# Conceptual Design of Spacesuit Hard Hip Joint

Authors: Wang Zhen-wei; Chen Peng; Lin Dan-lan
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

Spacesuit hip joint plays an important role on astronaut activities, such as planetary walking and surveying. This paper proposes a conceptual design of hard hip joint in consideration of the coupling effect of spacesuit hip joint and astronaut thigh. Firstly, lower extremity activities are introduced to illustrate the mobility of hard hip joint, such as walking, kneeing, and abduction. A conceptual design of hard hip joint is explained in detail, including geometric structure, components, design parameters, and mechanism models. Secondly, a 3-linkage coupling mechanism model is built up by synthesizing that conceptual design of hard hip joint. An equiangular dual-perpendicular representation method is brought out to parameterize that mechanism model of hard hip joint. Particularly, four geometric constraints are, respectively, given out to avoid impact between the hip joint and the thigh and to ensure the continuity of thigh motion. Finally, motion equations of hip joint parts are established by using coordinate transformation and vector representation. A case study is conducted to verify the correctness of the proposed representation method and that coupling mechanism model.


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## Body

## 1. Introduction

Hip joint is an important component in spacesuit, which enables mobility of astronauts wearing pressurized spacesuits to complete missions, including walking and surveying. Hip joint is briefly classified into soft hip joint and hard hip joint. Generally, hard hip joint is made of some rigid materials, such as aluminum alloy and stiffness composite. Soft hip joint is made of soft materials, such as nylon and fabric. General spacesuit is pressurized with gas to a certain pressure level making it stiff in the vacuum of space [1]. For soft hip joint, the stiffen spacesuit greatly increase unnecessary energy expenditure and impeded mobility. Additional factors also include change in suit column and fabric stiffness as the joints bend [2]. Relatively, hard hip joint can avoid above problems because deformation of rigid material caused by pressurized gas is very little. The major advantage of hard joint is that torque moment derived by bearing friction is less than other kinds of hip joint. Therefore, hard hip joint has been widely applied to some spacesuits, such as the Z series spacesuit [3] and Mark III spacesuit [4]. On the other hand, the primary problem of hard joint is that the placement of bearings causes programming and potentially unnatural movement and stances. Presently, it is one of technical bottlenecks.Spacesuit field involves many topics widely, such as mobility and agility [5], human-suit interaction [6], astronaut injury estimation [7], joint torque testing [8], human movements [9, 10], and gait simulation [11, 12]. Among those topics, hip joint mobility is one of the basic research focuses. Around joint mobility, many kinds of spacesuit joint had been developed, including flat-pattern joint, bellow-type joint, and rotation-bearing joint. Now, there are many research achievements in motion analysis and test [13, 14], computational methods [15, 16], human factors [17, 18], and joint development [19, 20]. As far as joint design is concerned, there is relatively less research on the improvement of hard joint design for many years. With the development of planetary explorations, a new spacesuit design is becoming more important, such as Mars exploration and lunar landing.Spacesuit is different from other classical spacecraft, which is characterized as the human-suit interactive spacecraft. As for spacesuit hip joint, there exists a strong coupling effect between hip joint and astronaut thigh. The coupling effect is embodied by impact, rubbing, etc. Therefore, the conceptual design of hip joint should pay attention to above coupling effect. To solve above problem, this paper brings forward an equiangular dual-perpendicular representation method and two related geometric constraints. Moreover, a 3-linkage coupling mechanism model is constructed based on a conceptual model of the hip joint. That coupling mechanism model is helpful for parametric design and motion analysis. To ensure the motion continuity of the joint and thigh, two additional geometric constraints are brought out and discussed in detail. Furthermore, motion equations of hip joint part are established using coordinate transformation and vector representation. Finally, a case study is conducted to verify the correctness of the proposed representation method and coupling mechanism model of the hip joint.


## 2. Conceptual Design of Hard Hip Joint

### 2.1. Conceptual Model of Hip Joint

Four basic lower extremity activities of designed spacesuit are illustrated in Figure1, including standing, walking, kneeing, and abduction. Those activities require the hip joint to be mobile and flexible. Although four activities appear different, they can be generalized as hip joint motion based on kinematics. Namely, every activity corresponds to specific rotation of the hip joint, but their rotational angles are different from each other. Also, Figure 1 shows some joints and components in designed spacesuit. This paper only discusses the hip joint, where it is made of aluminum alloy and designed to connect the waist part with the thigh part. When a suited astronaut moves the thigh, related hip joint will be driven to move synchronously. Thus, low extremity activities can be performed. Around the hip joint, the following sections will discuss its components, geometric structure, and design parameters.Figure 1
Four basic lower extremity activities of designed spacesuit.Conceptual models of the hip joint and its components are constructed, as shown in Figure2. From a view of mechanism design, the hip joint can be regarded as a rigid assembly with 3 degrees-of-freedom, where the hip joint consists of briefs part, right hip joint, left hip joint, and rotation bearings. Bearings are not displayed due to no effect on conceptual design. The structure of left and right hip joint is designed to be the same. It is composed of two parts—upper part and lower part. Both parts are connected together by a rotation bearing. Similarly, briefs part is, respectively, connected with two hip joint parts by rotation bearings. Due to rotation bearings, low part and high part can rotate independently. In other words, thigh motion can be decomposed into relative rotations of upper and lower parts. That is the basic thought for the conceptual design of the hip joint.Figure 2
Conceptual models of the (a) hip joint and (b) its components.


### 2.2. Geometric Structure of Hip Joint Part

To reduce manufacturing cost, a similar geometric structure is used to design upper part and lower part, as shown in Figure3. As a whole, it is a shell body with uniform thickness of 5 mm. With neglect of thickness, the geometric structure is formed by two tangent circles with inclined angle ai and one ruled surface sweeping from upper circle to lower circle. Where P is the tangent point of two circles, Ol and Ou are the corresponding center point of lower circle and upper circle. Through those three points, a central plane is constructed to parameterize joint part. Moreover, a local coordinate frame xiyizi is also established to express the geometric shape of joint part. In a central plane, the geometric structure can be represented by parameters ri, ri+1, and ai completely, where ri and ri+1 are the radius of upper circle and lower circle, respectively. Subscript i is the index of joint part, i.e., i=1 or 2, and number 1 stands for upper part. These parameters are determined according to design requirements.Figure 3
Geometric structure and parameters of (a) joint part and (b) link.
(a)
(b)Due to large stiffness, hip joint part is treated as a rigid body theoretically. Thus, the geometric structure can be synthesized to be one link from pointOl to point Ou based on the mechanism theory. Moreover, a rotation joint is placed at point Ol to connect other joint part. Generally, link is not required to be perpendicular to two circles. On the contrary, rotation joint must be vertical to lower circle to ensure relative rotation of hip joint part. As a result, upper and lower parts can be, respectively, represented as links and parameterized by sets {r1, r2, and a1} and {r2, r3, and a2}. Considering rigidity of human bone, we also simplify thigh as one link with a spherical joint. Although this simplification causes error, it is helpful for analytical solution of joint motion. Above all, joint parts and thigh can be regarded as link under a condition of enough stiffness.


### 2.3. Coupling Mechanism of Hip Joint Assembly

By combining the thigh with the hip joint, a hip joint assembly is constructed, as shown in Figure4(a). It consists of astronaut torso, thigh, briefs part, and hip joint. Generally, spacesuit mounted on astronaut should be by internal textile belt. A fixed belt in the waist joint is used to restrict lateral location. Thus, briefs part location is deterministic related to astronaut. The location limit is not our research objective. We do not discuss it anymore.Figure 4
Models of (a) hip joint assembly (b) 3-linkage coupling mechanism.
(a)
(b)By assembling above links, a 3-linkage coupling mechanism model of hip joint assembly can be built up, as shown in Figure4(b). It consists of three links, one cylindrical joint, two rotation joints, and ground base, where links 1, 2, and 3 correspond to upper part, lower part, and thigh, respectively. Astronaut torso is modeled as ground base. Link 1 connects with ground base by rotation joint 1 and with link 2 by rotation joint 2. Relative rotation of link 1 and link 2 can be measured by angles θ1 and θ2. Particularly, link 3 is connected with link 2 by cylindrical joint and with ground base by spherical joint. In order to ensure motion consistency between human thigh and hip joint, link 3 needs to be located at original point o. In other words, spherical joint and rotational joint 1 are coincident to original point o. All links are measured and expressed in reference coordinate frame xyz. More details about coordinate frame xyz will be discussed in Section 3.2.


## 2.1. Conceptual Model of Hip Joint

Four basic lower extremity activities of designed spacesuit are illustrated in Figure1, including standing, walking, kneeing, and abduction. Those activities require the hip joint to be mobile and flexible. Although four activities appear different, they can be generalized as hip joint motion based on kinematics. Namely, every activity corresponds to specific rotation of the hip joint, but their rotational angles are different from each other. Also, Figure 1 shows some joints and components in designed spacesuit. This paper only discusses the hip joint, where it is made of aluminum alloy and designed to connect the waist part with the thigh part. When a suited astronaut moves the thigh, related hip joint will be driven to move synchronously. Thus, low extremity activities can be performed. Around the hip joint, the following sections will discuss its components, geometric structure, and design parameters.Figure 1
Four basic lower extremity activities of designed spacesuit.Conceptual models of the hip joint and its components are constructed, as shown in Figure2. From a view of mechanism design, the hip joint can be regarded as a rigid assembly with 3 degrees-of-freedom, where the hip joint consists of briefs part, right hip joint, left hip joint, and rotation bearings. Bearings are not displayed due to no effect on conceptual design. The structure of left and right hip joint is designed to be the same. It is composed of two parts—upper part and lower part. Both parts are connected together by a rotation bearing. Similarly, briefs part is, respectively, connected with two hip joint parts by rotation bearings. Due to rotation bearings, low part and high part can rotate independently. In other words, thigh motion can be decomposed into relative rotations of upper and lower parts. That is the basic thought for the conceptual design of the hip joint.Figure 2
Conceptual models of the (a) hip joint and (b) its components.


## 2.2. Geometric Structure of Hip Joint Part

To reduce manufacturing cost, a similar geometric structure is used to design upper part and lower part, as shown in Figure3. As a whole, it is a shell body with uniform thickness of 5 mm. With neglect of thickness, the geometric structure is formed by two tangent circles with inclined angle ai and one ruled surface sweeping from upper circle to lower circle. Where P is the tangent point of two circles, Ol and Ou are the corresponding center point of lower circle and upper circle. Through those three points, a central plane is constructed to parameterize joint part. Moreover, a local coordinate frame xiyizi is also established to express the geometric shape of joint part. In a central plane, the geometric structure can be represented by parameters ri, ri+1, and ai completely, where ri and ri+1 are the radius of upper circle and lower circle, respectively. Subscript i is the index of joint part, i.e., i=1 or 2, and number 1 stands for upper part. These parameters are determined according to design requirements.Figure 3
Geometric structure and parameters of (a) joint part and (b) link.
(a)
(b)Due to large stiffness, hip joint part is treated as a rigid body theoretically. Thus, the geometric structure can be synthesized to be one link from pointOl to point Ou based on the mechanism theory. Moreover, a rotation joint is placed at point Ol to connect other joint part. Generally, link is not required to be perpendicular to two circles. On the contrary, rotation joint must be vertical to lower circle to ensure relative rotation of hip joint part. As a result, upper and lower parts can be, respectively, represented as links and parameterized by sets {r1, r2, and a1} and {r2, r3, and a2}. Considering rigidity of human bone, we also simplify thigh as one link with a spherical joint. Although this simplification causes error, it is helpful for analytical solution of joint motion. Above all, joint parts and thigh can be regarded as link under a condition of enough stiffness.


## 2.3. Coupling Mechanism of Hip Joint Assembly

By combining the thigh with the hip joint, a hip joint assembly is constructed, as shown in Figure4(a). It consists of astronaut torso, thigh, briefs part, and hip joint. Generally, spacesuit mounted on astronaut should be by internal textile belt. A fixed belt in the waist joint is used to restrict lateral location. Thus, briefs part location is deterministic related to astronaut. The location limit is not our research objective. We do not discuss it anymore.Figure 4
Models of (a) hip joint assembly (b) 3-linkage coupling mechanism.
(a)
(b)By assembling above links, a 3-linkage coupling mechanism model of hip joint assembly can be built up, as shown in Figure4(b). It consists of three links, one cylindrical joint, two rotation joints, and ground base, where links 1, 2, and 3 correspond to upper part, lower part, and thigh, respectively. Astronaut torso is modeled as ground base. Link 1 connects with ground base by rotation joint 1 and with link 2 by rotation joint 2. Relative rotation of link 1 and link 2 can be measured by angles θ1 and θ2. Particularly, link 3 is connected with link 2 by cylindrical joint and with ground base by spherical joint. In order to ensure motion consistency between human thigh and hip joint, link 3 needs to be located at original point o. In other words, spherical joint and rotational joint 1 are coincident to original point o. All links are measured and expressed in reference coordinate frame xyz. More details about coordinate frame xyz will be discussed in Section 3.2.


## 3. Parameter Calculation of Hip Joint

### 3.1. Parameterization of Hip Joint

Generally, there are many requirements on spacesuit, such as motion, safety, strength, and weight. This paper is mainly aimed at impacting and motioning continuity. To solve above two problems, an equiangular dual-perpendicular method is firstly brought out to parameterize a conceptual model of hard hip joint, as illustrated in Figure5, where upper part and lower part are aligned with their central planes. Initial angles of two joint parts equal to zero, θ1=θ2=0, where links 1, 2, and 3 are represented by segment lines JC, CE, and JE, respectively. It can be seen that lengths of segment lines PJ, PC, and PE are equal to radiuses r1, r2, and r3, respectively. Next, we will introduce four geometric constraints in order to avoid impact and ensure motion continuity.Figure 5
An equiangular dual-perpendicular method.To avoid impact between the hip joint and the thigh, two geometric constraints must be satisfied. The first constraint is that both segment linesJE and JC are, respectively, perpendicular to segment lines PE and PC. The second constraint is that segment line PC is a diagonal line of angle ∠JPE. It can be concluded that above constraints enable segment line JE to be perpendicular to the lower circle of lower part all the time. In other words, impact will not happen between the hip joint and the thigh. Moreover, above two constraints cause inclined angle a1 to be equal to inclined angle a2, a1=a2.In order to ensure continuous motion of the thigh, two geometric constraints are proposed in addition. The first constraint is that segment lineCE and CD is symmetric about segment line CJ. The second constraint is that point D must lie in coordinate axis z. Under condition of above two constraints, motion continuity means that segment line JE can rotate around point J freely. Namely, segment line JE is a fixed point rotation. According to kinematics, rotation around fixed point J can be discomposed into a planar rotation around point J and a rotation around coordinate axis z. Obviously, the latter rotation is continuous. So, main difficulty is how to realize the continuity of planar rotation. As mentioned above, two additional constraints enable segment line JE to rotate along planar arc lr. As a result, motion continuity is realizable according to above geometric constraint.Above all, those problems can be solved by introducing four geometric constraints, including two perpendicular constraints, one symmetric constraint and one coincident constraint.Based on above four constraints, structure parameters of hip joint and briefs part can be obtained. Next, we derive the structure parameters of hip joint. In Figure5, parameters r1 are determined by design requirement and parameter αd is given as the angle limit of thigh motion. Both parameters are known. Through geometric analysis, it is concluded that parameters r2 and r3 can represent the geometric structure of hip joint completely. Thus, the main work is to calculate points E and C. In a reference coordinate system xyz, coordinates of points E and C can be obtained by using vector representation, as shown as equation (1), where trigonometric functions cosine() and sine() are, respectively, abbreviated as letters c and s. For example, sinαd=sαd. Both abbreviated letters s and c will be used in following equations.
(1)rC=xCyCzC=−r1sαd420r12sαd2,rE=xEyEzE=−r1sαd420r12sαd.


### 3.2. Parameterization of Briefs Part

Intuitively, the geometric structure of briefs part is a spherical shell. Neglecting its shell thickness, it consists of three tangent circles, as shown in Figure6. Those circles correspond to connecting interfaces with waist part, left hip joint, and right hip joint, respectively. Points T1, T2, and T3 are tangent points. Moreover, two global coordinate frames XYZ and X′Y′Z′ are, respectively, established in the center points of waist part and spherical shell. Left and right circles are designed to have the same inclined angle αs. Also, a reference coordinate frame xyz is established by three points T1, Oj, and T3. Angle between coordinate planes YZ and yz is measured by parameter βs. From Figure 6, it can be seen that structure parameters of briefs part include the radius rs of spherical shell and two angle parameters αs and βs.Figure 6
Model and parameters of briefs part.Among above three parameters, parametersαs and αs are known because they can be determined by design requirements. Therefore, two radiuses rw and rs are needed to be calculated, where parameter rw is the radius of the waist part. From View A-A in Figure 6, radius rs can be obtained, as shown as
(2)rs=r1sαs.In global coordinate frameX′Y′Z′, position vector RS of any point S in the left circle of briefs part can be obtained by vector algebra, as shown as
(3)RS=r1cαs2sαs−r1sαssγsβs−r1cβscγ−r1cαssγ+1r1cαs2sαs−r1sαssγcβs+r1sβscγ,where parameter γ is the rotation angle of point S about coordinate axis x. According to tangent constraint between waist circle and left circle, angle γ can be obtained by coordinate transformation, as shown as
(4)tanγ=−sαscβssβs.By combining equations (3) and (4), position vector RS of any point S in global coordinate frame X′Y′Z′ can be rewritten as
(5)RS=r1cαs2cβs1sαs+cβs2sβs2+sαscβs2−r1sαscαs1sαs+cβs2sβs2+sαscβs2r1cβscαs2sαs−sβscβs2+sαs2.Furthermore, radiusrw can be calculated by using coordinates x and y of vector RS, as shown as
(6)rw=r1cαs1sαs+cβs2sβs2+sαscβs2×sβscαs2+sαs2.From equations (1) and (5), it can be seen that parameters of hip joint and briefs part are analytically calculated under the condition of given parameters r1, αd, αs, and βs. It is known that expressing a vector in different coordinate frame may bring more convenience for calculation. This paper derives translation vector Ro and transformation matrix Mo between coordinate frames xyz and XYZ, where parameter Ro is the position vector of center point J in reference coordinate frame xyz and parameter Mo is a transformation matrix between coordinate frames xyz and XYZ. Through geometric analysis, both parameters Ro and Mo are shown as
(7)Ro=r1cαs2sβssαs−r1cαsr1sβs2+sαscβs2,Mo=−cβs0sβs−sαssβs−cαs−sαscβscαssβs−sαscαscβs.


## 3.1. Parameterization of Hip Joint

Generally, there are many requirements on spacesuit, such as motion, safety, strength, and weight. This paper is mainly aimed at impacting and motioning continuity. To solve above two problems, an equiangular dual-perpendicular method is firstly brought out to parameterize a conceptual model of hard hip joint, as illustrated in Figure5, where upper part and lower part are aligned with their central planes. Initial angles of two joint parts equal to zero, θ1=θ2=0, where links 1, 2, and 3 are represented by segment lines JC, CE, and JE, respectively. It can be seen that lengths of segment lines PJ, PC, and PE are equal to radiuses r1, r2, and r3, respectively. Next, we will introduce four geometric constraints in order to avoid impact and ensure motion continuity.Figure 5
An equiangular dual-perpendicular method.To avoid impact between the hip joint and the thigh, two geometric constraints must be satisfied. The first constraint is that both segment linesJE and JC are, respectively, perpendicular to segment lines PE and PC. The second constraint is that segment line PC is a diagonal line of angle ∠JPE. It can be concluded that above constraints enable segment line JE to be perpendicular to the lower circle of lower part all the time. In other words, impact will not happen between the hip joint and the thigh. Moreover, above two constraints cause inclined angle a1 to be equal to inclined angle a2, a1=a2.In order to ensure continuous motion of the thigh, two geometric constraints are proposed in addition. The first constraint is that segment lineCE and CD is symmetric about segment line CJ. The second constraint is that point D must lie in coordinate axis z. Under condition of above two constraints, motion continuity means that segment line JE can rotate around point J freely. Namely, segment line JE is a fixed point rotation. According to kinematics, rotation around fixed point J can be discomposed into a planar rotation around point J and a rotation around coordinate axis z. Obviously, the latter rotation is continuous. So, main difficulty is how to realize the continuity of planar rotation. As mentioned above, two additional constraints enable segment line JE to rotate along planar arc lr. As a result, motion continuity is realizable according to above geometric constraint.Above all, those problems can be solved by introducing four geometric constraints, including two perpendicular constraints, one symmetric constraint and one coincident constraint.Based on above four constraints, structure parameters of hip joint and briefs part can be obtained. Next, we derive the structure parameters of hip joint. In Figure5, parameters r1 are determined by design requirement and parameter αd is given as the angle limit of thigh motion. Both parameters are known. Through geometric analysis, it is concluded that parameters r2 and r3 can represent the geometric structure of hip joint completely. Thus, the main work is to calculate points E and C. In a reference coordinate system xyz, coordinates of points E and C can be obtained by using vector representation, as shown as equation (1), where trigonometric functions cosine() and sine() are, respectively, abbreviated as letters c and s. For example, sinαd=sαd. Both abbreviated letters s and c will be used in following equations.
(1)rC=xCyCzC=−r1sαd420r12sαd2,rE=xEyEzE=−r1sαd420r12sαd.


## 3.2. Parameterization of Briefs Part

Intuitively, the geometric structure of briefs part is a spherical shell. Neglecting its shell thickness, it consists of three tangent circles, as shown in Figure6. Those circles correspond to connecting interfaces with waist part, left hip joint, and right hip joint, respectively. Points T1, T2, and T3 are tangent points. Moreover, two global coordinate frames XYZ and X′Y′Z′ are, respectively, established in the center points of waist part and spherical shell. Left and right circles are designed to have the same inclined angle αs. Also, a reference coordinate frame xyz is established by three points T1, Oj, and T3. Angle between coordinate planes YZ and yz is measured by parameter βs. From Figure 6, it can be seen that structure parameters of briefs part include the radius rs of spherical shell and two angle parameters αs and βs.Figure 6
Model and parameters of briefs part.Among above three parameters, parametersαs and αs are known because they can be determined by design requirements. Therefore, two radiuses rw and rs are needed to be calculated, where parameter rw is the radius of the waist part. From View A-A in Figure 6, radius rs can be obtained, as shown as
(2)rs=r1sαs.In global coordinate frameX′Y′Z′, position vector RS of any point S in the left circle of briefs part can be obtained by vector algebra, as shown as
(3)RS=r1cαs2sαs−r1sαssγsβs−r1cβscγ−r1cαssγ+1r1cαs2sαs−r1sαssγcβs+r1sβscγ,where parameter γ is the rotation angle of point S about coordinate axis x. According to tangent constraint between waist circle and left circle, angle γ can be obtained by coordinate transformation, as shown as
(4)tanγ=−sαscβssβs.By combining equations (3) and (4), position vector RS of any point S in global coordinate frame X′Y′Z′ can be rewritten as
(5)RS=r1cαs2cβs1sαs+cβs2sβs2+sαscβs2−r1sαscαs1sαs+cβs2sβs2+sαscβs2r1cβscαs2sαs−sβscβs2+sαs2.Furthermore, radiusrw can be calculated by using coordinates x and y of vector RS, as shown as
(6)rw=r1cαs1sαs+cβs2sβs2+sαscβs2×sβscαs2+sαs2.From equations (1) and (5), it can be seen that parameters of hip joint and briefs part are analytically calculated under the condition of given parameters r1, αd, αs, and βs. It is known that expressing a vector in different coordinate frame may bring more convenience for calculation. This paper derives translation vector Ro and transformation matrix Mo between coordinate frames xyz and XYZ, where parameter Ro is the position vector of center point J in reference coordinate frame xyz and parameter Mo is a transformation matrix between coordinate frames xyz and XYZ. Through geometric analysis, both parameters Ro and Mo are shown as
(7)Ro=r1cαs2sβssαs−r1cαsr1sβs2+sαscβs2,Mo=−cβs0sβs−sαssβs−cαs−sαscβscαssβs−sαscαscβs.


## 4. Motion Equations of Hip Joint

### 4.1. Coupling Mechanism Model of Hip Joint Assembly

Firstly, we try to construct the coupling mechanism model of hip joint assembly in global coordinate frameXYZ. In equations (1), (3), and (5), those vectors are derived in reference coordinate frame xyz. However, many gait experiments measure thigh motion in coordinate frame XYZ. To establish motion equation correctly, we must represent hip joint assembly in coordinate frame XYZ. By assembling hip joint with briefs part, a global coupling mechanism model is constructed in coordinate frame XYZ, as shown in Figure 7(b). It looks like a four-linkage mechanism, where briefs part is regarded as a link from origin point O to center point o. Briefs link is fixed in coordinate frame XYZ. To express thigh motion, we define unit vector RT to represent link 3 in global coordinate frame XYZ, as shown in Figure 7(a), where link 3 is measured by angle parameters φ1 and φ2. Similarly, two parameters θd and θu represent relative rotation of link 2 and link 1, respectively.Figure 7
Mechanism models of (a) thigh and (b) hip joint assembly.
(a)
(b)


### 4.2. Motion Equations of Hip Joints

Next, we will derive motion equations of upper part (link 1) and lower part (link 2) in coordinate framexyz. More details are shown in Figure 5, where unit vector Rt is illustrated to represent segment line JE. Obviously, it can be measured by two parameters θ and ψ. For a given angle θ, segment line JE moves to segment line JG. From point G, a parallel line GH is drawn to intersect line ED at point H. Point W is the projection of point G in coordinate plane x1y1. Then, circle C1 is formed through point W. From point H, vertical line HM intersects with circle C1 at points K and M. It is concluded that rotation of upper part can be measured by parameter θ1=∠WoM. In View A, motion trajectory of lower part can be expressed by circle C2. Then, vertical line HK is drawn to intersect with circle C2 at point K. Relative rotation of lower part can be calculated by parameter θ2=∠Ho2K.From circleC1, it can be seen that length lk of line Ko1 equals to length lw of line Wo1. In coordinate frame xyz, coordinates of points K and G can be obtained in a form of vectors RK and RG, as shown as
(8)RK=−r1sαd/222cθ2+1r1sαd2sαd4sθ2−r1sαd2sαd42cθ2+cαd42,RG=−r1sαd2sθ0r1sαd2cθ.Because oflk=lw, parameters θ1 and θ2 can be derived, as shown as
(9)sθ1=1sαd/4cθ−cαd/21+cθ,cθ2=cαd/42−cθsαd/42,θ∈−αd20,sθ1=π−1sαd/4cθ−cαd/21+cθ,cθ2=2π−1+cαd/42cθsαd/42,θ∈0αd2.Secondly, we will derive thigh motion equation in coordinate framexyz. In Figure 7, unit vector RT can be written as equation (10) in coordinate frame XYZ.
(10)RT=sφ2sφ1−sφ2cφ1cφ2.By coordinate transformation, unit vectorRT can be expressed as vector Rt in coordinate frame xyz, as shown as
(11)Rt=MoRT.By substituting equations (7) and (10) into equation (11), vector Rt can be rewritten as
(12)Rt=sβscφ2−cβssφ1sφ2cαscφ1−sαssβssφ1sφ2−sαscβscφ2sαscφ1+cαssβssφ1sφ2+cαscβscφ2.From Figure5, it can be seen that vector RT can also be written as vector Rts in a form of parameters θ and ψ.
(13)Rts=−sθcψ−sθsψcθ.According toRt=Rts, parameters θ and ψ can be calculated by
(14)tanψ=cαscφ1−sαssβssφ1sφ2−sαscβscφ2sβscφ2−cβssφ1sφ2,cosθ=cαssβssφ1+sαscφ1sφ2+cαscβscφ2.Based on the relationship between planar rotation and fixed axis rotation, parametersθd and θu can be derived in coordinate frame XYZ, as shown as
(15)θu=θ1+ψ,θd=θ2.Above all, motion equations (equation (15)) of lower part and upper part have been established analytically. Of course, thigh status (θ, ψ) can be calculated under condition of given ration angles (θd, θu) of hip joint.For convenience of programming, computation procedure is given to solve design parameters and motion equations based on known conditions, as shown in Figure8, where parameters r1, αd, αs, and βs are known as mentioned before. Parameters rc, rE, and rw, respectively, represent the geometric structure of hip joint parts and briefs part. Parameters θu and θd are the rotation angle of upper part and lower part. Parameters φ1 and φ2 express thigh motion, which are input for equation (15). Firstly, we determine parameters r1, αd, αs, and βs according to design requirement. Then, using equations (10) and (15) solve hip joint motion for any given thigh parameters. Finally, structural parameters are calculated by equations (1) and (6).Figure 8
Computation procedure.


### 4.3. A Case Study

Based on above computation procedure, related program was developed to carry out a case study, where design parameters are set tor1=113mm, αd=80°, αs=20°, and βs=5°. Thigh data had been downloaded from the website [21]. Only flexion/extension data are adopted. According to above computation procedure, motion equations of hip joint can be computed and plotted in Figure 9. It can be seen that both curves of parameters θu and θd are smooth and continuous. They are consistent with theoretical expect. Above all, our proposed representation method and coupling model are correct and feasible.Figure 9
Motion curves of upper part (green solid curve) and lower part (blue solid curve) and thigh (red solid curve).


## 4.1. Coupling Mechanism Model of Hip Joint Assembly

Firstly, we try to construct the coupling mechanism model of hip joint assembly in global coordinate frameXYZ. In equations (1), (3), and (5), those vectors are derived in reference coordinate frame xyz. However, many gait experiments measure thigh motion in coordinate frame XYZ. To establish motion equation correctly, we must represent hip joint assembly in coordinate frame XYZ. By assembling hip joint with briefs part, a global coupling mechanism model is constructed in coordinate frame XYZ, as shown in Figure 7(b). It looks like a four-linkage mechanism, where briefs part is regarded as a link from origin point O to center point o. Briefs link is fixed in coordinate frame XYZ. To express thigh motion, we define unit vector RT to represent link 3 in global coordinate frame XYZ, as shown in Figure 7(a), where link 3 is measured by angle parameters φ1 and φ2. Similarly, two parameters θd and θu represent relative rotation of link 2 and link 1, respectively.Figure 7
Mechanism models of (a) thigh and (b) hip joint assembly.
(a)
(b)


## 4.2. Motion Equations of Hip Joints

Next, we will derive motion equations of upper part (link 1) and lower part (link 2) in coordinate framexyz. More details are shown in Figure 5, where unit vector Rt is illustrated to represent segment line JE. Obviously, it can be measured by two parameters θ and ψ. For a given angle θ, segment line JE moves to segment line JG. From point G, a parallel line GH is drawn to intersect line ED at point H. Point W is the projection of point G in coordinate plane x1y1. Then, circle C1 is formed through point W. From point H, vertical line HM intersects with circle C1 at points K and M. It is concluded that rotation of upper part can be measured by parameter θ1=∠WoM. In View A, motion trajectory of lower part can be expressed by circle C2. Then, vertical line HK is drawn to intersect with circle C2 at point K. Relative rotation of lower part can be calculated by parameter θ2=∠Ho2K.From circleC1, it can be seen that length lk of line Ko1 equals to length lw of line Wo1. In coordinate frame xyz, coordinates of points K and G can be obtained in a form of vectors RK and RG, as shown as
(8)RK=−r1sαd/222cθ2+1r1sαd2sαd4sθ2−r1sαd2sαd42cθ2+cαd42,RG=−r1sαd2sθ0r1sαd2cθ.Because oflk=lw, parameters θ1 and θ2 can be derived, as shown as
(9)sθ1=1sαd/4cθ−cαd/21+cθ,cθ2=cαd/42−cθsαd/42,θ∈−αd20,sθ1=π−1sαd/4cθ−cαd/21+cθ,cθ2=2π−1+cαd/42cθsαd/42,θ∈0αd2.Secondly, we will derive thigh motion equation in coordinate framexyz. In Figure 7, unit vector RT can be written as equation (10) in coordinate frame XYZ.
(10)RT=sφ2sφ1−sφ2cφ1cφ2.By coordinate transformation, unit vectorRT can be expressed as vector Rt in coordinate frame xyz, as shown as
(11)Rt=MoRT.By substituting equations (7) and (10) into equation (11), vector Rt can be rewritten as
(12)Rt=sβscφ2−cβssφ1sφ2cαscφ1−sαssβssφ1sφ2−sαscβscφ2sαscφ1+cαssβssφ1sφ2+cαscβscφ2.From Figure5, it can be seen that vector RT can also be written as vector Rts in a form of parameters θ and ψ.
(13)Rts=−sθcψ−sθsψcθ.According toRt=Rts, parameters θ and ψ can be calculated by
(14)tanψ=cαscφ1−sαssβssφ1sφ2−sαscβscφ2sβscφ2−cβssφ1sφ2,cosθ=cαssβssφ1+sαscφ1sφ2+cαscβscφ2.Based on the relationship between planar rotation and fixed axis rotation, parametersθd and θu can be derived in coordinate frame XYZ, as shown as
(15)θu=θ1+ψ,θd=θ2.Above all, motion equations (equation (15)) of lower part and upper part have been established analytically. Of course, thigh status (θ, ψ) can be calculated under condition of given ration angles (θd, θu) of hip joint.For convenience of programming, computation procedure is given to solve design parameters and motion equations based on known conditions, as shown in Figure8, where parameters r1, αd, αs, and βs are known as mentioned before. Parameters rc, rE, and rw, respectively, represent the geometric structure of hip joint parts and briefs part. Parameters θu and θd are the rotation angle of upper part and lower part. Parameters φ1 and φ2 express thigh motion, which are input for equation (15). Firstly, we determine parameters r1, αd, αs, and βs according to design requirement. Then, using equations (10) and (15) solve hip joint motion for any given thigh parameters. Finally, structural parameters are calculated by equations (1) and (6).Figure 8
Computation procedure.


## 4.3. A Case Study

Based on above computation procedure, related program was developed to carry out a case study, where design parameters are set tor1=113mm, αd=80°, αs=20°, and βs=5°. Thigh data had been downloaded from the website [21]. Only flexion/extension data are adopted. According to above computation procedure, motion equations of hip joint can be computed and plotted in Figure 9. It can be seen that both curves of parameters θu and θd are smooth and continuous. They are consistent with theoretical expect. Above all, our proposed representation method and coupling model are correct and feasible.Figure 9
Motion curves of upper part (green solid curve) and lower part (blue solid curve) and thigh (red solid curve).


## 5. Conclusion

Around a coupling effect of hip joint and thigh, this paper proposes an equiangular dual-perpendicular method to design a conceptual model of hip joint. Geometric structure and coupling constraints are discussed in detail. Specifically speaking, two perpendicular constraints are given out to avoid impact between hip joint and thigh. Symmetric and coincident constraints are brought out to realize thigh continuous motion. Based on above constraints, a 3-linkage coupling mechanism model is built up. Motion equations of hip joint are derived by using coordinate transformation and vector representation. Meanwhile, related computation procedure is formed to solve structural parameters and motion parameters, respectively. Finally, a case study is conducted to verify the proposed representation method and motion equations. Results show that the conceptual design of hip joint is correct and feasible.



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*Source: 3742123-2019-09-17.xml*