Design, Fabrication, and Evaluation of a New

IEEE/ASME TRANSACTIONS ON MECHATRONICS VOL. 6, NO. 3, SEPTEMBER, 2001

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Design, Fabrication, and Evaluation of a New Haptic Device Using a Parallel Mechanism Jungwon Yoon, Jeha Ryu* , Member, IEEE *Corresponding Author

Abstract-- This paper present design, fabrication, and evaluation of a new 6 degree-of-freedom haptic device for interfacing with virtual reality by using a parallel mechanism. The mechanism is composed of three pantograph mechanisms that are driven by ground-fixed servomotors, three spherical joints between the top of the pantograph mechanisms and the connecting bars, and three revolute joints between the connecting bars and a mobile joystick handle. Forward and inverse kinematic analyses have been performed and the Jacobian matrix is derived. Performance indices such as GPI (Global Payload Index), GCI (Global Conditioning index), Translation and Orientation workspaces, and Sensitivity are evaluated to find optimal parameters in design stage. The proposed haptic mechanism has better load capability (low inertia, high bandwidth, etc) than those of the pre-existing haptic mechanisms due to the fact that motors are fixed at the base. It has also wider orientation workspace mainly due to a RRR type spherical joint. A control method is presented with gravity compensation and with force feedback by a F/T sensor to compensate for the effects of unmodeled dynamics such as friction and inertia. Also, dynamic performance has been evaluated for force characteristics such as maximum applicable force, static-friction force, minimum controllable force, and force bandwidth by experiments. Virtual wall simulation with the developed haptic device has been demonstrated. Index Terms-- haptic device, parallel mechanism, pantograph, gravity compensation, performance indices, dynamic performance, RRR type spherical joint.

V

I. INTRODUCTION

IRTUAL REALITY (VR) application is widely spreading in the areas of engineering, medical operation, teleoperation, welfare, and entertainment with the rapid development of computer technology. Haptic devices which feedback kinesthetic or tactile sensation to interactive users are also becoming indispensable to enhance the feeling of immersion in a VR system [1]. For kinesthetic sensation, many researchers have proposed different types of haptic devices such as a tool type, exoskeleton type, and serial robot type. Among these, a tool type device on the desk has been more widely accepted than the other types because of large Manuscript received March 30, 2000. This research was supported in part by the Brain Korea21 research fund from the Ministry of Education. The authors are with the Department of Mechatronics, Kwangju Institute of Science and Technology (KJIST), 1 Oryong-dong, Puk-gu, Kwangju 500-712, Korea(E-mail: [email protected]).

bandwidth, safeness, and compactness. An ideal haptic device is required to have large workspaces, low inertia, high stiffness, low friction, and high control bandwidth and so on. It is, however, almost impossible to construct a haptic device satisfying all these requirements. Therefore, every effort must be made on optimally designing haptic devices by reducing moving inertia, by enlarging workspaces, and so on. Moreover, a device developer should provide users with evaluated kinematic and dynamic performance of the haptic device so that users may take care of the limitations of the haptic device for their specific application. Parallel mechanisms have been used for tool type haptic devices because they have the characteristics of low inertia, high rigidity, compactness, and precise resolution compared with serial mechanisms. However, some of the haptic devices based on parallel mechanisms that have been developed so far still have disadvantages such as large inertia, difficult forward kinematics, and small workspaces. Long and Collins [2] and Iwata [3] proposed 6-dof tool type haptic devices with a parallel mechanism which has 3-pantograph linkages, each of which is attached to a mid-point of an equilateral base triangle through a passive revolute joint. Woo et al. [4] made a similar force feedback device for telesurgery. This device has five bars instead of pantograph linkages for easier construction. In the above two devices, top of each pantograph and five bar mechanism is connected to one vertex of a mobile platform through a three-dof ball-and–socket joint. These devices, however, have important disadvantages of having large inertia because rotary motors are not fixed to base and of having small orientation workspace due to restricted rotation range of spherical joints. Tsumaki et al. [5] developed a 6-dof haptic device that has an orientation gimbal mechanism on top of a 3-dof modified DELTA mechanism. This device has compact size and wider orientation workspace but still has the problem of large motor inertia because three motors are located above the DELTA mechanism. Millman et al. [6] developed a Stewart platform type device with 4-dof motion, which has 3-dof translation and 1-dof roll angle orientation motion. This device has good resolution and high stiffness due to the advantage of the parallel mechanism but cannot make tilt angles that are important for versatile virtual reality application. Some researchers presented dynamic performance evaluation of various haptic devices. Howe and Kontarinis [7]


have used a 2-dof vertical planar device, with a force bandwidth exceeding 100Hz over a 5N range, in teleoperation experiments. Adelstein and Rosen [8] developed a 2-dof spherical mechanism which can be controlled with high fidelity up to 48Hz at a sustained tip force of 20N. Ellis et al. [9] developed a 2-dof planar haptic device and presented experimental evaluation methods of dynamic performances. Their mechanism has maximum output force of 56N, passive static-friction force of 1.7N, minimum force of 0.4N, and force bandwidth of 80Hz at forces exceeding 50N. Moreyra and Hannaford [10] suggested a method to characterize and experimentally measure the dynamic performance of haptic display devices. They introduced a dimensionless measure of Structural Deformation Ratio (SDR) to quantify some aspects of the high frequency performance. Carignan and Cleary [11] pointed out that the quality of haptic devices can be measured in terms of impedance accuracy and resolution (or fidelity) and investigated several control methodologies for improving dynamic quality. Colgate and Brown [12] suggested the dynamic range of achievable impedance (Z-width) as a measure of performance. This paper presents design, fabrication, and evaluation of a new 6 degree-of-freedom haptic device for interfacing with virtual reality by using a parallel mechanism. The mechanism is composed of three pantograph mechanisms that are driven by ground-fixed servomotors, three spherical joints between the top of the pantograph mechanisms and the connecting bars, and three revolute joints between the connecting bars and a mobile joystick handle (see Fig. 1). The low moving inertia gives better dynamic bandwidth as compared to the existing devices and the RRR type spherical joint assures a wider orientation workspace. Forward and inverse kinematic analyses have been performed and the Jacobian matrix is derived. Static performance indices such as GPI (Global Payload Index), GCI (Global Conditioning index), Translation and Orientation workspaces, and Sensitivity are evaluated to find optimal geometric parameters at the design stage. This paper also presents a control method and experimental evaluation of dynamic performances such as minimum controllable force, static-friction force, force bandwidth, and maximum applicable force. In addition, stable ranges of virtual wall parameters such as maximum achievable stiffness and damping coefficients are obtained by virtual wall simulation with the developed haptic device. This paper is organized as follows; Chapter 2 presents kinematic analyses including inverse, forward, and Jacobian analyses. Chapter 3 presents design analyses for optimal design of the parallel mechanism. Rearrangeability of the mechanism is also discussed for more versatile usage. Chapter 4 compares kinematic performance indices of the proposed device with those of the existing devices. Chapter 5 presents a control system with gravity, friction, and inertia compensation. Chapter 6 presents evaluation of dynamic performance. Chapter 7 shows virtual wall simulation results and Chapter 8 presents conclusions and discussions of the current research.

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Fig. 1.

Proposed Haptic Device based on Parallel Mechanism

II. KINEMATIC ANALYSES A. Mechanism Description The proposed haptic device with a parallel mechanism is shown in Fig. 1. The new mechanism is composed of three pantograph mechanisms that are driven by six base-fixed servomotors and that stand perpendicularly to the base plate, three RRR type spherical joints between the top of the pantograph mechanism and the connecting bars, and three revolute joints between the connecting bars and a mobile platform. Since each pantograph mechanism is confined to a fixed vertical plane due to motors fixed at the base plate, the revolute joints between the base and the pantograph mechanisms in the pre-existing devices [2-4] should be moved to the locations between the top plate and the connecting bar, which is the main characteristics of the new haptic mechanism. In order to analyze the proposed mechanism, kinematic parameters are shown in Fig. 2. The fixed global reference frame ( X b , Yb, Z b ) is located at the bottom center of the base plate. The mobile reference frame ( X o ' , Yo ' , Z o ' ) is located at the top center of the top plate, where the X o' -axis is in the plane of the top plate and is directed toward the first revolute joint. Each pantograph local reference frame ( X pi , Y pi , Z pi ) is


3 Qi = T ( x, y, z ) R (φ ,θ , ϕ )Tx ( R1 ) R z (γ i ) R x (−90°) R z (α i )Tx ( L3 )[0001]T

located at an active revolute joint, where the X pi -axis is directed perpendicular to each pantograph plane. The axisymmetric position of pantographs on base plate with radius R0 are given by the angles δ i (0, 2π/3, and 4π/3 radians), which specify the rotation angles about Z b -axis from X b -axis. Note that each pantograph has 2-dof motion on the Ypi − Z pi plane.

Notice also that even though the

spherical joints look to be located on the top of pantograph mechanisms (see Fig. 2) in the following kinematic analyses, actual center of a spherical joint is located in an offset distance ( SL1 in Fig.1). However, the following kinematic analyses assume that the center of spherical joint is just on the top of a pantograph mechanism because the offset distance does not affect the kinematic analyses results if the base radius R0 is replaced by R0 '− SL1 . Lower links of pantographs are denoted by L1 and upper links by L2 . A connecting bar length is denoted by L3 . Circle radius of the top plate is defined by R1 . The points M i , Qi , and Ei denote the positions of active, spherical, and revolute joints, respectively. The three revolute joints are attached to the top plate and centered on a circle of radius R1 with the angle

γi

(0, 2π/3,

and 4π/3 radians), which specifies rotation angles about the Z o ' -axis from the X o ' -axis.

(1)

where T ( x, y, z ) specifies translation of the top plate origin and R(φ ,θ , ϕ ) is the orientation matrix of the top plate, which is described by three successive Euler angles. Note in (1) that Qi includes unknown revolute joint angle α i at Ei position. Since Qi position must be on the pantograph Y pi − Z pi plane, the following constraint equations of the plane Y pi − Z pi should be satisfied:

cos(δ i )(Qxi − R0 cos(δ i )) + sin(δ i )(Qyi − R0 sin(δ i )) = 0

(2)

Note geometrically that the Qi points are obtained by the intersection of the circle of radius L3 from revolute joint Ei with the pantograph plane Y pi − Z pi . Thus, inserting Qxi and Q yi components from

(1) into

(2) gives an equation with

the unknown revolute joint angle α i in a closed form. Then, by calculating (1) with the computed α i , it is possible to find inverse kinematics of 2-dof pantograph as follows; From Fig. 2, the distance between Qi and M i is expressed as Lmi = ( M xi − Q xi ) 2 + ( M yi − Q yi ) 2 + ( M zi − Q zi ) 2

The spherical joint positions

p

Qi

(3)

with respect to the

pantograph local frame ( X pi , Y pi , Z pi ) can be derived as p

Qi = R z−1 (δ i )(Qi − M i )

(4)

where the angles δ i (0, 2π/3,and 4π/3 radians) specify the rotation from the global reference frame to the pantograph local frame. From Fig. 2, the intermediate angles θ pi and

θ mi are then given by θ pi = cos −1 (

p Q yi L2mi + L12 − L22 ) , θ mi = cos −1 ( ) Lmi 2 Lmi L1

(5)

Finally, active joint angles are given by θ1i = θ mi − θ pi , θ 2i = θ mi + θ pi Fig. 2.

Kinematics Model

B.

Inverse Kinematic Analysis The inverse kinematics computes the active joint angles θ1i and θ 2i of the pantographs given the position and

orientation ( x, y, z , φ ,θ , ϕ ) of the top plate. The position Qi of a spherical joint can be represented by using (4 × 4) homogeneous matrices in the global reference frame ( X b , Yb, Z b ) as

(6)

C.

Forward Kinematic Analyses Forward kinematic analysis determines the position and orientation of the ( X o ' , Yo ' , Z o ' ) frame with respect to the ( X b , Yb, Z b ) frame given actuated angles θ1i and θ 2i . The first

thing for solving forward kinematics is to find revolute joint angels α i at Ei points. These angles can be obtained by the fact that the distances between Qi points in the global reference frame and the distances between o 'Qi (α i ) points in the mobile reference frame are the same. The following steps are taken for forward kinematics: (i) In the global reference frame, the points Qi should be


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calculated by using pantograph forward kinematics given the active joint angles ( θ1i and θ 2i ) as Qi = R z (δ i )Tx ( R0 ) R x (θ 1i )T y ( L1 ) R x (θ 2i − θ1i ) T ( L )[0,0,0,1]T y

(7)

Qi = R z (γ i )Tx ( R1 ) R x (−90°) Rz (α i )T y ( L3 )[0,0,0,1]T

(8)

(ii) To solve the unknown angle α i , the following three equality conditions are used: o'

o'

| Q1 − Q2 | = | Q1 − Q2 | |o ' Q2 − o 'Q3 | = | Q2 − Q3 | o'

(9)

o'

| Q3 − Q1 | = | Q3 − Q1 |

Nonlinear equations (9) can be solved for the variable α i by using the Newton-Raphson’s numerical method. (iii) Using the computed angle, α i , the following vectors can be derived in the mobile reference frame ( X o' , Yo' , Z o' ) : From Fig. 2, the unit vector defining the direction of the o' u s axis expressed in the ( X o ' , Yo ' , Z o ' ) frame is given as o'

u s =( o 'Q1 − o 'Q3 ) /|o ' Q1 − o 'Q3 |

(10)

In addition, the o' ws axis that is normal to the plane defined by the three points Q1 , Q2 , Q3 can be found by the vector cross product between o'

θ = sin −1 (Too ' [1,3])

o'

Q1 o 'Q2 and

o'

Q1 o 'Q3 :

ws =( o 'Q1 o 'Q2 ×o ' Q1 o 'Q3 ) /|o ' Q1 o 'Q2 ×o ' Q1 o 'Q3 |

Then the third axis o'

o'

v s is given by

o'

o'

o'

o' o'

v s [1]

v s [ 2] v s [3]

o' o'

ws [1] w s [ 2] w[3]

o'

0

(15)

D.

Deriviation of Jacobian The Jacobian matrix can be easily derived by using the concept of reciprocal screws [13-14]. For each right sub-chain of the pantograph mechanism, the top platform twist T is a linear combination of the joint–screws T = θ&1, i S1, i + θ&3, i S 3, i + θ&s , i S s , i + α& i Sα , i

(16)

where θ&1,i is the active joint rate, θ&3, i and α& i are the rates of the passive revolute joints, and θ&s, i S s, i is a twist about the spherical three-system. From (16), the angular velocities of the passive joints are eliminated by the concept of the reciprocity stated as follows; Two screws S m and S n are said to be reciprocal if S 'mT S n = 0, m ≠ n

where S 'T is

defined as

(17)

[a3 , a 4 , a5 , a1 , a 2 , a3 ] when S

= [a1 , a 2 , a3 , a 4 , a5 , a6 ]T and S 'T is the

transpose of a screw

S ' . Let S '1, i be the unit screw that is reciprocal to all the

joint screws excluding the S1,i screw. In general, since the other five screws

S 3, i

,

Sα , i ,

and S s ,i

are linearly

S '1,i is uniquely specified. The reciprocity

relationship yields θ&1,i by taking the virtual product of

(12)

Thus, the intermediate transformation matrix from the mobile frame ( X o ' , Yo ' , Z o ' ) to the spherical joint frame ( X s , Ys , Z s ) can be described as o'

Too ' [1,2] ) cosθ

S '1, i with (16):

o'

v s =( ws × u s ) /| ws × u s |

⎡ o ' u s [1] ⎢ o' u [ 2] s To ' = ⎢⎢ o ' s u [3] ⎢ s ⎢⎣ 0

ϕ = sin −1 (−

,

where Too' [i,j] is the i,j-th component of the matrix Too' .

independent, (11)

,

x = Too ' [1,4], y = Too ' [2,4], z = Too ' [3,4]

2

Then in the mobile reference frame, points o 'Qi (α i ) that include unknown revolute joint angle, α i , can be represented by o'

φ = tan 2 −1 ( −Too' [2,3], Too' [3,3])

0

Q1[1] ⎤ ⎥ o' Q1[2]⎥ o' Q1[3]⎥ ⎥ 1 ⎥⎦

θ&1, i =

Similarly, the intermediate transformation matrix

θ&2,i =

Then, forward kinematic solutions are derived as

[ S ' 2,i ]T T

(19)

[ S ' 2,i ]T S 2,i

Rearranging the expressions for θ&1,i and θ&2, i (i=1,2,3) into a single transformation, we obtain

Tos from

& = JTT Θ

the

base frame to the spherical joint frame ( X s , Ys , Z s ) is computed. (iv) The (4 × 4) homogeneous transformation matrix from the base to mobile reference frames is obtained as Too ' = Tos (Tos' ) −1

(18)

[ S '1, i ]T S1, i

Similarly, for each left sub-chain, θ&2, i is given by

o'

(13)

[ S '1, i ]T T

(14)

where

& = [θ& ,θ& ,θ& ,θ& ,θ& ,θ& ]T Θ 1,1 2,1 1, 2 2, 2 1,3 2,3

(20) ,

and

J

is

a (6 × 6) matrix which represents inverse kinematic Jacobian. When WR, i and WL,i represents [ S '1,i ]T S1,i and [ S '2,i ]T S 2,i , respectively, J is described as J =[

S '1,1 S ' 2,1 S '1, 2 S ' 2, 2 S '1,3 S ' 2,3 T , , , , , ] WL,1 WR ,1 WL,2 WR, 2 WL,3 WR,3

(21)


5 plate plane or are vertically erected, and when all the connecting bars are perpendicular to pantograph planes. III.

Fig 3.

DESIGN ANALYSES

In haptic device design, there are many mechanical design requirements such as small inertia, large stiffness, small backlash, and compactness in order to achieve good static and dynamic characteristics such as workspace, force transmissibility, isotropy of the force and motion, backdrivability, high force bandwidth, and high force dynamic range [15]. In order to fulfill these requirements, this chapter presents design analyses to obtain an optimum mechanical architecture with respect mainly to such static performances as the larger workspace, larger force transmissibility, better isotropy, and smaller sensitivity with respect to articular variable noises. In addition, we will discuss rearrangement of our design for more versatile usage for different situations.

Reciprocal Screws

The reciprocal screw S '1,i is a zero pitch screw passing through the spherical joint position Qi and acts along the line of intersection of the N i -plane which contains point Qi and axis of Sα , i , S axisα ,i with the M i -plane which contains point Qi and axis of S 3, i , S axis3,i (see Fig. 3). Let ni be a unit normal vector defined by the cross product of Ei Qi and Sα , i to N i -plane.

Then,

ni = Ei Qi × S axisα , i

And let q1, i be a vector normal to M i -plane.

(22) Then

q1, i = Ai Qi × S axis3, i

(23)

A.

RRR Type Spherical Joint Design The proposed parallel mechanism has three spherical joints. However, a conventional ball-and-socket type spherical joint has rotational limitation as well as difficulties in connecting three bars (two L2 links and one L3 link). Therefore, it is necessary to design a spherical joint that allows largest possible rotations and easy installation. A spherical joint satisfying these requirements is a RRR type joint shown in Fig. 4(a). As shown in Fig. 4 (b), however, this joint does not allow full rotation about the y-axis. It is, therefore, necessary to maximize this rotational angle for larger workspace without sacrificing other performance indices which will be discussed in the following section III. B.

So, a unit vector t1,i along the direction of the screw axis S '1, i is represented by

t1,i =

ni × q1,i

(24)

| ni × q1,i |

Then, a screw that is reciprocal to all screws except S1,i is given as S '1,i = [t1,i , Qi × t1,i ]T

(25)

(a)

Similarly, a screw S '2, i that is reciprocal to all screws except S 2,i is obtained as q 2, i = Bi Qi × S 4, i , t 2,i =

Inserting

S '1, i

and

S '2,i

ni × q 2,i | ni × q 2,i |

, S ' 2,i = [t 2,i , Qi × t 2,i ]

(26)

(i=1, 2, 3) into (21), therefore,

completes the derivation of Jacobian. Note that singular configurations are generated when S '1,i and S '2,i are reciprocal to S1,i and S 2, i . Some of the singularities which have been found so far occur; when the pantograph mechanisms are either lowered down to the base

(b) Fig. 4. RRR Type Spherical Joint Modeling. (a) Spherical joint detail (b) Spherical joint rotation about y-axis


From Fig. 4(b), d is expressed as d= L−

B.

p 2 sin θ

(27)

Using the relationship: tan θ = t/d and sine formula, θ is represented by

θ = sin −1 (

p

t ) − tan −1 (− ) L (2 L) + (2t ) 2

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2

where p is assumed to be less than

(2 L) 2 + (2t ) 2

(28)

.

As θ becomes small, the rotation range of the L3 link increases, which in turn increases the constant orientation workspace projected in the x-y plane as shown in Fig. 5. It turns out, however, that if the angle θ is smaller than 20°, the RRR type spherical joint does not significantly increase the constant orientation workspace, while degrading other performance indices that are presented in the following section. Therefore, the θ angle is designed to be 20° by appropriately selecting L, p, and t parameters in (28).

Parameter design for optimal static performance Using the performance indices such as GPI (Global Payload Index)[16], GCI (Global Conditioning Index)[17], S (Sensitivity) [18], and COW (constant orientation workspace), optimum geometric design parameters can be found for the proposed mechanism. The constant orientation workspace (COW) is defined as the three-dimensional region that can be attained by an end-effector point when the mobile platform is kept at a constant orientation. Therefore, large values of COW will give wide range of motion. When the input motor torques have unity magnitude, the extreme values of the payload at the end–effector are || F ||= σ , || F ||= σ max

max

min

min

(29)

where σ max and σ min represent the largest and the smallest singular values of the inverse Jacobian matrix. Then, GPI (Global Payload Index) is defined as

∫

∫

∫

∫

GPI max = σ max dw / dw , GPI min = σ min dw / dw

(30)

where w is the area of the constant orientation workspace of the robot . This GPI index represents force transmission capability. Therefore, the larger GPI, the bigger force transmission in the end-effector. The global conditioning index (GCI) is define as GCI = (a)

(b)

1

∫ CN dw / ∫ dw , 0

≤ GCI ≤ 1

(31)

where CN is the condition number of a manipulator Jacobian at a manipulator configuration. This index represents the isotropy of manipulation over the entire workspace. Therefore, the closer the GCI to unity value, the more even feel through the workspace. Sensitivity (S) of endpoint position with respect to perturbations in articular variables is defined as the sum of the absolute values of all Jacobian elements in a single row over the COW. For the first row of the Jacobian, for example, sensitivity in x-coordinate direction is defined as Sx =

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∂x

∫ ∑ | ∂qi |dw / ∫ dw

,

i =1

(q = θ11 , θ 21 , θ12 , θ 22 , θ13 , θ 23 )

(c) Fig. 5. Constant Orientation Workspace Variation with respect to angle θ .

(a) θ = 30° (b) θ = 20°

(c) θ = 10°

(32)

This index may be an important measure of kinematic performance because amplification of uncertainty in articular variable is undesirable from the precision standpoint. Therefore, the lower Sensitivity, the better attenuation of actuator perturbations at the end-effector. The variations of performance indices with respect to design variable changes are summarized in Table 1, in which the ranges of design parameters are; 0.5 ≤ L1 / L2 , ( L1 + L2 ) / L3 , R0 / R1 ≤ 1.5 and 0 ≤ SL1 ≤ 20mm.


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In this table, the upward arrow indicates increase of geometric parameters (or ratio) or increase of performance indices. From this table, we concluded that the conditions on the optimum parameters are; L1 = L2 , ( L1 + L2 ) = L3 , R0 = R1 , SL1 = value for θ = 20° . With this design, the constant orientation workspace is larger than 300mm diameter circle in the x-y plane, the maximum payload is over 40N, and the maximum tilt angle is over 50° . (c )

TABLE 1. VARIATIONS OF PERFORMANCE INDICES Fig. 6. (d) Case4

IV.

C.

Rearrangements of the proposed mechanism Even though we arrange the designed mechanism as shown in Fig. 1, this arrangement can be changed drastically to different arrangements (see Fig. 6) with only reorienting actuators, with a minor change of upper revolute joints, and/or by removing some of the links. This is possible because all passive joints of the proposed parallel mechanism are revolute joints and the direction of the revolute joint can be changed arbitrarily. Furthermore, some links can be taken out of the existing structure to make a simpler architecture (case 1 and case 2) with 3-dof. Using these properties, other arrangement such as case3 can be assembled by change of direction of the pantographs and upper revolute joints from the original mechanism. Note that in case 2 and case 3 the upper revolute joints have changed their rotation directions in the rearranged installation. In addition, a pantograph with 2-dof motion can be changed simply to a five-bar mechanism (case4). These rearrangements of the proposed mechanism are good for more versatile usage of a haptic mechanism in different situations where different performance is required.

(a)

(b)

(d)

Possible Rearrangements. (a) Case1 (b) Case2 (c) Case3

STATIC PERFORMANCE COMPARISON WITH EXISTING DEVICES

In order to compare static performances of the proposed mechanism with the existing haptic mechanisms of similar architecture, workspaces, GPI, GCI, and sensitivity are compared. KAIST master and UCI Hand Controller are selected for comparison (see Fig. 7). They are composed of three five bar mechanisms or three pantographs which are connected to the top plate with spherical joints and which are connected to base with passive revolute joints. Therefore motors are not fixed to the base plate, which results in large constant orientation workspace and large moving inertia. For fair comparison, the geometric parameters of each mechanism are adjusted to maintain equal height of the end-effecter, and equal base and top plate radii.

Fig. 7.

A.

Comparison of Haptic Mechanisms

Workspace Comparison The COW was determined numerically with an inverse kinematics search algorithm in which joint rotation limits are taken into account. A typical ball-and-socket type spherical joint is assumed to have 60 degree of cone angle. The constant orientation workspaces of the KAIST master and the UCI Hand Controller that have the spherical joint range of 60° are about twice larger than that of the proposed mechanism due to the fact that the pantographs and five bars are rotating from base plate. On the other hand, the translational motion of the proposed mechanism is determined mainly by the upper parts between the top plate and the pantographs, which results in smaller COW. Note that the COW shapes of KAIST


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master and UCI controller are very different from those in [4] and [2] because of limited rotation angle of the ball-and-socket type spherical joint. The larger the spherical joint angle, the closer the shapes to that of the proposed mechanism.

(a)

(a)

(b) Fig. 9. Orientation Workspace at a Nominal Position ( Ox = 0, O y = 0, Oz = 230mm ). (a) Orientation

(b)

Workspace

in

Cylindrical Coordinates( (1)New haptic device (2) UCI hand controller ) (b) Projected Orientation Workspaces( (1) New haptic device (2) UCI hand controller )

B.

(c) Fig. 8 Constant Orientation Workspace of Top Plate Center-Point ( φ = 0,θ = 0, γ = 0) ).(a) New haptic device (b)KAIST master (c) UCI

hand controller

Another kind of workspace is the orientation workspace that is defined as the set of all attainable orientations of the mobile platform about a fixed point. The analysis of the orientation workspace may be based on the use of a modified set of Euler angles [19]. Fig. 9(a) shows the orientation workspaces in a cylindrical coordinate system. It was found that UCI hand controller and KAIST master have similar shape and size of orientation workspace due to limitation of spherical joint angle. The projected orientation workspace defined as the set of possible directions of the approach vector (represented by two tilt angles φ , θ ) of the mobile platform may also be obtained [19]. Fig. 9(b) shows samples of projected orientation workspaces. The orientation workspaces of the existing mechanisms are about 40% smaller than that of the proposed mechanism mainly due to the spherical joint type.

Other Performances Comparison Other performance indices that have been compared are the GPI, the GCI, and the sensitivity. These have been analyzed over the constant orientation workspace ( φ = 0,θ = 0, γ = 0 ) for each mechanism. In Table 2, GPImax, GPImin, and GCI are shown for three mechanisms. The GPI of the proposed mechanism is about twice as large as that of the others. Therefore, the proposed mechanism can support double payloads compared to others. However, the GCI of each mechanism is very similar. On the other hand, the proposed mechanism is the least sensitive to the actuator perturbations at each direction. V. HAPTIC DEVICE CONTROL A.

System Hardware All links of the haptic device are made of light aluminum frames in order to reduce link inertia. Rotary motors are directly connected to lower bars of pantographs without reduction gears. This direct drive can achieve low friction and high bandwidth and eliminate backlash so that the haptic device is easily backdrivable by the operator. This construction is possible due to the fact that all 6 motors are fixed at the common base, which allows use of high power motors without increasing moving inertia. In addition, in order to reduce joint frictions miniature ball bearings are inserted at


9

every revolute joint. TABLE 2. GPI, GCI, SENSITIVITY COMPARISON

where We used a 6-axis motor controller which is connected through ISA slot to the Pentium III-450 PC which performs haptic rendering algorithm with a C program, in which calculations of forward kinematics and Jacobian, communication, and feedback control are performed at the rate of 600 Hz. Through the motor controller, the commanded torque is changed to voltage by D/A converter and this value is fed to the AC servomotor (maximum output torque of 0.951Nm) with pulse width modulation (PWM) amplifier. Motion of a motor is sensed by an encoder that gives 2048 pulse in one revolution. We used the Assurance Technologies mini-40FT model, a 6-axis force/torque sensor with 16-bit analog-to-digital conversion in increments of 0.02N (x and y-axis) and 0.14N(z-axis). This sensor can detect up to ±40 N (x and y-axis) and ±120 N (z-axis). This sensor is used to determine the differential force between the mechanism and the human operator so that a feedback loop could be used to compensate for unmodeled components of the device dynamics. Gravity Compensation In order to precisely transfer the simulated force from the virtual reality to the operator the haptic device should have negligible gravity force, friction, and inertia. Since the weight of the proposed haptic device is about 300g, which is heavy according to the pilot study [9], an operator may feel fatigue after 30-minute operation. Therefore gravity effect must be compensated in the haptic device control. The motor torques required for compensating gravitational forces of the haptic mechanism can be obtained from the derivatives of the gravitational potential energy term that is represented as

V platform = M p Z p

V2i = (1 / 2) M 1L1 sin θ 2i , V4i = (1 / 2) M 2

V1i = (1 / 2) M 1L1 sin θ1i

,

,

V3i = (1 / 2) M 2 ( L1 sin θ1i + Lmi sin θ 3i ) ,

( L1 sin θ 2i + Lmi sin θ 3i ) ,

V5i = M s Lmi sin θ 3i

,

V6i = M 3 Z 3i , θ 3i = (θ1i + θ 2i ) / 2 and where M p , M 1 , M 2 , M s , M 3 are weights of the

mobile platform, pantograph lower bar, pantograph upper bar, small connection bar in Fig. 4 (a), and the connecting bar Z p , Z 3i are ( L3 ), respectively, and where center-of-gravity heights of the platform and the connecting bar, respectively. The distance Lmi between the M i and θ i points in Fig. 2 is computed from the following quadratic equation: Lmi 2 − 2 L1Lmi cos{(θ 2i − θ1i ) / 2} + L21 − L22 = 0

(34)

(33) can be simplified as V = (1 / 2)( M 1 + M 2 ) L1 (sin θ1i + sin θ 2i ) + ( M 2 + M s ) Lmi sin θ 3i + M 3 Z 3i + M p Z p

(35)

Therefore, the motor torques from the gravitational forces of the haptic mechanism are computed as

B.

3

V = V platform +

6

∑∑Vij i =1 j =1

(33)

τ gki =

∂V ∂θ ki

= ( M 1 + M 2 ) L1 cos θ ki / 2 + ( M 2 + M s )(

∂Lmi sin θ 3i + ∂θ ki

∂Z p ∂Z 3i +Mp ∂θ ki ∂θ ki (k=1, 2 ; i=1, 2, 3)

Lmi cos θ 3i / 2) + M 3

where

∂Lmi −∂Lmi = ∂θ1i ∂θ 2i

(36)

is computed in a closed form from

partial derivatives of the equation (34). Once the forward kinematics in section II.C are solved numerically, the other partial derivatives in (36) may be computed approximately by a finite difference method as


∂Z p ∆Z p ∂Z 3i ∆Z 3i ≅ , ≅ ∂θ ki ∆θ ki ∂θ ki ∆θ ki

(37)

C.

Friction and Inertia Compensation To control force precisely, the friction and inertia of the haptic device as well as external disturbances from unmodeled operator’s dynamics should be compensated by the force/torque sensor in the feedback loop. The total control torque input in this case can be represented by τ = τ g + J T {K p ( Fdes − F ) + K d

d ( Fdes − F ) + Fdes } (38) dt

where J is the Jacobian matrix and Fdes is the desired force from VR simulation. D.

Step Input Responses Two step input responses in the x-direction with and without force sensor feedback were measured so as to investigate the open and closed loop performance of the haptic device. In this experiment, a step input of 2N in the x-axis is commanded to the haptic device while an operator grasped the haptic device handle firmly in order not to move the handle (i.e. zero velocity at the grasped position). An open loop step input response in a gravity compensation plus feed-forward control in Fig. 10(a) shows that a constant force was maintained close to the desired force within about 15% error that is caused by variation of human hand force. A closed loop step input response in Fig. 10(b) is shown with a PD feedback (Kp=0.1, Kd=0.05; gains are obtained empirically by tuning process for minimizing steady state and overshoot errors) plus feed-forward control. The overshoot and the steady state error were relatively small compared to those of the open loop response. Note that even though the hand force of an operator was varied, the differential force between the operator and the haptic device maintained a constant force, which showed disturbance rejection effect of the force feedback. Since the open-loop step response is not far worse than the closed-loop step response, the open loop impedance control by using only the gravity compensation plus the feed-forward loop may be applied to some applications which do not need high fidelity of force control. An open loop control can lower significantly the price of a haptic device by removing an expensive force/torque sensor.

(a)

(b)

Fig. 10. Step Input Responses. (a) Open loop step response loop step response

(b) Closed

10 VI.

DYNAMIC PERFORMANCE EVALUATION

Until now, experimental performance evaluation of only small number of 6-dof haptic devices has been reported [9]. We have evaluated experimentally dynamic performance of force characteristics such as force bandwidth, static-friction breakaway force, and extremum controllable forces. Ellis et al. [9] suggested two constraint conditions in the general task of free and unimpeded motion of the wrist in the plane for haptic device design: human operator constraints and mechanical constraints. As for the human operator constraints, maximum and minimum values of the controllable motion of a human operator are: Workspace of 15 cm × 15 cm , Minimum force of 1N, Peak force of 40N, and Force bandwidth of 50Hz. As for the mechanical constraints arising from considerations of the mechanical interaction of the human with the device, any haptic device (1) should keep low apparent mass and inertia for less operator fatigue, (2) must have an internal friction level of at most 5% of the peak force, (3) should have high native mechanical stiffness, and (4) should have no kinematic singularities in the operating workspace. We will evaluate the performance of our haptic device with respect to these constraint conditions even though the human operator constraints studied by Ellis et al. are for planar free wrist motion. For force characteristics, we evaluated maximum applicable force, static-friction breakaway force, and minimum controllable force. The maximum applicable force was measured by the force sensor attached to handgrip when the computer commanded a constant force in an open loop control mode while the operator keeps the handle unmoved. The static-friction breakaway force is defined as the minimum open-loop force increment when the change of the end-effector position was position resolution of the end-effector. This force is measured by incrementally increasing command force when the haptic device stands alone by the gravity compensation control without the human operator. The minimum controllable force was measured by commanding a closed-loop proportional controller to maintain a zero differential force on the operator handle. This procedure was to move the device by voluntary hand motion, and record the resulting force and position. Force bandwidth in haptic device is affected by a number of factors such as stiffness, inertia, damping, friction, actuator limiting, contact, sensor/actuator collocation, gains, and operator impedance [20]. The dynamic bandwidth of the proposed mechanism is expected to be higher than that of the UCI controller and the KAIST master because moving inertia is significantly reduced by the base-fixed motor design even though the mechanism stiffness may be somewhat decreased due to more links and joints in RRR type spherical joint design. The force bandwidth was measured by the device response while the human operator grasping the handle as they would when interacting with virtual environments. The computer commanded the device to maintain an offset-sinusoidal differential force. Then, the peak-to-peak


amplitudes were recorded at each frequency in the range of 1-120Hz. The experimental results for force characteristics were as follows; a maximum applicable force was different for each axis. The maximum force in the z-axis is 40N and the force in the x and y-axis is 20N. The static-friction breakaway force was 1.5N in the z-axis direction. Friction force, therefore, was within 5% of the maximum applicable force. Fig. 11 shows the minimum controllable forces and torques. The force that induces the device motion is less than 0.4N (in the z-axis direction) and 0.3N(in the x-axis and y-axis directions) as shown Fig. 11(a). The torques are less than 0.01Nm as shown in Fig. 11(b). The position and orientation are changing slightly during this minimum controllable force test with the max speed of about 15cm/s and 0.2 rad/s, as drawn in Fig.11(c) & (d).

11 Table 3 summarizes the performance characteristics of the proposed device. Comparing to the constraints suggested by Ellis et al. [9], the proposed haptic device sufficiently achieved the human operator and mechanical constraints. TABLE 3. NEW HAPTIC DEVICE CHARACTERISTICS

VII.

(a)

VIRTUAL WALL SIMULATION

This chapter presents a virtual wall simulation by using the developed haptic device to observe the closed-loop response of virtual environment simulation. Virtual wall simulation has been used as a representative task featuring both very high impedance (when in contact with the wall) and very low impedance (when out of contact) [21,22]. The virtual wall can be modeled as an equivalent spring-damper system. The force from the wall can be written as

(b)

& ] Fdes = −ς [ K (X h − X wall ) + γBX h

(c)

where ζ and γ are unilateral constraints, K is the virtual wall spring constant, and B is the virtual wall damper coefficient. The unilateral constraints ζ and γ are given as

(d)

Fig. 11. Minimum Controllable Force. (a) force Fx, Fy, Fz Ty, Tz (c) position x, y, z (d) orientation φ ,θ , ϕ

(b) toque Tx,

In Fig. 12, the force bandwidth of the x-axis direction is shown to be about 70 Hz at the cross over frequency of –3dB below the DC level. This response is fast enough for some haptic applications, satisfying the maximum force bandwidth (50Hz) from the pilot study [9] for the human wrist free motion.

Fig. 12.

X-axis Force Bandwidth

(39)

⎛1 X > X

⎞

h wall ⎟⎟ , γ ς = ⎜⎜ ⎝ 0 X h ≤ X wall ⎠

⎛1 X& h > 0 ⎞ ⎟ =⎜ ⎜ 0 X& ≤ 0 ⎟ h ⎝ ⎠

(40)

The unilateral constraint ζ ensures that the force is exerted to the end-effector only when the handle penetrates the wall. Similarly, γ ensures that the damper exert no force when the handle is being moved away from the wall. A total control block diagram is represented in Fig. 13. The difference between the position X h of the haptic device driven by a human operator and reference value X wall that is the position of a virtual wall is represented by X. Then, the value of X is used to generate a virtual wall reaction force by the (39) for the human operator. In this block diagram, the dotted box represents nonlinear kinematics/dynamics. In this virtual wall simulation system, both control of the haptic device and haptic rendering are performed by the haptic controller with Pentium III-450 processor which is equipped with 6-axis motor controller. The virtual wall environment that is modeled by the WTK (World Tool Kit) software is simulated in a Pentium II-dual 350 processor NT system with a high-speed graphic board. The rendering of the virtual


12

environment is performed at the rate of about 40Hz. Both computers are interfaced by a RS-232 communication at the rate of 57600bps.

(b)

Fig. 13.

Control Block Diagram including Virtual Environment

Through this simulation we can find achievable wall impedance while keeping the system stable. We define instability as the situation in which vibration occurs at the boundary of a virtual wall during wall contact operation. The maximum achievable wall stiffness without inducing vibration is measured to be about 400N/m when no wall damping exists while the maximum achievable wall damping is measured to be about 1500 N/m ⋅ sec . If there are both stiffness and damping, the maximum achievable stiffness with a damping value of 1000 N/m ⋅ sec is increased to 600N/m for a stable virtual wall simulation. Fig. 14 shows results of a simulation of moving down along the virtual wall with the parameters (K=250N/m and B=10 N/m ⋅ sec ). In this simulation, an operator first approached the wall in the x-direction while grasping the haptic device. After contacting the virtual wall, the operator moved down in the z-direction while contacting the virtual wall (see Fig. 14(a)). During this operation, the operator could maintain contact with the surface of the virtual wall due to the force feedback toward the operator through the haptic device (see Fig. 14(b)). At this experiment, the virtual wall feels like smooth mud as depicted in Fig. 14(c).

(a)

(c) Fig. 14.

Virtual Wall Simulation Results

VIII.

CONCLUSIONS AND DISCUSSIONS

This paper proposed a new tool type haptic device based on a parallel mechanism for improving static and dynamic quality of haptic interface in virtual reality simulation. This mechanism has small moving inertia because motors are fixed to the base plate and has larger orientation workspace because there is only small rotational limitation in the RRR type spherical joint. Through an optimal design and kinematic analyses, it has been shown that the proposed haptic device has better static performances such as wider orientation workspace, lesser sensitivity with respect to actuator perturbations, and higher force transmission capability compared to the mechanisms of similar mechanical architecture. However, the constant orientation workspace is smaller. In addition, possible rearrangements of the proposed mechanism are discussed for more versatile usage. This paper also presented control methods, performance evaluation, and virtual wall simulation. It has been shown that open-loop gravity compensation plus feed-forward control may be good for less precise control because this device has low friction and small backlash as manifested by the result of the open loop step input response. Meanwhile, the force feedback control to compensate the effects of inertia and friction using a F/T sensor may be used for more precise and stable control. By experiments, dynamic performance characteristics such as force bandwidth, minimum controllable force, and static-friction force have been evaluated. This new haptic device has force bandwidth of 70Hz, the maximum force of 40N in the z-direction, minimum controllable force of 0.3N in the x and y directions, and wide orientation workspace. Finally, we demonstrated a virtual wall simulation using the proposed haptic device. The measured maximum achievable wall stiffness (about 600N/m) may be low for applications


requiring harder contact operation. This low wall stiffness may be increased by; higher sampling rate (currently 600Hz) by superior PC, use of material with higher stiffness, higher rate of communication, use of admittance control instead of impedance control, and so on. Even though the proposed haptic device has not been designed for any specific application, users can choose it for their particular application by understanding the static and dynamic performances evaluated in this paper.

13 [19] I. A Bonev, and J. Ryu, “A New Approach to Orientation Workspace Analysis of 6-DOF Parallel Manipulators”, Mechanism and Machine Theory, Vol. 36, pp.15-28, 2001. [20] T. Brooks, “Telerobotic Response requirements”, Proc. IEEE Int. Conf. on System, man, and Cybernetics, NY. pp.113-120, 1990. [21] J. E. Colgate, P. E. Grafing, M. C. Stanley, and G. Schenkel, “Implementation of Stiff Virtual Walls in Force-Reflecting Interfaces”, Proc. IEEE VR Annual Inter. Symp. (VRAIS), New York, pp.202-208, 1993. [22] S. E. Salcudean and T. D. Vlaar, “On the Emulation of Stiff Walls and Static Friction with Magnetically Levitated Input/Output Device”, ASME J. of Dynamic systems, Measurement, and Control, Vol. 119, pp.127-132, 1997.

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Jungwon Yoon received the B.S. degree in precision mechanical engineering in 1998 from the Chonbuk National Univ., Chonju, Korea, and the M. S. degree in the Department of Mechatronics in 2000 from Kwangju Institute of Science and Technology (KJIST), Kwangju, Korea, where he is currently pursuing the Ph.D. degree. His research interests include the design, analysis, and control of parallel mechanisms with applications to virtual reality haptic devices and medical systems.

Jeha Ryu received the B.S degree in 1982 from the Seoul National Univ., Seoul, Korea, the M. S. degree in 1984 from the Korea Advanced Science and Technology (KAIST), Seoul, Korea, and the Ph. D. degree in 1991 from the Univ. of Iowa, Iowa City, U.S.A., all in mechanical engineering. From 1992 to 1994, he had worked as a master engineer in the simulation lab of BMY Combat Systems, York, PA. In 1994, he joined the Department of Mechatronics, Kwangju Institute of Science and Technology (KJIST), Kwangju, Korea, where he has been an associate professor since 1999. In 1999, he received a best-educator award from KJIST. His research interests are kinematics, dynamics, and control of mechatronics systems such as robot manipulators, vehicle systems, and haptic joysticks for interfacing with virtual reality systems. He published more than 40 international journal and conference papers in these areas. Dr. Ryu is a member of ASME and IEEE.