Design Optimization for Parallel Mechanism Using on Human Hip ...

2012 IEEE International Conference on Robotics and Automation RiverCentre, Saint Paul, Minnesota, USA May 14-18, 2012

Design Optimization for Parallel Mechanism Using on Human Hip Joint Power Assisting Based on Manipulability Inclusive Principle Yong Yu* and Wenyuan Liang Abstract— This paper summarizes the design optimization of a parallel mechanism using on human hip joint power assisting. Manipulability Inclusive Principle (MIP) evaluation criterion for evaluating assisting mechanism’s assisting feasibility and assisting effect is proposed. The design of parallel assisting mechanism and building kinematical Jacobian are discussed. Moreover, as an important part of this paper, in order to finding out a architecture, which can satisfy assisting feasibility and realize higher assisting efficiency, more assisting ability and better feature on assisting isotropy, design optimization MIP is shown in this paper. Index Terms— Parallel Mechanism; Power Assisting Robot; Mechanical kinematical Jacobian; Manipulability;

I. INTRODUCTION Currently, there are two major types of power assisting robot [1]-[2]. The first type is desktop assist system, and the other type is wearable assisting system. Compared to desktop assisting system, wearable power assisting robot becomes an attractive method of power assisting since it can be adapted to a wide range of applications. Power assisting robots, which are usually conceived as systems including upper extremities, lower extremities, or both, is also called as exoskeletons, where Hardiman project [3], BLEEX [4], Wearable Power Suit [5], HAL [6] and so on are such famous robots among present research. In our research [7], the assisting robot focus solely on human hip joint 3-DOF power assisting. In order to realize hip joint 3DOF power assisting, a 6-DOF 3UPS parallel mechanism [10]-[11], which can keep its kinematic instantaneous center on hip joint center, is adopted. It is our vision that it will provide a versatile assisting support for our daily life to help operator’s thigh move flexibly. To accomplish this assisting system, three critical features should be executed: mechanical architecture design, a high-sensitive sensor, and an effective control scheme. In this paper, we focus on the assisting robot mechanical architecture design optimization which is also the basic of the three critical features above. For the design optimization of mechanism, researchers usually focus on the symmetry, workspace and isotropy, where these works are shown in [12]-[13] and so on; some other researchers [14] also have performed their optimization work based on the robotics This work was supported by NSFC (Grant No. 60875047), 863 Program of China (Grant No. 2006AA040204) and NSFY (Grant No. 61005054). Yong Yu is with Graduate School of Science and Engineering, Kagoshima University, Kagoshima, Japan. [email protected] Wenyuan Liang is with the Graduate School of Science and Engineering, Kagoshima University, Japan; Institute of Intelligent Machine, Chinese Academy of Science; and Department of Automation, University of Science and Technology of China. [email protected]

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manipulability theory [8] to obtain the configuration with better isotropy. However, different than the purposes in [12]-[14] which are just optimizing the object as a whole parallel mechanism, the assisting system including two parts: assisting mechanism and human body. If we only optimize it by considering human body and assisting mechanism as a whole parallel mechanism, we would lose the most important information for the assisting mechanism: assisting feasibility and assisting effect. Hence we need to consider not only the workspace and isotropy, but also the most important characteristic of assisting mechanism: assisting feasibility and assisting effect. To author’s knowledge, no evaluation criterion, which can assess the designed assisting mechanism owns the assisting feasibility and effect for operator whether or not, has been performed for assisting robot. Hence, this paper is organized as: in Section 2, Manipulability Inclusive Principle (MIP) is proposed as an evaluation criterion to evaluating assisting mechanism’s assisting feasibility and assisting effect. Section 3 introduces the assisting mechanism design. According with MIP, we need to build the kinematical Jacobian both for robot-human system and human thigh. This work is presented in Section 4. In order to find out the structure which can satisfy MIP and realize better assisting effect, design optimization based on MIP is shown in Section 5. Conclusions are presented in Section 6. II. PROBLEMS During the working of assisting system, human are acting as active joints; then the actuators installed on the assisting mechanism also need to act actively to provide assisting for human thigh. The basic characteristic of the assisting robot is that it can provide necessary assisting for human. Different factors, such as mechanical structure, types of joints, actuators and so on, will have influence on the final assisting result. Surely the final assisting result is directly related to the assisting design, sensor and control method. We can develop the final assisting result in terms of assisting design. For a certain assisting mechanism, assistance includes two ideas: assisting feasibility and assisting effect. However, there is no any evaluation criterion to evaluate the assisting feasibility and assisting effect during the assisting mechanism design and optimization. Satisfying the assisting feasibility ensure the assisting mechanism can shadow human’s movements well to be able to provide assisting. The assisting effect includes these aspects: assisting efficiency, assisting ability and assisting isotropy.

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Intuitively, after the assisting mechanism is mounted to human, while human moves at its maximum velocity on certain orientation, the assisting can still shadow human movements well ensure the assisting possess the assisting feasibility to assist on this orientation. The problem of assisting feasibility becomes comparing the maximum velocity of human and robot-human system (robot-human system, composed of assisting mechanism and human, is driven by the assisting mechanism). The velocity manipulability ellipsoid [15] is a useful means of quantifying end-effector’s moving velocity on each DOF, and it includes two ideas: the magnitude and orientation of each DOF. Along the orientation of the major axis of the ellipsoid, the end-effector can move at largeer velocity, while along the orientation of minor axis small velocity is obtained. So on certain orientation, if the manipulability magnitude of robot-human system is larger than human’s, the maximum velocity of robot-human system is larger than human’s to ensure the assisting mechanism shadow human’s movement well on this orientation. Further, while the manipulability magnitude of robot-human system is larger than human’s on all the orientations, it can shadow human’s movements well to assist on all the orientations. Therefore, in this paper the Manipulability Inclusive Principle (MIP) is firstly proposed for evaluating assisting mechanism’s assisting feasibility and assisting effect. It also has two ideas. Firstly, while the manipulability ellipsoid of robothuman system includes human’s totally, the assisting mechanism satisfies the assisting feasibility to provide assistance. Secondly, while the orientation of robot-human system’s manipulability ellipsoid aligns with human’s better, it means the assisting mechanism owns higher efficiency to apply its largest velocity on human normal moving orientation; the manipulability ellipsoid of robot-human is much larger, its assisting ability is larger by reacting quicker; better feature on assisting isotropy can ensure the assisting mechanism have similar assisting ability on each DOF.

Fig. 1. Manipulability Inclusive Principle for Human Thigh and Robothuman System

In this paper, human hip joint has 3 DOF just as shown in Fig. 1(a). While operator wears the assisting robot, the DOF

of robot-human system’s end-effector (Fig. 1(b-1)) should still be the same as human thigh. Then in order to assist human thigh on 3 DOF, human thigh’s manipulability ellipsoid must be totally inclusive by the manipulability ellipsoid of robot-human system just like Fig. 1(c-3). Otherwise, if human thigh’s manipulability ellipsoid is partly inclusive by robot-human system’s as shown in Fig. 1(c-2), the assisting mechanism can not shadow human thigh’s movements well on the exclusive orientations. On the contrast, while human thigh moves towards to the exclusive orientations, assisting mechanism will drag thigh’s movment. In the case of none inclusive, thigh’s movements will be dragged whatever orientation it move towards. The inclusive results also have influence on the assisting effect. So in order to make the whole workspace satisfy assisting feasibility and realize better assisting effect, in this paper, the design optimization is based on MIP. As to discuss and compare the manipulability of robot-human system and human thigh, the velocity manipulability ellipsoid can be obtained with the following mathematic ways. J = U · S ·V.

(1)

Therefore the above equation can generate a manipulability ellipsoid which represents end-effector’s kinematical characteristic. The principal axes of ellipsoid are aligned with the matrix U, and the manipulability length of each principal axis is diagonal with matrix S. With the computing results, we draw ellipsoid locating its center at the location of the robot end-effector in workspace so that the globe characteristics of the robot kinematic can be represented in the whole. III. ASSISTING MECHANISM DESIGN ARCHITECTURE As well known, hip joint [9] is an important part for human. Whether walking, running or jumping, all these movements cannot realize without the hip joint. Here we use parameters {θ , φ , ϕ} to denote its 3 DOF: θ denotes the rotational angle on flexion/extension, φ denotes the rotational angle on adduction/abduction, and ϕ denotes the rotational angle on lateral/medial rotation. In our design, human thigh is considered as one part of parallel mechanism, thus a 6DOF 3UPS parallel assisting system is proposed (seeFig. 2). This parallel mechanism consists of 3 driving chains, and 2 strips of bandage. Each of the driving chains contains a universal joint, a prismatic joint and a spherical joint, the extremities of driving chains are separately connected with the bandage at waist and thigh. In this assisting mechanism architecture, there are 8 links (3 prismatic joints have 6 links and 2 bases can be considered as 2 links), 9 joints (3 spherical joints, 3 prismatic joints and 3 universal joints), and totally 18 joint DOFs (spherical joint has 3 DOFs, prismatic joint has 1 DOF, and universal joint has 2 DOFs). According to the well-known Chebychev-Grubler-Kutzbach formula, the number of the degree of this parallel mechanism is 6; when considering human thigh as one part of this system, one link and one spherical joint are added to this parallel mechanism.

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Because this parallel system is not a redundant mechanism, its degree of freedom is 3.

constraint equations to the parallel system can be yielded as:  −−−→ −−−→ − −− → A B2 + A1 B3 = 2A1 B1   −−−→ 2  −1−−→ (A1 B1 − A1 O1 ) =| B1 O1 |2 , (4) − −− → −−−→  (A1 B2 − A1 O1 )2 =| B2 O1 |2   −−−→ −−−→ 2 2 (A1 B3 − A1 O1 ) =| B3 O1 |

where in (4), | B1 O1 |, | B2 O1 | and | B3 O1 | are constant. The first equation in (4) can be expressed as 3 scalar equations, so actually there are 6 scalar equations in (4). All the constraint equations are about these 9 parameters: θ11 , θ12 , d13 , θ21 , θ22 , d23 θ31 , θ32 and d33 . So through differing (4), the velocity of each joint is included in the results. For simplicity, the results can be denoted as: ki1 θ˙11 + ki2 θ˙2 + ki3 d˙13 + ki4 θ˙21 + ki5 θ˙22 + ki6 d˙23 + ki7 θ˙31 + ki8 θ˙32 + ki9 d˙33 = 0, i = 1, 2, · · · , 6. Fig. 2.

Model and Architecture of Human-robot Parallel Mechanism

In this section the mechanical architecture of assisting mechanism is proposed, and the rest it to confirm the sizes with optimization which is based on MIP. According to MIP, we need to compare the manipulability of robot-human system and human thigh, so the kinematical Jacobian for robot-human system and human thigh should be built. IV. BUILDING KINEMATICAL JACOBIAN Right now, there are many well established modeling methods [10]-[12] to conduct mechanical kinematic analysis. However, in our research, there are only three active joints’ information, which are provided by the three actuators, is known, leading to that it is hardly to build the Jacobian which can be used to control easily by those well established methods. Hence, a direct Jacobian modeling method based on differing the constraint equations [7] is adopted. A. Relation between Active Joints and Passive Joints To build direct kinematical Jacobian, the most crucial part is to solve out the relation between active joints and passive joints. By the way of differential coefficient, we can effectively and quickly to build the relation between active and passive joints. As shown in Fig. 2, the coordinate O1 − XY Z is located at the center of hip joint; {Ai : (xi , yi , zi ), i = 1, 2, 3} are the center of universal joints, where we use {θi1 , θi2 } to represent the angle of universal joint’s two revolute joints; {di3 } are the length of the prismatic joint; and the spherical joints’ center are {Bi }. Hence, the vector from O1 to Bi are: T −−→  O1 Bi = X(i+4) Y(i+4) Z(i+4) , i = 1, 2, 3. (2) With the serial robotics, (2) can be expressed as: −−→ O1 Bi =

−ci1 ci2 di3 + xi −si1 ci2 di3 · cos γi − si2 di3 · sin γi + yi −si1 ci2 di3 · sin γi + si2 di3 · cos γi + zi

! ,

(3)

where si j represents sin θi j , ci j represents cos θi j , γi is defined as shown in Fig. 2. Combining these expressions, the

(5)

Different than serial mechanism, for parallel mechanism selecting different joints as active joints the final kinematical performance will be different. Through some calculation, we learned that while {θ11 , θ22 , θ32 } are selected as active joints (as shown in Fig. 2), the kinematical performance of endeffector can reach more ideal effect. So dividing the items which are separately about the velocity of passive joints and active joints into the equation’s two sides, hence the velocity of passive joints can be expressed as:  T  T θ˙12 θ˙21 θ˙31 d˙13 d˙23 d˙33 = K3 · θ˙11 θ˙22 θ˙32 . (6) In this paper the items in matrix S are defined as:   .. S(1) S(n + 1) · · · .   .. .. S =  .. .  . S(n)

. S(2n)

··· ···

. S(n × m)

n×m

B. Velocity of Driving Chains’ End-positions In this robot-human system, the bandage (end-effector) applied to human thigh is driven by the 3 active actuators to achieve the aim of power assisting as shown in Fig. 2. Therefore the velocity of end-effector is directly relative to the velocity of the end-positions of driving chains on the thigh bandage which are defined as (2). {B1 : (X5 ,Y5 , Z5 )}, as one of the end-positions, is determined by the variables of {θ11 , θ12 , d13 } as seen in (3). Therefore its differing result is:  T  T X˙5 Y˙5 Z˙ 5 = N1 · θ˙11 θ˙12 d˙13 . (7) Equation (7) includes two passive joints: θ12 and d13 . According to (6), we have the following relation, " ˙ # " ˙ # " # θ11 θ˙12 d˙13

= K4 ·

θ11 θ˙22 θ˙32

, K4 =

1 K3 (1) K3 (4)

Then naturally,  T  T X˙5 Y˙5 Z˙ 5 = M1 · θ˙11 θ˙22 θ˙32 ,

0 K3 (7) K3 (10)

0 K3 (13) K3 (16)

. (8)

M1 = N1 · K4 .

(9)

As the same way taken above, for the other two endpositions, {B2 : (X6 ,Y6 , Z6 )} and {B3 : (X7 ,Y7 , Z7 )}, there are:  T  T X˙ Y˙ Z˙ = M2 · θ˙11 θ˙22 θ˙32 , (10) T  6 6 6 T  X˙7 Y˙7 Z˙ 7 = M3 · θ˙11 θ˙22 θ˙32 .

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V. DESIGN OPTIMIZATION BASED ON MANIPULABILITY THEORY

C. Direct Parallel Mechanism Kinematical Jacobian Bandage, which is applied to human thigh and as the end-effector in this parallel system, needs (X˙5 , Y˙5 , Z˙ 5 , β˙ ) four parameters to describe its movement, where (X˙5 , Y˙5 , Z˙ 5 ), which represents end-effector’s 2 planar DOF, can describe the motion of flexion/extension and abduction/adduction; β˙ represents thigh’s rotation along the longitudinal axis, and β is defined as (11). sin β = (Z6 − Z7 )/2l, l =| B1 B2 |=| B1 B3 | .

In order to find out the structure which can satisfy assisting feasibility and realize better assisting effect in the whole expected workspace, design optimization based on MIP is shown in this section. Before comparing the manipulability, we need to normalize these two Jacobian at first. iT h ˙ ˙ ˙ ˙ β Y5 Z5 X5 (X˙5 )max (Y˙5 )max (Z˙ 5 )max (β˙ )max h ˙ iT θ˙32 θ11 θ˙22 = A−1 · J · A · 1 1 3 (θ˙11 )max (θ˙22 )max (θ˙32 )max

(11)

For the purpose of getting the relation of velocity between active joints and end-effector, it is need to differ (11). Combining with (9) and (10), the differing results for β is: β˙ =

1 ˙ 2l cos β {[M2 (3) − M3 (3)] · θ11 + [M2 (6)

. − M3 (6)] · θ˙22 + [M2 (9) − M3 (9)] · θ˙32 }

2l cos β

2l cos β

(13)

2l cos β

D. Kinematical Jacobian for Hip Joint In order to compare the manipulability of robot-human system and human thigh, here we also need to develop human thigh’s kinematical Jacobian with the same end-effector. Human thigh, as a part of human-robot system, can be built as a 3-DOF module which includes a spherical joint and a link. When thigh moves, the joint center keeps still but the axes will rotate, hence we use Roll-Pitch-Yaw model to describe thigh’s locomotion. (X5 ,Y5 , Z5 ) is also a point on human thigh, while in static condition its coordinate is (x5 , y5 , z5 ). As shown in Fig. 2, the rotation matrixes along axes Z, Y , X are presented as: 

cos θ R1 =  sin θ 0 

1 R3 =  0 0

− sin θ cos θ 0

0 cos ϕ sin ϕ

 0 0 , 1



cos φ 0 R2 =  − sin φ

 sin φ 0 , cos φ

0 1 0

 0 − sin ϕ  . cos ϕ

Hence the rotational matrix is: RPY = R1 · R2 · R3 . Then the extremities can be represented as: [Xi Yi Zi ]T = RPY · [xi yi zi ]T ,

i = 5, 6, 7.

(14)

Similarly, (14) is the function of (θ , φ , ϕ). By differing (14), there is  T  T X˙i Y˙i Z˙ i = Mi−1 · θ˙ φ˙ ϕ˙ , i = 5, 6, 7. (15) Eventually, the relation of velocity between hip joint’s 3 DOF and end-effector can be denoted as: h iT  T X˙5 Y˙5 Z˙ 5 β˙ = J2 · θ˙ φ˙ ϕ˙ , " M4 J2 = M5 (3)−M6 (3) M5 (6)−M6 (6) 2l cos β

2l cos β

# M5 (9)−M6 (9) 2l cos β

(16) .

h

(12)

Eventually, the velocity relation between active joints and end-effector can be denoted as: iT h  T X˙5 Y˙5 Z˙ 5 β˙ = J1 · θ˙11 θ˙22 θ˙32 , " # M1 J1 = M2 (3)−M3 (3) M2 (6)−M3 (6) M2 (9)−M3 (9) .

iT β˙ Y˙5 Z˙ 5 X˙5 ˙ ˙ ˙ ˙ (X5 )max (Y5 )max (Z5 )max (β )max h iT φ˙ ϕ˙ θ˙ = A−1 3 · J2 · A2 · (θ˙ )max (φ˙ )max (ϕ) ˙ max

(17) ,

(18) ,

where A1 = diag((θ˙11 )max ,(θ˙22 )max ,(θ˙32 )max ), A2 = ˙ max ), A3 = diag((X˙5 )max , (Y˙5 )max , diag((θ˙ )max , (φ˙ )max , (ϕ) (Z˙ 5 )max , (β˙ )max ), and diag(·) represents a diagonalmatrix. In this assisting mechanism optimization, the input maximum velocity is confirmed at first, where all of them are equal to π/3. A3 is the maximum expected output velocity, where in this paper it is considered as human thigh’s maximum velocity. In this paper, since the mechanical structure and actuators are confirmed, after optimization, the size of the assisting mechanism can be determined. Associated with the normalized Jacobian and with singular decomposition value, there are: Ji0 = Ui · Si ·Vi , i = 1, 2, (19) −1 0 where J10 = A−1 3 · J1 · A1 , J2 = A3 · J2 · A2 . Therefore through the equations above, we can obtain Manipulability Orientation of Robot-human System (MO-rhs), and Manipulability Orientation of Human Thigh (MO-ht). Then we can also generate Manoeuvrability Ellipsoid of Robot-human System (ME-ths), and Manipulability Ellipsoid of human thigh (MEht). With these results, we draw manipulability ellipsoids locating their centers at the location of the robot endeffector in workspace so that the globe characteristics of end-effector’s velocity can be represented in the whole. During our learning on the optimization of this parallel mechanism, we scan sizes space: {O1 O2 ∈ [100, 250], A1 A2 ∈ [60, 130], O1 A1 ∈ [40, 80], O2 B1 ∈ [50, 80], B2 B3 ∈ [15, 30], γ1 ∈ [−π/9, π/9], γ2 ∈ [0, π/3], γ3 ∈ [−π/3, 0]}, where the unit of length is [mm], trying to find out the architecture posing 3 special kinematical characteristics: higher efficiency, more power assisting ability, and better feature on assisting isotropy. Confirming the parameters should consider the globe workspace assisting results. Hence 7 human thigh postures which can cover the expected workspace are considered. These 7 postures Wk (θ , φ , ϕ) are: W1 (0◦ , 0◦ , 0◦ ), W2 (−20◦ , 0◦ , 0◦ ), W3 (−40◦ , 0◦ , 0◦ ), W4 (0◦ , −15◦ , 0◦ ), W5 (0◦ , −30◦ , 0◦ ), W6 (−10◦ , −5◦ , −5◦ ), W7 (−20◦ , −10◦ , −10◦ ), where {W1 } is the posture while human thigh stands straight up, {W2 ,W3 } represent the motion of flexion/extension, {W4 ,W5 } represent the motion

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TABLE I R ESULTS FOR 7 S ELECTED P OSTURES ON P ROTOTYPE M ODEL

W1 W2 W3 W4

(t11 , (3.14, (3.10, (3.05, (2.30,

t12 , 1.79, 1.82, 1.94, 3.63,

t13 ) 0.70) 0.73) 0.73) 0.60)

W5 W6 W7

(t11 , (1.67, (3.13, (2.33,

t12 , 5.92, 1.57, 1.32,

inclusive by ME-rhs. The assisting mechanism can satisfy assisting feasibility at the 7 selected postures, and then for the assisting effect:

t13 ) 0.68) 0.81) 0.85)

• as for the assisting efficiency, the MO-rhs aligns with MO-ht better, the intersection angle qk between these two orientations is smaller. Then the efficiency is higher, so we use (η1 )k to estimate the assisting efficiency. As−−−→ −−−→ −−−→ suming the MO-ht and MO-rhs are [( j11 )k , ( j12 )k , ( j13 )k ] −−−→ −−−→ −−−→ and [( j21 )k , ( j22 )k , ( j2n )k ]. q0 is the threshold value which the intersection angle qk should not exceed.

of adduction/abduction, and {W6 ,W7 } represent the motion of lateral/medial rotation. A. Manipulability of Human Thigh In this part, we will discuss the manipulability of human thigh moving without assisting mechanism as a standard for comparison. Then we can compare the manipulability of a certain architecture and human thigh’s to judge the power assisting feasibility and effect. During human thigh usual movements, its expected workspace is limited in {−75◦ ≤ θ ≤ 40◦ , − 30◦ ≤ φ ≤ 10◦ , − 20◦ ≤ ϕ ≤ 20◦ }. By considering the singular value decomposition about normalized J20 , as the hip joint is considered as a spherical joint, the result shows at any posture human thigh’s DOF has the same manipulability magnitude for each DOF, where the the magnitude is equal to 1. Let the product of each DOF’s manipulability magnitude represent the volume of the manipulability ellipsoid, so for human thigh it has permanent value in the expected workspace just as shown in Fig. 3.

(η1 )k = 1 − qk /q0 , 0 ≤ (η1 )k ≤ 1, −−−→ −−−→ qk = max{|< ( j1i )k , ( j2i )k >|}, −−−→ ( j ji )k = [(U j (n(i − 1) + 1))k · · · (U j (ni))k ]T ,

(20)

−−−→ −−−→ where < ( j1i )k , ( j2i )k > represents the intersection angle between these two vectors, i = 1, 2, 3, j = 1, 2. • for the assisting ability, while the manipulability magnitude on which orientation is larger, the assisting ability on this orientation is larger. Hence we use the volume of the manipulability ellipsoid to represent the total assisting ability. Let ((t11 )k , (t12 )k , (t13 )k ) is the manipulability magnitude of the robot-human system on each DOF. By considering that the manipulability ellipsoid may degenerate to ellipse or become a hyperellipsoid, (η2 )k , which represents the volume of the manipulability ellipsoid and is used to evaluate the assisting ability, is denoted as: (η2 )k = (t11 )k · (t12 )k · (t13 )k /v0 , 0 ≤ (η2 )k ≤ 1, (21) (S1 )k = diag((t11 )k , (t12 )k , (t13 )k ),

Fig. 3.

where v0 is also the threshold value the volume of the manipulability ellipsoid. The reason of setting v0 is: while the manipulability ellipsoid is oblate, (η2 )k may be also quite large, however, oblate ellipsoid means the assisting ability on each DOF is evidently different. • in order to make the assisting ability on each DOF be similar, while the manipulability magnitude of each DOF is closer, the assisting isotropy is better, where this factor is estimated with (η3 )k . Here we make a sort order for ((t11 )k , (t12 )k , (t1n )k ), let (tˆ11 )k ≥ (tˆ12 )k ≥ (tˆ1n )k .

Manipulability Ellipsoid Volume Contour for Human Thigh

Therefore, in this part, the manipulability of human thigh is obtained. Since the manipulability ellipsoid of human thigh becomes a ball with the radius of 1, while the value of the least manipulability magnitude of robot-human system exceeds 1, the human thigh’s ellipsoid can be totally inclusive. Before optimization, we also have considered an initial prototype with the sizes: O1 O2 = 210, (A1 A2 )x = 0, (A1 A2 )y = 60, (A1 A2 )z = 45, (O1 A1 )x = 60, (O1 A1 )y = 0, (O1 A1 )z = 120, O2 B1 = 80, B2 B3 = 20, γ1 = 0, γ2 = π/2, γ3 = −π/2. Some results are shown in Table I. However, the results can not meet our demand since it can not cover the manipulability ellipsoid of human thigh. Hence in the next part, optimization algorithm based on MIP is shown. B. Optimization Algorithm For all the selected postures, while its least manipulability magnitude on each DOF is larger than 1, ME-ht can be totally

(η3 )k =

(tˆ12 )k (tˆ13 )k
+ /2, 0 ≤ (η3 )k ≤ 1. (tˆ11 )k (tˆ11 )k

(22)

During optimization, 7 human thigh postures (including human thigh’s 3-DOF activities) are considered, hence the cost function for posture Wk and total workspace are: 7

ηk = (η1 )k + (η2 )k + (η3 )k , k = 1, · · · , 7; η =

∑ ηi .

(23)

k=1

In order to meet our demand, the threshold values are set as: q0 = 10◦ , v0 = 3.5, and (tˆ13 )k /(tˆ11 )k ≥ 0.5.

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TABLE II R ESULTS FOR 7 S ELECTED P OSTURES AFTER OPTIMIZATION

W1 W2 W3 W4 W5 W6 W7

(t11 ,t12 ,t13 ) (1.59,1.35,1.47) (1.54,1.33,1.42) (1.40,1.30,1.27) (1.52,1.48,1.29) (1.39,1.50,1.16) (1.43,1.29,1.62) (1.39,1.19,1.71)

qk 0◦ 2.3◦ 5◦ 0◦ 0◦ 6.7◦ 9.4◦

t11 t12 t13 3.14 2.91 2.32 2.90 2.42 2.99 2.82

ˆ

( ttˆˆ12 , ttˆ13 ) 11 11 (0.93,0.85) (0.92,0.86) (0.91,0.90) (0.97,0.85) (0.93,0.77) (0.88,0.80) (0.81,0.70)

ηi 0.9 0.79 0.66 0.88 0.79 0.74 0.5

C. Manipulability of Optimized Robot-human System After optimization, with the size parameters of: O1 O2 = 210, (A1 A2 )x = 0, (A1 A2 )y = 100.5, (A1 A2 )z = 60, (O1 A1 )x = 60, (O1 A1 )y = 0, (O1 A1 )z = 130.5, O2 B1 = 60, B2 B3 = 36.3, γ1 = 0, γ2 = π/4, γ3 = −π/4, where the unit of length is [mm], this architecture can achieve: higher assisting efficiency, more assisting ability and better feature on assisting isotropy. Let the postures where the ME-ht is totally inclusive by ME-rhs is defined as admissible workspace of robot-human system. It means in this admissible workspace the assisting robot can satisfy assisting feasibility to realize 3-DOF power assisting. The admissible workspace for this architecture is shown in Fig. 4. In this figure, P2 is located in the admissible workspace, it shows the case that ME-ht is totally inclusive by ME-rhs; P1 is a posture which the end-effector can reach, but for this posture the ME-ht is partly inclusive by ME-rhs. Fig. 4 shows the workspace is {−78◦ ≤ θ ≤ 78◦ , −34◦ ≤ φ ≤ 20◦ , − 20◦ ≤ ϕ ≤ 20◦ }. Compared to the expected workspace, this admissible workspace can cover the expected workspace. Enhancing the maximum velocity of the actuators can enlarge the admissible workspace.

Fig. 4. Admissible Workspace of Robot-human System (−78◦ ≤ θ ≤ 78◦ , −34◦ ≤ φ ≤ 20◦ , − 20◦ ≤ ϕ ≤ 20◦ ). The unit of the reference is [mm].

Table II also shows the least manipulability magnitude is larger than 1. A more visualized results are shown in Fig. 5. In Fig. 5, in the admissible workspace, all the ME-ht is totally inclusive by ME-rhs. Additionally, this architecture also can provide more power assisting during our optimization. While human thigh stand up straight, the manipulability magnitude for each DOF are: (1.59, 1.35, 1.47). In this kind of parallel mechanism, the bottom of the workspace has the largest volume of manipulability ellipsoid. In Fig. 6, the volume

Fig. 5. Some Examples about ME-ht and ME-rhs in Admissible Workspace

of manipulability ellipsoid at bottom is 3.14, very close to 3.5. During the optimization, it is found the manipulability ellipsoid volume descends much slower just as shown in Fig. 6.

Fig. 6. Manipulability Ellipsoid Volume Contour for Optimized Robothuman System

Figure 7 shows the MO-rhs and MO-ht while assisting robot moves with human thigh. It indicate that the robothuman system has the same DOF as human thigh. In other words, human thigh can keep its own DOF while operator wears this parallel assisting robot. Human thigh’s MO-ht represents human thigh’s normal moving orientations. At this posture, the largest intersection angle between them is about 5◦ . Table II shows, among the 7 selected postures, the largest intersection angle is not over 10◦ . Much more visualized results are shown in Fig. 8, where robot-human system’s and human thigh’s manipulability orientation of lateral/medial rotation align completely. In Fig. 8 (a), the MO-rhs aligns with MO-ht completely. It means on the motion of adduction/abduction, assisting mechanism can apply its maximum velocity on human’s normal moving orientations. In Fig. 8 (b), although for this architecture the flexion/extension and abduction/adduction’s highest efficiency directions are not on the human normal directions, the largest angle between MO-rhs and MO-ht does not exceed 10◦ . Hence, this architecture still has high efficiency on the human normal directions. On the assisting isotropy, our demand is (tˆ13 )k /(tˆ11 )k ≥

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Fig. 9.

Fig. 7.

Manipulability Orientation Comparison Example

Parallel Hip joint Power Assisting Mechanism

fixed location, types of joint, mechanical structure and so on. Based on MIP, the optimization algorithm can find out the assisting mechanism which satisfies assisting feasibility and realize higher assisting efficiency, more assisting ability and better assisting isotropy in the expected workspace. R EFERENCES

Fig. 8. Manipulability Orientation Comparison Orientation. (a) shows the motion of adduction/abduction, and (b) is the motion of flexion/extension.

0.5. In Table II, among the selected 7 postures, there is (tˆ13 )k /(tˆ11 )k ≥ 0.7. It means this optimized assisting mechanism will not have an evident assisting ability difference on each DOF. Finally, the real structure is shown as Fig. 9. VI. CONCLUSIONS In this paper, based on Manipulability Inclusive Principle (MIP), design optimization for hip joint 3-DOF parallel assisting mechanism is proposed. By comparing the manipulability of human and robot-human system, the MIP can evaluate assisting mechanism’s assisting feasibility and assisting effect. In order to ensure the proposed assisting mechanism satisfy assisting feasibility, the manipulability ellipsoid of human must be inclusive by robot-human system’s. In this paper we just optimize the size of the 3UPS parallel assisting mechanism. The MIP is also related to actuator,

[1] M. Vukobratovic, V. Circ, and D. Hristic, 1972, “Contribution to the Study of Active Exoskeletons,” Proceedings of the 5th IFAC Congress, Paris. [2] M. Vukobratovic, D. Hristic, Z. Stojiljkovic, ”Development of Active Anthropomorphic Exoskeletons.” Medical and Biological Engineering, pp. 66-80, 1974. [3] B. J. Makinson and General Electric C. O., ”Research and Development Prototype for Machine Augmentation of Human Strength and Endurance, Hardiman I Projict,” General Electric Report, S-71-1056, 1971. [4] Adam Zoss, H. Kazerooni and Andrew Chu, ”On the Mechanical Design of the Berkeley Lower Extremity Exoskeleton (BLEEX),” Proceeding of the 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems, pp. 3132-3139, 2005. [5] Yamamoto Keijiro, Hyodo Kazuhito, Ishi Mineo and Matsuo Takashi, ”Development of power assisting suit for assisting nurse labor, ” JSME International Journal, Series C: Mechanical Systems, Machine Elementsand Manufacturing, Vol. 45, No. 3, pp. 703-711, 2002. [6] Suzuki Kenta, Mito Gouji, Kawamoto Hiroaki, Hasegawa Yasuhisa, and Sankai Yoshiyuki, ”Intention-Based Walking Support for Paraplegia Patients with Robot Suit HAL,” Advanced Robotics, vol. 21, num. 12, pp. 1441-1469(29), 2007. [7] Yong Yu, Wenyuan Liang and Yunjian Ge, ”Jacobian Analysis for Parallel Mechanism Using on Human Walking Power Assisting,” Proceedings of the 2011 IEEE International Conference On Mechatronics and Automation, pp. 282-288, 2011. [8] Haruhiko Asada, ”Dynamic Analysis and Design of Robot Manipulators Using Inertia Ellipsoids”, Proceedings of the 1984 IEEE International Conference On Robotics and Automation, pp. 94-102, 1984. [9] Wikipedia, http://en.wikipedia.org/wiki/Hip. [10] Mircea Badescu and Constantinos Mavroidis, ”Workspace Optimization of 3-Legged UPU and UPS Parallel Platforms With joint Constraints”, Jouranl of Mechanical Design, Vol. 126, pp. 291-300, 2004. [11] Joshi, S. A., ”Comparison Study of a Class of 3-DOF Parallel Manipulators”, Ph.D. Dissertation, University of Maryland, College Park, May, 2002 [12] Gosselin, C., and Angeles, J., ”The Optimum Kinematic Design of a Spherical Three-Degree-of-Freedom Parallel Manipulator”, ASME J. Mech. Transm., Autom. Des., pp. 202-207, 1989. [13] Richard E. Stamper, Lung-Wen Tsai, and Gregory C. Walsh, ”Optimization of a Three DOF Translational Platform for Well-Conditioned Workspace”, Technical Research Report, T.R. 97-71. [14] Chin Pei Tang and Venkat N. Krovi, ”Manipulability-based configuration evaluation of cooperative payload transport by mobile manipulator collectives”, Robotica, vol. 25, pp. 29-42, 2007. [15] Yoshikawa T. ”Manipulability of robotic mechanisms”. Int. J. Robotics Res., Vol. 4(2), pp. 3-9, 1985.

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