Small Scale Parallel Manipulator Kinematics for Flexible Snake Robot Application

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Communications in Control Science and Engineering (CCSE) Volume 2, 2014

Small Scale Parallel Manipulator Kinematics for Flexible Snake Robot Application Raisuddin Khan*1, Md. Masum Billah2, Mitsuru Watenabe3, Amir Akramin Shafie4 Depart ment of Mechatronics Engineering, International Islamic University Malaysia, 53100 Kuala Lumpur, Malaysia *1

2

3

4

raisuddin@iium.edu.my; mdmasum.b@live.iium.edu.my; nabe@yahoo.com; aashafie@iium.edu.my

Abstract A small-scale parallel manipulator is designed in this paper. The kinematic analysis of the manipulator is also elucidated for the development of multilinked snake robot. A compliant central colum is used to connect two parallel platforms of Incompletely Restrained Positioning Mechanism (IRPM). The compliant column allows the configuration to achieve 3 DOFs with 3 tendons of active materials connected between the upper and loer platform of the mechanism. In particular, this investigation f ocuses on the angular deflection of the upper platform with respect to the lower platform. The application here is a imed at developing an active linkable module that can be connected to one another so as to f orm a “snake robot” of sorts. For an arbitrary angular displacement of the platforms, the corresponding length of each tendon can be determined through inverse kinematics. From the experimental result, the extreme bending of the central column plane of 30° angular displacement with the of the horizontal axis.

Keywords 3 DOF; Stewart Platform; Tendon

Introduction During the last two decades research continued on parallel robots, and today many, especially the planar parallel manipulators have found their way into practical applications such as positioning devices, motion generators, an d [1] ultra-fast pick and place robots . P erhaps the most well-known and widely used parallel robot is the Stewart platform, which gained prominence due to its use in flight simul ators. The traditional Stewart platform has given rise to other variations due to modification either of the act uators or joints. Of these variants the ones of interest here are firstly those using compliant members, an d secondl y pl atforms using tendon, or cabl e actuation.

In Korea Choi et al proposed a passi ve compliant Stewart platform to accurately measure both position and force on [2] each member of the platform . Though the compliant members were not used as active actuators, their use in [3] Stewart platform open s up new grounds. Moon and Kota , took this concept and applied it to active members. They added compliant joints to piston-like linear actuators and incorporated a central constraining leg when fewer than 6 DOFs were required.

The idea of the constraining central leg, introduces another method of configuring the Stewart platform. In the area of tendon-driven or cable-suspended platforms, a few configurations have been proposed. The best known cable[4, 5] , where the end-effector sits on a platform suspen ded plan ar parallel robot is the RoboCranes developed at NIST suspen ded from a fi xed frame using 6 cables to achieve 6 DOF. Issues regarding the workspace an d design of general [6] cable-based planar robots were addressed by Fattah and Agrawal , where the pl atform was not only suspended but also constrained by cables from above and below. As only tension force coul d be tolerated by a ten don, analysi s needs be performed to keep the tendon under tension. Sometimes the end-effector is suspen ded by ten dons and use of the [7] gravity force or any other passive force en sure operation of the platform . Another more applicable sol ution for high acceleration applications is to use redun dant act uators, and to resolve the redundancy to ensure positive tension in [8] all the tendon s. This can be performed in a fully–constrained or over–constrained moving platform . However, [9] [10, 11] . analysis of the geometry or possible workspace is very difficult From the advantages of the compliant members and of the tendon-act uated pl anar parallel mechanisms, a design is suggested h ere where both the compliant aspect as well as tendon act uation could be used to develop simpl e miniature Stewart platforms that could be used as linkable modules for the locomotion of flexible robots.

Proposed Model [12]

One of the more comprehensive treatments given to tendon-driven Stewart platforms is that of Verhoeven et al. .

In their formul ation, they divi ded such platforms into two major types; the first being the “Compl etely Restrained Positioning Mechanism” (CRPM), where t endons are attach ed both above and below the pl atform. This makes

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perfect sense since tendons are only able to exert t ensile forces and not compressive, and therefore are paired into opposing sets, much like antagonistic muscles foun d in limbs of living organisms. However, another cable i s necessary to balance forces an d effective torques to maintain stable positions. The second type of platforms is known as the “Incompletely Restrained Positioning Mechanism” (IRPM) where ten dons are positioned only below or above the platform. In this configuration an external force must be added to reverse the tensile actuation of the wires. Th e most common example of such a force is gravity, as in the RoboCran e where the platform is suspended by cabl e attached to fixed points above it. In this configuration, as the wire is relaxed or released, the w eight of the platform will force it to traverse to the desired position. Since gravity acts vertically in the –z direction, it does not generat e torque about z and hence an extra cable is not necessary to stabilize the platform, allowing n cabl es to produce n [13, 14] .

DOF The proposed design, here as shown in Fig. 1, also falls into the IRPM, but rather than relying on gravity to provide the ext ernal force, a compliant member is added to balance the external forces by the cords at any orientation of the platforms. That is, unlike the RoboCran e while it can only function with cable above the platform, the new configuration will allow the platform to operate with cables below it, much like the traditional Stewart platform, or the whole configuration can be tilted to be supported from any direction.

FIG. 1 PROPOSED MODEL CONFIGURATION

In the proposed Stewart pl atform, both the upper and lower platforms are of configurations of equilateral triangles with cables attached to the vertices. Each cable attaches each of the vertices of the lower platform to the vertices directly above the upper platform. In order for the upper platform not to collapse onto the fixed base platform, a compliant central col umn must be added. The three tendons thus enable the platform to exhibit 3 DOF. The joints between the column an d the upper an d the low er platforms are fixed. The central column at the point of attachments with the platforms will therefore always form right angle with the corresponding platform regardless of orientation.

Moments an d forces are transferred into the column at both ends; hence the col umn can be modeled as a cantilever beam with moment and force acting upon the free en d. Furthermore, the column needs to be able to compress upon itself as well as bend in the desired direction to achieve 3 DOF. If the column were to bend only, then only 2 DOF i s achievable. Th us, a spring of a fairly l arge di amet er is proposed as the central support. The spring is able to undergo compression as well as bending, an d the large diameter will prevent the spring from buckling out of position when a bending force is applied together with compression. With this configuration, the three ten dons can be act uated indepen dently to achieve the desired angle angles between the platforms.

To determine the output , three generalized coordinates are selected. Th e first is the height of the central column, h , effectively a transl ation along z, which can be compressed by applying tension on all three ten dons as shown in Fig. 1.

The second and third coordinate are the angles formed by the upper platform and the lower platforms. Rather than composing this angles from rotations aroun d x an d y, spherical-like coordinates are adopted. That is, in the x-y plane, angle θ denotes the anti-clockwise angle measured from the positive x- axis. The projection of the bent central column onto the x-y plane is therefore assigned the vari able d. The secon d angle φ thus denotes the angle between the plane of the upper platform and d. This angle differs from the spherical coordinate equivalent, as the vertex of this angle is not at the origin.

Given the coordinates of h, θ, and φ in Fig. 2, the Cartesian coordinat es x, y, z, can be determined, which is in turn used to calculate the length of the three cables. Before performing the inverse kinematics assumptions are made in order to simplify the calculations. The assumption made here is that for an y desired position of the end-effector, the

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central col umn shall always bend to assume a circular arc of radius of curvature R. Though in reality the column may not necessarily form a circular arc. Th e validity of this assumption shall be investigated upon constructing a physical model. z y

Op (x,y,z) R

Ob (0,0,0)

φ

θ d

x FIG. 2 PROPOSED MODEL BENDING GEOMETRY

Inverse Kinematics For every position of the upper platform, lengths of each of the ten dons are determined using inverse kinematics.

Since both the lower and upper pl atforms have three points of attachment, the length from one point on the lower platform to the corresponding point on the upper pl atform can be determined by t aking the norm of the vector. To simplify the vector calculations, reference points are introduced: one on the center of the base, Ob, and one on the upper pl atform Op. The points where the tendons are attached to the base are named as B1, B2, and B3. As these points are at the vertices of a equilateral triangle, that the points are lying exactly on a circle of radius r. Th us the position vectors with respect to Ob are,

 r cos 30°    −r cos 30° 0    RB1 =  r  , RB2 =  −r sin 30° , RB3 =  −r sin 30°  0      0 0

(1)

Similarly the points on the upper platform with respect to Op are,

0   r cos 30°    −r cos 30°  RP1 =  r  , RP2 =  −r sin 30° , RP3 =  −r sin 30°  0      0 0

(2)

Before continuing further with the determination of cable lengths, the transformation from one set of coordinates to another must be delineated. From the coordinates determined earlier as h , θ, and φ, the first value that must be calculated is that of the radius of curvature, R, of the central spring, given by,

R=

180h

πθ

(3)

Then the coordinates (x, y, z) is determined to be,

= x R (1 − cos φ ) sin θ = y R(1 − cos φ ) cos θ z = R sin φ

(4)

As for the positioning mechanism, in order to determine the vector from B1 to P 1, for example, when the end- effector is positioned at [ x, y, z ] , the original vectors plus the transformation vector from will be added.

T

    B1 P1 = − RB1 + Ob O p + RP1 Which, in column vector form, would be,

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(5)

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  0   x  0  B1 P1 = −r  +  y  +  r   0   z  0 

(6)

The orientation of the upper platform is then broken down into roll, pitch, yaw angles. As there is no rot ation about z axis in this application, only the pitch an d the yaw angles need to be determined from the input coordinates of θ an d φ. Assigning α as rotation about x axis an d β as rotation about y axi s, the following is calculated,

 tan φ    sin θ   tan φ  β = tan −1    cos θ 

α = tan −1 

(7)

Using angles α and β, the position vectors of the platform in terms of the transforming coordinates would be,

 cos β  RPt  0 =  − sin β

sin α sin β cos α sin α cos β

cos α sin β  − sin α  Pi cos α cos β 

(8)

 l3 = B3 P3

(9)

For any position and orientation, then, the lengths are given by,   l1 = B1 P1 , l2 = B2 P2 , Thus the val ues for l 1 can be expressed by the following qeuations,

 0   x   cos β l1 =  −r  +  y  +  0  0   z   − sin β

sin α sin β cos α sin α cos β

cos α sin β  0  − sin α   r  cos α cos β  0 

(10)

Or, 1

(11)

l1 ([r sin α sin β + x]2 + [r cos α − r + y ]2 + [r sin α cos β + z ]2 ) 2 = Similarly l2 an d l3 can be determined by,

l2 = ([−r cos 30 + x + r cos β cos 30 − r sin α sin β sin 30]2 + [r sin 30 + y − r cos α sin 30]2 1

+[ z − r sin β cos 30 − r sin α cos β sin 30]2 ) 2 l3 = ([−r cos 30 + x + r cos β cos 30 − r sin α sin β sin 30]2 + [r sin 30 + y − r cos α sin 30]2 1

+[ z − r sin β cos 30 − r sin α cos β sin 30]2 ) 2 Then adjusting each of the cables to the calcul ated lengths woul d result in the desired position of the upper platform.

Fig. 3 an d Fig. 4 show the simulated result for different configuration of the upper platform and the central col umn.

Snake

Tendons Length vs Different Configuration of Upper Plarform 0.16 Tendon1 Tendon2 Tendon3

0.14

0.04 0.1

Length (m)

Tendon Length (m)

0.12

0.08 0.06

0.03 0.02 0.01 0 0.04

0.04

0.04 0.02

0.02

0.02

0

0 0

1

2 Different Configuration of Upper Plarform

3

FIG. 3 LENGTHS OF THE 3 TENDONS FOR DIFFERENT CONFIGURATION

-0.02

-0.02 Y

-0.04

-0.04

X

FIG. 4 BENDING OF THE CENTRAL BEAM IN DIFFERENT CONFIGURATION

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Experiment An experiment has been carried out to demonstrate the effectiveness of the proposed design. The platform that w e use for the experi ments is one of the link of Smart Mat erial Act uated Robotic Snake I (SMARS-I) [15] as shown in Fig. 5. The platform consists of a flexible central column and three EAP tendons are attached for ten don-driven Stewart platforms actuator. The act uator are pow ered externally using el ectric supply.

In this experi ment, the link represents the 3DOF movement according to the proposed kinematic analysis. Fig. 5 shows six angular di splacements between two parallel platform attached at the two ends of the actuator against different voltages in different tendons.

If 3 t endons' lengths are same the central compliant column only shortens along the z-axis of the actuator and the top an d bottom platforms remain approximately parallel. In the following step, adjust ment of the lengths of the three tendons help change the angul ar position between the two platforms in a 3D space. Thus the actuator has a wide scope of 3D work space. Fig. 6 shows the 30째 angle between the plane of the upper and lower platform.

1

2

3

4

5

6

FIG. 5 PROTOTYPE PRESENTS THE ACTUAL PROCESS OF ACTUATOR MOVEMENT (SMARS-I)

x

120째

O

y

FIG. 6 BENDING GEOMETRY OF THE PROTOTYPE

Conclusion This research shows the design an d kinematic analysi s of a small-scale parallel manipulator. In particular, this investigation focuses on the ang ular defl ection of the upper platform with respect to the lower platform. A manipulator consisting of an IRPM mech anism that contains a compliant central colum connected between two parallel pl atforms is capable to demonstrated angular di splacment between the platforms. The angular displ acement of 30째 between the platforms were achived experimentally. It was also found that the curvature of the central compliant col um assumes approxi mately a shape of a circul ar arc within this 30째 ang ular deformation.

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ACKNOWLEDGMENT

This research i s supported by Sci ence Fund, Ministry of Science an d Technology (MOSTI), Mal aysi a.

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