
velocity vectors for left and right chains, which are obtained by decomposing
the five-bar mechanism from point *C*; ${\mathit{\tau}}_{\text{R}}={\left[{k}_{\mathrm{3}}\mathrm{\Delta}{\mathit{\theta}}_{\mathrm{3}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{4}}\mathrm{\Delta}{\mathit{\theta}}_{\mathrm{4}}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{5}}\mathrm{\Delta}{\mathit{\theta}}_{\mathrm{5}}\right]}^{\text{T}}$, ${\mathit{\tau}}_{\text{L}}={\left[{k}_{\mathrm{1}}\mathrm{\Delta}{\mathit{\theta}}_{\mathrm{1}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}{k}_{\mathrm{2}}\mathrm{\Delta}{\mathit{\theta}}_{\mathrm{2}}\right]}^{\text{T}}$ indicate the torques on the
joints of left and right chains, respectively.

Substituting Eqs. (2) and (6) into (7), and eliminating ${\dot{\mathit{q}}}_{\text{L}}$, yields:

$$\begin{array}{}\text{(10)}& \sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}{\mathbf{G}}_{i}^{\text{T}}{\mathit{w}}_{i}+{\left[{\mathbf{J}}_{\text{L}}\right]}^{\text{T}}\mathit{F}+{\left[{\mathbf{J}}_{u}\right]}^{\text{T}}{\mathit{\tau}}_{\text{R}}+{\mathit{\tau}}_{\text{L}}=\mathrm{0}\end{array}$$

Differentiating with respect to time, the relationship between the payload force and the infinitesimal incremental changes of the joints is obtained:

$$\begin{array}{ll}\text{(11)}& {\displaystyle}\sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}& {\displaystyle}\mathit{\delta}{\mathbf{G}}_{i}^{\text{T}}{\mathit{w}}_{i}+\sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}{\mathbf{G}}_{i}^{\text{T}}\mathit{\delta}{\mathit{w}}_{i}+\mathit{\delta}{\mathbf{J}}_{\text{L}}^{\text{T}}\mathit{F}+{\mathbf{J}}_{\text{L}}^{\text{T}}\mathit{\delta}\mathit{F}{\displaystyle}& {\displaystyle}+\mathit{\delta}{\mathbf{J}}_{u}^{\text{T}}{\mathit{\tau}}_{\text{R}}+{\mathbf{J}}_{u}^{\text{T}}\mathit{\delta}{\mathit{\tau}}_{\text{R}}+\mathit{\delta}{\mathit{\tau}}_{\text{L}}=\mathrm{0}\end{array}$$

where the gravity of links are constant, such that *δ**w*_{i}=0.

The first term of the equation can be rewritten:

$$\begin{array}{ll}\text{(12)}& {\displaystyle}\sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}\mathit{\delta}{\mathbf{G}}_{i}^{\text{T}}{\mathit{w}}_{i}& {\displaystyle}=\sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}\left(\sum _{j\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{2}}{\displaystyle \frac{\partial {\mathbf{G}}_{i}^{\text{T}}}{\partial {\mathit{\theta}}_{j}}}\mathit{\delta}{\mathit{\theta}}_{j}\right){\mathit{w}}_{i}{\displaystyle}& {\displaystyle}=\sum _{i,=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}\left(\sum _{j\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{2}}{\displaystyle \frac{\partial {\mathbf{G}}_{i}^{\text{T}}}{\partial {\mathit{\theta}}_{j}}}{\mathit{w}}_{i}\mathit{\delta}{\mathit{\theta}}_{j}\right)\\ {\displaystyle}& {\displaystyle}=\sum _{i\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{1}}^{n\phantom{\rule{0.125em}{0ex}}=\phantom{\rule{0.125em}{0ex}}\mathrm{4}}\left({\mathbf{H}}_{i}^{\text{T}}\otimes {\mathit{w}}_{i}\right)\mathit{\delta}{\mathit{q}}_{\text{L}}\end{array}$$

The Hessian matrix **H**_{i} is defined as an influence coefficient
matrix of the second-order partial derivatives of a scalar-valued function
(Huang, 1985; Hudgens and Tesar, 1991; Huang et al., 1997). In this study,
Hessian matrix involved in the stiffness analysis denotes the derivative of
the transposition of several coefficient matrices with respect to the angular
changes of joints in the independent chain, including angular velocity
matrices of two links, the mapping relationship between two chains and the
forward Jacobian. Furthermore,

$$\begin{array}{}\text{(13)}& {\displaystyle}& {\displaystyle}\mathit{\delta}{\mathbf{J}}_{\text{L}}^{\text{T}}\mathit{F}=\left({\mathbf{H}}_{\text{L}}^{\text{T}}\otimes \mathit{F}\right)\mathit{\delta}{\mathit{q}}_{\text{L}}\text{(14)}& {\displaystyle}& {\displaystyle}\mathit{\delta}{\mathbf{J}}_{u}^{\text{T}}{\mathit{\tau}}_{\text{R}}=\left({\mathbf{H}}_{u}^{\text{T}}\otimes {\mathit{\tau}}_{\text{R}}\right)\mathit{\delta}{\mathit{q}}_{\text{L}}\text{(15)}& {\displaystyle}& {\displaystyle}{\mathbf{J}}_{u}^{\text{T}}\mathit{\delta}{\mathit{\tau}}_{\text{R}}=-{\mathbf{J}}_{u}^{\text{T}}{\mathbf{K}}_{\text{R}}\mathit{\delta}{\mathit{q}}_{\text{R}}=-{\mathbf{J}}_{u}^{\text{T}}{\mathbf{K}}_{\text{R}}{\mathbf{J}}_{\text{L}}\mathit{\delta}{\mathit{q}}_{\text{L}}\text{(16)}& {\displaystyle}& {\displaystyle}\mathit{\delta}{\mathit{\tau}}_{\text{L}}=-{\mathbf{K}}_{\text{L}}\mathit{\delta}{\mathit{q}}_{\text{L}}\end{array}$$

where ${\mathbf{K}}_{\text{L}}=\left[\begin{array}{ll}{k}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& {k}_{\mathrm{2}}\end{array}\right]$, ${\mathbf{K}}_{\text{R}}=\left[\begin{array}{lll}{k}_{\mathrm{3}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {k}_{\mathrm{4}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& {k}_{\mathrm{5}}\end{array}\right]$ is the stiffness matrix of the joints separately in the left and right chains.

Substituting Eqs. (10)–(14) into Eq. (9) and yields:

$$\begin{array}{ll}\text{(17)}& {\displaystyle}& {\displaystyle}\mathit{\delta}{\mathit{q}}_{\text{L}}={\displaystyle}& {\displaystyle}{\left({\mathbf{J}}_{u}^{\text{T}}{\mathbf{K}}_{\text{R}}{\mathbf{J}}_{\text{L}}+{\mathbf{K}}_{\text{L}}-\sum _{i=\mathrm{1}}^{\mathrm{4}}\left({\mathbf{H}}_{i}^{\text{T}}\otimes {\mathit{w}}_{i}\right)-\left({\mathbf{H}}_{\text{L}}^{\text{T}}\otimes \mathit{F}\right)-\left({\mathbf{H}}_{u}^{\text{T}}\otimes {\mathit{\tau}}_{\text{R}}\right)\right)}^{-\mathrm{1}}{\mathbf{J}}_{\text{L}}^{\text{T}}\mathit{\delta}\mathit{F}\end{array}$$

Due to the mapping relationship, the displacements of the joints in the left
link *δ**q*_{L} can be substituted by a function of the
incremental displacement of the end effector ${\mathbf{J}}_{\text{L}}^{-\mathrm{1}}\mathit{\delta}{\mathit{D}}_{\text{C}}$.

Therefore,

$$\begin{array}{}\text{(18)}& \mathit{\delta}{\mathit{D}}_{\text{C}}={\mathbf{J}}_{\text{L}}\mathit{\delta}{\mathit{q}}_{\text{L}}\end{array}$$

The stiffness matrix of the complaint mechanism is obtained by rearranging the Eq. (16)

$$\begin{array}{ll}\text{(19)}& {\displaystyle}\mathbf{K}& {\displaystyle}={\displaystyle \frac{\mathit{\delta}\mathit{F}}{\mathit{\delta}{\mathit{D}}_{\text{C}}}}{\displaystyle}& {\displaystyle}={\left({\mathbf{J}}_{\text{L}}^{\text{T}}\right)}^{-\mathrm{1}}\left({\mathbf{J}}_{u}^{\text{T}}{\mathbf{K}}_{\text{R}}{\mathbf{J}}_{\text{L}}+{\mathbf{K}}_{\text{L}}-\sum _{i=\mathrm{1}}^{\mathrm{4}}\left({\mathbf{H}}_{i}^{\text{T}}\otimes {\mathit{w}}_{i}\right)\right.\\ {\displaystyle}& {\displaystyle}-\left({\mathbf{H}}_{\text{L}}^{\text{T}}\otimes \mathit{F}\right)-\left({\mathbf{H}}_{u}^{\text{T}}\otimes {\mathit{\tau}}_{\text{R}}\right)){\mathbf{J}}_{\text{L}}^{-\mathrm{1}}\end{array}$$

Based on Eq. (17), the stiffness characteristics of the compliant five-bar
mechanism respect to its variables are discussed via simulations, where the
lengths of links are normalized to the length of base frame; the variation
range of the initial angle is decided according to installation space and the
stiffness value of the unstudied torsional springs is defined as
20 N mm rad^{−1}. As shown in Fig. 4, (a) suggests the relation of links
and the mechanism stiffness; (b) indicates that the stiffness value varies
drastically as tends to and the relative influence of each joint stiffness to
the system is indicted in (c).

This compliant five-bar mechanism is a complex spring system, and the
stiffness analysis exploits the motion of the end effector. In this research
of the lower extremity exoskeleton, we concern about the misalignment problem
which expresses as the radial wobble movement of the output point *C*, and
this exoskeleton should not hinder extension/flexion motions, which means the
output can freely move in circumferential direction. In this section, the
directional stiffness characteristics of the end effector of the mechanism is
discussed.

Suppose that the stiffness matrix of the mechanism is **K**. When
misalignment happens, an infinitesimal force $\mathit{\delta}{\mathit{f}}_{\text{C}}={\left\{\mathit{\delta}{\mathit{f}}_{xC},\mathit{\delta}{\mathit{f}}_{yC}\right\}}^{\text{T}}$ is
exerted on the end effector, and the corresponding infinitesimal movement
*δ**D*_{C} is produced, which can be distributed into
radial and circumferential components of displacement. And the relationship
between them gives

$$\begin{array}{}\text{(20)}& \mathit{\delta}{\mathit{f}}_{\text{C}}=\mathbf{K}\mathit{\delta}{\mathit{D}}_{\text{C}}=\left[\begin{array}{ll}{K}_{\mathrm{11}}& {K}_{\mathrm{12}}\\ {K}_{\mathrm{21}}& {K}_{\mathrm{22}}\end{array}\right]\left\{\begin{array}{l}\mathrm{cos}{\mathit{\alpha}}_{\text{D}}\\ \mathrm{sin}{\mathit{\alpha}}_{\text{D}}\end{array}\right\}\mathit{\delta}{D}_{\text{C}}\end{array}$$

where *δ**D*_{C} indicates the displacement distance; *K*_{ij}
indicates elements of the compliant matrix **K** and *α*_{D}
indicates the angular separation of the disquisitive direction.

Thus, the directional stiffness of the end effector point *C* in the
*α*_{D} direction is defined as

$$\begin{array}{ll}\text{(21)}& {\displaystyle}{\mathbf{K}}_{Cd}& {\displaystyle}={\displaystyle \frac{\mathit{\delta}{\mathit{f}}_{\text{C}}}{\mathit{\delta}{\mathit{D}}_{\text{C}}}}={\displaystyle \frac{{\left[{\left(\mathit{\delta}{f}_{xC}\right)}^{\mathrm{2}}+{\left(\mathit{\delta}{f}_{yC}\right)}^{\mathrm{2}}\right]}^{\frac{\mathrm{1}}{\mathrm{2}}}}{\mathit{\delta}{D}_{\text{C}}}}{\displaystyle}& {\displaystyle}=\left({\mathrm{cos}}^{\mathrm{2}}{\mathit{\alpha}}_{\text{D}}\left({K}_{\mathrm{11}}^{\mathrm{2}}+{K}_{\mathrm{21}}^{\mathrm{2}}\right)+\mathrm{2}\mathrm{sin}{\mathit{\alpha}}_{\text{D}}\mathrm{cos}{\mathit{\alpha}}_{\text{D}}\right.\\ {\displaystyle}& {\displaystyle}{\left.\left({K}_{\mathrm{11}}{K}_{\mathrm{12}}+{K}_{\mathrm{21}}{K}_{\mathrm{22}}\right)+{\mathrm{sin}}^{\mathrm{2}}{\mathit{\alpha}}_{\text{D}}\left({K}_{\mathrm{12}}^{\mathrm{2}}+{K}_{\mathrm{22}}^{\mathrm{2}}\right)\right)}^{\frac{\mathrm{1}}{\mathrm{2}}}\end{array}$$

The radial stiffness K_{R} and
the circumferential stiffness K_{C} of the mechanism is
obtained by substituting into the corresponding directional angle values.

To investigate the stiffness constraint of the end effector of closed-chain
mechanism, the gravitation of each link also need to be taken into account.
And ${\mathit{V}}_{i}={\left({\dot{x}}_{{c}_{i}},{\dot{y}}_{{c}_{i}},{\mathit{\omega}}_{i}\right)}^{\text{T}}$ indicate the three-dimensional velocity of link *i*, in
which *c*_{i} represents center point of mass of each link, and since the
velocity relationship of two point on the same rigid body: ${\dot{x}}_{{c}_{i}}={\dot{x}}_{{O}_{i}}-{y}_{{c}_{i}}{\mathit{\omega}}_{i}$ and ${\dot{y}}_{{c}_{i}}={\dot{y}}_{{O}_{i}}+{x}_{{c}_{i}}{\mathit{\omega}}_{i}$, an expression of the velocity is
obtained as:

$$\begin{array}{}\text{(22)}& {\mathit{V}}_{i}={\mathbf{\Gamma}}_{i}{\mathit{T}}_{i}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}(i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4})\end{array}$$

where ${\mathbf{\Gamma}}_{i}=\left(\begin{array}{lll}\mathrm{1}& \mathrm{0}& -{y}_{{c}_{i}}\\ \mathrm{0}& \mathrm{1}& {x}_{{c}_{i}}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right)$ is the transformation matrix of analyzing positions;
${\mathit{T}}_{i}={\left({\dot{x}}_{{O}_{i}},{\dot{y}}_{{O}_{i}},{\mathit{\omega}}_{i}\right)}^{\text{T}}$ is the instantaneous motion of link *i*, and *O*_{i}
indicates the point attached to the link and instantaneously coincident with
the original point.

Each link's rotor *T*_{i} is related to the joints' angular motions:

$$\begin{array}{}\text{(23)}& {\mathit{T}}_{i}={\mathbf{S}}_{i}{\dot{\mathit{q}}}_{\text{L}}\phantom{\rule{0.25em}{0ex}}(i=\mathrm{1},\mathrm{2})\phantom{\rule{0.25em}{0ex}}\text{and}\phantom{\rule{0.25em}{0ex}}{\mathit{T}}_{i}={\mathbf{S}}_{i}{\dot{\mathit{q}}}_{\text{R}}\phantom{\rule{0.25em}{0ex}}(i=\mathrm{3},\mathrm{4})\end{array}$$

As what have been discussed before, ${\mathbf{J}}_{\text{L}}=({\mathbf{J}}_{\mathrm{1}},{\mathbf{J}}_{\mathrm{2}})$ is the Jacobian of the left chain, and likely,
${\mathbf{J}}_{\text{R}}=\left({\mathbf{J}}_{\mathrm{3}},{\mathbf{J}}_{\mathrm{4}},{\mathbf{J}}_{\mathrm{5}}\right)$ is the Jacobian of the right chain. **J**_{i} $(i=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4},\mathrm{5})$
are the vectors representing the mapping coefficient between velocity of the
end effector and the angular velocities of corresponding joints, and
${\mathbf{S}}_{\mathrm{1}}=({\mathbf{J}}_{\mathrm{1}},\mathbf{0}),\phantom{\rule{0.25em}{0ex}}{\mathbf{S}}_{\mathrm{2}}=({\mathbf{J}}_{\mathrm{1}},{\mathbf{J}}_{\mathrm{2}}),\phantom{\rule{0.25em}{0ex}}{\mathbf{S}}_{\mathrm{3}}=(\mathbf{0},{\mathbf{J}}_{\mathrm{4}},{\mathbf{J}}_{\mathrm{5}}),\phantom{\rule{0.25em}{0ex}}{\mathbf{S}}_{\mathrm{4}}=(\mathbf{0},\mathbf{0},{\mathbf{J}}_{\mathrm{5}})$.

Substituting Eqs. (21) into (20) gives

$$\begin{array}{}\text{(24)}& \left\{\begin{array}{l}{\mathit{V}}_{i}={\mathbf{\Gamma}}_{i}{\mathbf{S}}_{i}{\dot{\mathit{q}}}_{{}_{\text{L}}}={\mathbf{G}}_{i}{\dot{\mathit{q}}}_{\text{L}}\phantom{\rule{0.25em}{0ex}}(i=\mathrm{1},\mathrm{2})\\ {\mathit{V}}_{i}={\mathbf{\Gamma}}_{i}{\mathbf{S}}_{i}{\dot{\mathit{q}}}_{\text{R}}={\mathbf{\Gamma}}_{i}{\mathbf{S}}_{i}{\mathbf{J}}_{u}{\dot{\mathit{q}}}_{\text{L}}={\mathbf{G}}_{i}{\dot{\mathit{q}}}_{\text{L}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}(i=\mathrm{3},\mathrm{4})\end{array}\right.\end{array}$$

where **G**_{1}, ${\mathbf{G}}_{\mathrm{2}}\in {\mathrm{\Re}}^{\mathrm{3}\times \mathrm{2}}$,
**G**_{3}, ${\mathbf{G}}_{\mathrm{4}}\in {\mathrm{\Re}}^{\mathrm{3}\times \mathrm{3}}$.

4 Optimization

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Optimization of stiffness is widely concerned in mechanical designs, such as increasing the isotropy of stiffness (Shin et al., 2013; Hao and Li, 2016), maximizing stiffness in one direction (Lee et al., 2011; Hao, 2017), and maximizing stiffness in all directions (Shin et al., 2007). In this research, we primarily concern about the magnitude and anisotropy of stiffness matrices, and furthermore optimize the mechanical parameters to improve the performance.

This part presents a parameter optimization of the planar mechanism. The stiffness distribution of the compliant mechanism is mainly related to the geometric dimensions, stiffness of each compliant joints and initial configuration. In order to provide with enough support in radial direction when the user wears the exoskeleton and stands erectly, this mechanism is supposed to only have slight deformation so as not to cause discomfort. In addition, this exoskeleton should have a relatively flexible feature to assist the user in bending and stretching knees, such that an optimization is performed to locate a larger system radial stiffness whereas the circumferential stiffness is relatively smaller.

The genetic algorithms (GAs) was applied to optimize the design parameters.
GAs are global search methods inspired by natural selection, and operate with
populations of individuals of an optimization problem (Chi and Zhang, 2012).
These populations evolve toward better solutions, and individuals in a
population are evaluated about their fitness that reflects how well the
optimization problem been solved. In this section, we applied the GAs in the
determination of parameters that effects the stiffness in radial and
circumferential direction respectively denoted by K_{R} and
K_{C}. Since the initial configuration of the mechanism is
symmetric, *l*_{1}=*l*_{4}, *l*_{2}=*l*_{3}, ${\mathit{\theta}}_{\mathrm{1}}=\mathit{\pi}-{\mathit{\theta}}_{\mathrm{5}}$
and the length of the base link is 50 mm with the consideration of size of
human knees. Thus, we only need to optimize the ratio value of the length of
each link to that of base frame in one chain, stiffness of each joint and the
initial angle *θ*_{1}. Therefore, the optimization process, minimizing
the ratio (${x}_{\text{R}}^{\text{C}})$ of *K*_{C} and
*K*_{R}, can be expressed as:

$$\begin{array}{}\text{(25)}& {x}_{\text{R}}^{\text{C}}={\displaystyle \frac{{K}_{\text{C}}}{{K}_{\text{R}}}}=f({x}_{\mathrm{1}},{x}_{\mathrm{2}},{k}_{i},{\mathit{\theta}}_{\mathrm{1}})\end{array}$$

where *x*_{1}, *x*_{2} indicate the ratio value of link *l*_{1}, *l*_{2} to
*l*_{5}.

According to the magnitude of the knee part of the exoskeleton and the stiffness requirements, optimization function subjects to the following constraint conditions:

$$\left\{\begin{array}{l}\mathrm{1}\le {x}_{\mathrm{1}}\le \mathrm{1.5}\\ \mathrm{1}\le {x}_{\mathrm{2}}\le \mathrm{1.5}\\ \mathrm{20}\le {k}_{i}\le \mathrm{300}\\ \mathrm{85}\le {\mathit{\theta}}_{\mathrm{1}}\le \mathrm{100}\end{array}\right.$$

The workspace, determined by the design of the mechanism, is related to dimension of human knee joint. As shown in Figs. 5–6, a sample is given by a 23-year-old Asia female of 165 cm in height. According to the definition of the coordination system in the article (Andriacchi et al., 1998), the coordination system of tibia and femur is built, and a two-dimensional tibiofemoral joint model is obtained by using 3-D Guidance driveBAY and trakSTAR.

The workspace of the mechanism should cover the whole path of the measuring
point *P* as shown in Fig. 7. During movement, this mechanism deforms toward
one side, and the displacement of the end effector can be decomposed into
radial motion *x*_{R} and circumferential motion *x*_{C}.

The MATLAB GA toolbox is implemented to finish this optimization. After 482 generations, the objective function Eq. (23) is convergent to minimum point as shown in Fig. 8 in which the fitness value dropped from the initial 0.59, and the best variables are revealed in Table 1.

According to the results of optimization, the stiffness distribution
diagrams are obtained via MATLAB. As shown in Fig. 9, the stiffness in two
directions has large discrepancy initially, and the radial stiffness is far
larger than the circumferential stiffness when the user just put on the
exoskeleton. And the mechanism become softer in radial direction during leg
flexion because the corresponding stiffness gets smaller when loaded, such
that the mechanism become elastic in radial direction and can respond freely
to the applied force on the point *C*.

Moreover, when the end effector reaches up to relative higher position,
radial stiffness abruptly increase for the mechanism is near to the singular
configuration. This characteristic meets our expectation for the mechanism,
since the weight of the whole exoskeleton is 1.6 kg, and due to the braces
sharing the most of it, only a fraction of the load is applied on the
five-bar mechanism. Substituting the optimized results into Eq. (22),
we received the directional stiffness: *K*_{R}=0.568 N mm^{−1}, *K*_{C}=0.057 N mm^{−1}. Therefore, the radial
stiffness is large enough to support the initial load, and the smaller
circumferential stiffness can let users bend their legs smoothly. Through
optimizing these variables, the mechanism performance is improved.

5 Experiments

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This section presents the experimental verification of the self-aligning performance of knee exoskeleton. It is difficult to detect the explicit trajectory of both ICRs of human knee and exoskeleton joint externally during extension or flexion. When operating a traditional wearable exoskeleton, the user receives force from the exoskeleton through the braces (Akiyama et al., 2012; Zanotto et al., 2015), and the tangential component of this force is undesirable, since it is accompanied by the human/robot misalignment. Therefore, the tangential interaction force can be utilized to effectively assess the self-alignment ability of the exoskeleton. Moreover, a motion measurement is conducted to validate the adaptation of the exoskeleton.

We adopted the filtering criterion based on a simple ergonomics principle (Accoto et al., 2014): a correct interaction requires forces to be applied perpendicularly to the human body segments since tangential forces (shear forces), besides being ineffective for motion generation, may cause discomfort or even tissue damage. Therefore, in the verification of being able to mitigate the misalignment, it is not definitely required to identify the explicit ICR of the human knee or exoskeleton if the tangential interaction forces conform to ergonomics principle. As shown in Fig. 10, we conducted experiments on the subject. According to the results of the optimization, selection of parts and assembly of the five-bar mechanism are completed, and a linear motor is parallelly connected to the exoskeleton mechanism and set up a closed-chain structure. Then this whole exoskeleton is tightly attached to human thigh and shank with four braces at a natural standing position, in order to prevent links from oscillating in the experiments. A pressure sensor JLBS-MD is mounted on the exoskeleton, Since the misalignment will cause tangential interaction force thigh between human thigh and exoskeleton rod, this sensor is used to measure the tension and compression force which can reflect the extent of human/exoskeleton misalignment. The self-adjusting ability of the compliant five-bar mechanism is evaluated by analyzing the results measured by the sensor.

The first group (I and II) studies the process of sitting down and standing up, and the second group (III and IV) studies the actual walking condition.

The experiments of the mechanism are conducted in practical situations, and knee flexion is performed in experiment I in which the user sits down from a standing posture. To assess the performance objectively, experiment II is measured by using a traditional reference exoskeleton mounted with hinge knee joint. Experiment III is under the circumstance that the user extends knees and stands up from a sitting posture, and IV is the corresponding contrast experiment of it. As depicted in Fig. 11.

In Fig. 11, II and IV indicate that the interaction force at the position of
thigh brace have similar tendencies whether in extension or in flexion,
which suggests this experiment is unaffected by human factors and it is
irrelevant to motion forms. Moreover, when the knee angle is about
90^{∘}, oscillating scatters appear much more turbulent. This is
because the influence of friction between human leg and exoskeleton
overpowered the misalignment when near squatting position.

And in the tests of reference exoskeleton, trends of the interaction force shown in Fig. 11 II and IV indicate that thigh receives increasing pressure at the brace position during knee flexion, whereas the tendency converts to decreasing pressure and increasing tension during knee extension. One assumed reason for this is the distance amount between these two attachment area of the leg has larger range of variation than the exoskeleton does, and it becomes longer than the device during extension as well as it gets shorter than the device during flexion. In the calculation, the attachment area of the thigh endure dramatic displacement and force change in the traditional exoskeleton experiment. Furthermore, based on the comparison of the results of measuring force of the reference and self-aligning exoskeletons, it clearly shows that measured curve of the self-aligning exoskeleton is less turbulent and smoother than the reference one, and the mechanism also narrow the range of interacting force by more than 85 %. Therefore, this experiment validates the device can effectively attenuate the misalignment between human and exoskeleton, and the design could to a great extent provide with gentle and comfortable experience for users.

As shown in Fig. 12, the user finished four gaits in each experiment, in which V is conducted by using the proposed exoskeleton, and VI is the contrast test. Human/exoskeleton interaction force is much more turbulent and reaches to extreme values at the boundaries between (a) stance phase and (b) swing phase. Because at these position, the angle of human knee respectively reaches to the most exceeding misalignment case. In contrast, from the comparison of the maximum ranges of peak-to-peak, the force drawing by using proposed exoskeleton reflects the human gaits in a much flatter way. Thus, the ability of self-aligning the compliant five-bar mechanism knee joint performs better than the traditional knee.

Given that both of the tangential interaction force and relative movement between human and exoskeleton reflect the misalignment, to validate the self-alignment of the mechanism, motion adaptation also needs to be investigated. In the process of walking, a displacement sensor is installed on the thigh link of exoskeleton at point Q, which is initially coincident with the measuring point P on the subject mentioned in Fig. 6. Furthermore, according to the trajectory of point P, we track the moving path of the corresponding point Q on the exoskeleton and detect the gap between paths of these two measuring points. Measurements of force and motion are drawn in the same figure.

As shown in Fig. 14, fitting curves of these two trajectories of measuring points on the subject and exoskeleton are ultimately consistent, and apparently the gap value is very small, which is in accordance with the force fluctuation as the pressure is getting larger at the same period, which indicates that the proposed knee exoskeleton sufficiently adapts the movement of human knee joint. Meanwhile, the tangential interaction force is small and has little fluctuation. The proposed exoskeleton is able to self-align to human knees.

6 Conlcusion

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In order to reduce the misalignment problem in using the traditional lower extremity exoskeletons, we propose a novel design, which provides an implicit self-alignment to human knee joints, and such ability is related to the structural design and human loads. Such that the kinematic characteristics is analyzed and the stiffness matrix is obtained, which provides a necessary basis for further optimization. The experiment investigation, including motion and force tests, validates the self-alignment capacity of knee exoskeleton design.

Most recent designs of upper limb tend to introduce passive auxiliary mechanisms to alleviate misalignment at the elbow and shoulder joints. Nevertheless, such mechanisms might not be necessary for the case of lower extremities exoskeletons, provided that a parallel compliant mechanism is utilized to significantly reduce the effects of knee misalignment, especially on the tangential interaction forces at the thigh/brace interfaces, and this approach effectively guarantee the comfort and safety for users.

Data availability

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Data availability.

A portion of the underlying research data are presented in the text, and the rest cannot be shared at this time as the data also form part of an ongoing study.

Author contributions

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Author contributions.

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Project no. 51475322, 51535008, 51775367, 51721003), the Tianjin Municipal
Science and Technology Department Program (Grant no. 17JCZDJC30300), and the
international collaboration programme under Grant no. B16034.

Edited by: Guangbo Hao

Reviewed by: two anonymous referees

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Short summary

In rehabilitation, the widely existing misalignment between human limb and exoskeleton causes discomfort even danger. To alleviate the misalignment in lower extremity exoskeleton, this study proposes a compliant parallel mechanism, which has the potential to automatically change its configuration to make exoskeleton knee joint fit the human joint movements. And the self-alignment performance of the mechanism is validated via conducting a series of experiments under different working conditions.

In rehabilitation, the widely existing misalignment between human limb and exoskeleton causes...

Mechanical Sciences

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