Jacobian Analysis of Limited DOF Parallel Manipulator using Wrench and Reciprocal Screw Principle

Jacobian Analysis of Limited DOF Parallel Manipulator using Wrench and Reciprocal Screw Principle

Hassen Nigatu1

Department of Mechanical & Vehicle Engineering

Adama Science & Technology University, Adama, Ethiopia

Ajit Pal Singp

Manufacturing Section Production Engineering Department

College of Engineering Defence University, Bishoftu, Ethiopia

1. Prabhu3

   Research Scholar, Department of Engineering & Technology Himalayan University Arunachal Pradesh, India.

   AbstractThis paper presents a new methodology of formulating Jacobian matrix for limited degree of freedom (DOF) parallel kinematic machine (PKM), which is a very important tool to relate the end-effector velocity with the joint rate velocity. Even if it is believed by many researchers that Jacobian matrix is critical to generating the trajectories of the prescribed geometry in the end-effector space, it was cumbersome to formulate in simple and descriptive form by partial differentiation. In this work screw mathematics is used to formulate the Jacobian matrix in simple and integrated form

   under a unified framework and it is proved that the resulted

   smart approach is to drive an input-output velocity relationship from which the Jacobian matrix of such a manipulator can be formulated [3, 5]. While this approach is valid for general purpose planar and spatial parallel manipulator for which the connectivity of each serial chain(limb) is equals to the mobility of the moving platform, it is not necessarily true for parallel manipulators with less than 6-DOF. For example, this approach leads to 3 3Jacobian matrix for the 3-UPU parallel manipulator assembled for pure translation [4, 6]. However, such a 3 3 Jacobian matrix

   Jacobian matrix is 6 6 which provides clear information about

   the architecture and singularity condition of the manipulator. Obtaining Jacobian matrix in unambiguous way is very crucial step to formulate and solve velocity, acceleration and motor torque equations with less computational burden. The 3PRS parallel kinematic machine is selected as an example to demonstrate the methodology. The numerical solution is obtained using MATLAB.

   KeywordsJacobian analysis; limited DOF; end-effector, joint space

   1. INTRODUCTION

      The development of limited-DOF (degree of freedom) PKMs (parallel kinematic machines) has been hot research area due to their advantage in terms of simple structure, lower cost and easy to control comparing with 6-DOF parallel manipulators. The appearance and application of limited DOF PKMs with coupling of translation and rotation provides an option for a bottleneck problem of the manufacture and

      cannot predict all possible singularities and cannot provide full information about the architecture.

      In what follows, the researcher develops a methodology for the Jacobian analysis of limited DOF parallel kinematic machines. The Jacobian matrix so derived provides full information about both architecture and constraint singularities. The 3-PRS parallel manipulator [5, 7] is used to illustrate the methodology.

   2. JACOBIAN ANALYSIS

      In this section generalized Jacobian for limited-DOF parallel manipulator is developed. A limited DOF manipulator possesses F-DOF, where F is between 0 and 6. Also it is considered that the moving platform is constrained by F number of limbs and each limb is driven by one actuator. Then apply the theory of reciprocal screw to find the Jacobian matrix that can relate the joint rate and end-effector velocity. In this regard, the instantaneous twist

      assembly for large components in aircraft and automobile ci

      industries. As an outstanding representation of the limited- DOF PKMs, the 3PRS (prismatic-revolute-spherical)

      S p j,iS j,i j1

      (1)

      manipulator has been applied to many areas because of compact architecture and excellent kinematic and dynamic

      Where ci

      is joint connectivity

      j,i is intensity and

      S

      j,i

      performances. For example, the famous sprint-Z3 head made by DS Technology Company in Germany [1, 2].

      It is well known that the Jacobian of 6-DOF general purpose parallel manipulator is 6 6 matrix, relating the end- effector linear and angular velocities to the six input joint rates. However, it is not clear as to what is the best way to express the Jacobian of limited-DOF parallel manipulator. A

      represents a unit screw associated with jth joint of the ith limb.

      The twist of the moving platform is defined as

      S p T vT T

      Where, is angular velocity of the moving platform and

      v is the linear velocity of a point in the moving platform

      which is instantaneously coincident with the origin of reference frame in which the screws are expressed.

      wrench of constraints and S T tc,1,i twist screw of the

      The C numbers of screw on a limb forms nth systems for

      constraint if each limb and generally Jc and Ja are Jacobian

      i

      which a one system of reciprocal screw exists. This reciprocal screw forms a Jacobian of constraint and it is reciprocal to all the joint screw of the ith limb from a (6-Ci) reciprocal screw system. Since this reciprocal screw can be identified let

      of constraint and Jacobian of actuation respectively.

   3. METHODOLOGY

      In this section the 3DOF parallel manipulator is used to

      S

      r, j,iS p

      denote the jth reciprocal screw of the ith limb.

      demonstrate the methodology (specifically named 3PRS high

      speed machine).

      Therefore taking the orthogonal product both sides of Eq. (1) with each of reciprocal screw gives

      r

      p

      S T j,iS 0 for j=1,2,3–6-Ci (2)

      Writing equation to once for each limb produces the following constraint matrix.

      In this manipulator there are three limbs which connect the moving platform at point Bi and to the fixed base at point Ai. Also each link in a limb is connected by three joints prismatic, revolute and spherical respectively as shown in the following Fig. 1.

      S T r,1,1

      S T r,1,2

      Jc .

      .

      S T r,6CF,F

      Each row in Jc represents a unit wrench of constraints imposed by the joint of a limb and it is known that its rank

      should be equal to 6 Ci to properly constrain the moving

      platform. Then the generalized Jacobian for 3 DOF manipulator will be

      S T wa,g ,1

      Fig. 1. Schematic diagram of 3PRS parallel robot.

      1 S T ta,g ,1

      Jc,1

      1

      Each limb connects the fixed base to the moving platform

      2

      S T wa,g ,2

      Jc,2

      by a prismatic joint followed by revolute joint and spherical

      J

      J Ja ,

      J a

      S T ta,g

      2

      .

      ,2 .

      , Jc ,

      .

      joint. A linear actuator drives each of the three prismatic joints. The connectivity of each limb is equal to five.

      c .

      Therefore the instantaneous twist, S p of the moving platform

      .

      .

      J

      can be expressed as a linear combination of 5 instantaneous screws.

      S T wa,gf, f

      S T ta,gf, f

      c,i

      sp d

      1,is1,i

      • 2,i

        s

        2,i

      • 3,i

        s

        3,i

      • 4,i

        s

        4,i

      5,i

      s

      5,i

      (4)

      S T wc,2,i

      S T wc,1,i, S T tc,1,i

      Where,

      s1

      (bi li ) s2

      bi s3

      S T wc,2,i, S T tc,2,i

      s1,i 0 , s2,i s

      , s3,i s ,

      2

      3

      Jc,i .

      .

      s

      4,i

      bi

      s

      • s4

      , and s

      5,i

      bi

      s

      s5

      .

      S T wc,6 – n,i

      4

      5

      S T wc,6 – n,i, S T tc,6 – n,i

      Where si is a unit screw along the ith direction of each

      (3)

      1

      Where, S T wa,g ,1 is the wrench of actuation,

      limb. These five screw forms a five system for which a one system of reciprocal screw exists. This reciprocal screw lies on the intersection of the two planes the first plane is

      1

      S T ta,g ,1is twist screw of actuation,

      S T wc,1,i

      is the

      perpendicular to the prismatic joint and the second plane is containing both the revolute and spherical joint. Fig. 2 shows

      the reciprocal screw which is orthogonal to all instantaneous screws except s1 and parallel to s2.

      Take the inner product of this reciprocal screw with for both sides of the twist screw (1) gives

      sp sr2,i d i (s1,i sr2,i )

      s

      T

      3,1

      (b1

      s3,1

      )T

      Jx s

      T 3,2

      (b2

      s3,2

      )T

      (8)

      s

      T 3,3

      (b3

      s3,3

      )T

      Again, from the right hand side of the inner product we obtain jq .

      sp sr2,i d1(s1,i sr2,i )

      s1 bi s3

      0 s

      3

      s Ts 0 0

      3,1

      J 0

      1,1

      s Ts 0

      Fig. 2. Representation of the unit wrench screw.

      q

      0

      3,2

      0

      1,2

      s Ts

      S r,1,i

      bi

      s2,i

      s

      (5)

      3,3

      1,3

      2,i

      Since this mechanism is not outer driving manipulator

      Taking the orthogonal product of both sides of equation

      one we found Jc .

      jq will not be identity matrix.

      S p

      S

      r,1,i

      0 , since the constraint wrench is reciprocal

      J Jx

      a

      Jq

      to all screw, the right hand side equation will be set to zero. The constraint Jacobian of the manipulator along the revolute joint axis will be found where its row represents a unit wrench

      s T

      3,1 T

      (b s )T

      1 3,1

      T

      constraining the degree of freedom of the moving platform. Since, 3DOF manipulator is a capable of three independent

      s3,1 s1,1

      s Ts

      (b s

      s3,1 s1,1

      )T

      DOF, but moves all in six, it is expected that the Jacobian of

      Ja 3,1

      1,2

      T

      2 3,2

      T (9)

      constraint would be composed of three rows.

      s3,2 s1,2

      s3,2 s1,2

      s T

      (b s

      )T T T

      J s

      2,1

      T

      1 2,1

      (b s

      )T

      (6)

      s3,3 s1,3

      s Ts

      (b3 s3,3)

      s Ts

      T

      c 2,2

      2 2,2

      3,3

      1,3

      3,3

      1,3

      s

      (b s )T

      Ja

      2,3

      3 2,3

      and we have seen that J is J .The generalized Jacobian

      This matrix represents the constraint imposed by the revolute joint. Then we look for the reciprocal screw for each limb which forms two systems. An additional basis screw which is reciprocal to the passive joint of the ith limb can be identified as zero pitch screw passing through the center of spherical joint. This reciprocal screw represents wrench of actuation and it is normal to the previous one system. This can be represented as follow,

      Jc

      for this manipulator will be 6 x 6.

      By combining the Jacobian of actuation and Jacobian of constraint we formulate the general Jacobian matrix as shown above which is very important to solve the velocity of the active joint.

      This 6 6 dimensional matrix characterizes the

      s

      r2,i

      bi s

      s

      3,i

      (7)

      instantaneous motion of the moving platform. The upper sub- matrix transforms the linear and angular velocity of the moving platform to the actuated joint rate. The constraint

      3,i

      singularity will be analyzed by evaluating the rank of Jc where as the architectural singularity occurs when the determinant of the overall Jacobian is equals to zero.

   4. NUMERICAL SOLUTION

      All numerical solutions shown are the sub and generalized Jacobian matrix of limited DOF parallel industrial robot and ranks of sub-matrices which tells us architectural and constraint singularity conditions.

      s T

      (b s )T

      3,1 T

      1 3,1

      T

      0.0009

      0.4251

      1.0000

      0.0195

      0.0000

      0.0000

      s3,1 s1,1

      a

      s Ts

      (b s

      s3,1 s1,1

      )T

      J 0.3691

      0.2126

      1.0000

      0.0078

      0.0144

      0.0002

      3,1

      1,2

      T

      2 3,2

      T

      0.3671

      0.2125

      1.0000

      0.0078

      0.0144

      0.0002

      s3,2 s1,1

      T

      s3,2 s1,2

      T

      0.5000

      0.8660 0

      0.0053

      0.0031

      0.0084

      c

      J s3,3 s1,3

      s Ts

      (b3 s3,3)

      s Ts

      J 0.5000

      0.8660

      0 0.0027

      0.0015

      0.0177

      s T

      3,3

      1,3

      (b s )T

      3,3

      1,3

      1.0000 0

      0 0 0.0031

      0.0084

      2,1

      1 2,1

      0.0000

      0 1.0000

      0.0958

      0.2165

      0.0000

      s

      T

      2,2

      (b s )T

      0.0000

      0 1.0000

      0.0958

      0.2165

      0.0000

      2 2,2

      T T

      s2,3

      (b3 s2,3)

      0 0.1685

      J

      1.0000

      0.2186

      0.0000

      0.0000

      Equation (10) tells us how we use this Jacobian to relate the moving platform with joint rate velocity.

      0.9470

      0.9470

      1.6402

      1.6402

      0 0.1318

      0 0.1318

      0.0761

      0.0761

      0.4458

      0.4458

      q JS p

      (10)

      1.9206

      0.0000 0

      0.0000

      (10)

      0.3086

      0.3678

      Where q is the joint rate velocity which can be represented as follows in matrix form

      0.9203 0 0

      Jq 0 0.9200 0

      0

      0 0.9206

      d1

      d2

      (11)

      Rank (Jc)=3 and rank (Ja)=3

      (11)

      d1

   5. CONCLUSIONS

3

and

S p is the twist screw which contains the linear and

In this paper it is shown that the Jacobian of small degree of freedom parallel kinematic machine can be driven using

angular velocity of the moving platform. It can also expressed

mathematically as

reciprocal screw which is not common for such application.

The generalized Jacobian (J) is 6 x 6 matrix consists of

vp

two sub-matrices, one associated with the constraint imposed

S P

p

(12)

by the joint and the other associated with actuator movement. A manipulator is said to be an architecture singularity when

Use the generalized Jacobian and expanding (10) will let to solve the active joint velocity in simple and unified form.

the actuation sub-matrix becomes rank deficient or the overall Jacobian becomes rank deficient while the constraint sub- matrix have full rank. On the otherhand the manipulator will

s

T 3,1

s Ts

(b1 s

3,1)

T

s Ts

be constraint singularity when the constraint sub-matrix becomes rank deficient. Therefore, conclude that one can get

s Ts

3,1

1,1

(b s

3,1

)T

1,1

d1

information about the manipulator sim(1p3ly)

like architectural

3,1

1,2

2 3,2

and constraint singularities from the Jacobian matrix by

s Ts

s Ts

d 2

obtaining it in unified and simple form.

3,2

1,1

3,2

1,2

s Ts

(b s )T

v0 d 3

3,3 1,3

3 3,3

s Ts

s Ts n 0

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3,3 1,3

(13)

3,3 1,3

s

T 2,1

(b s )T

0

0

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   (b s )T

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   T T

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