Kinematic Design Optimization of Planer-link Mechanism based Manipulator

Kinematic Design Optimization of Planer-link Mechanism based Manipulator

Dr. J. M. Prajapati

Department of Mechanical Engineering, The M S University of Baroda, Vadodara, India

Abstract This paper presents optimized kinematic design of a planer mechanism (4-link) based planar manipulator is presented in this paper. In case of the parallel manipulator, there is only one location defined between force and motion where local mobility index is one. In this paper, for maximum local mobility index, optimum link lengths of the manipulator that is function of the location of the input link are obtained. Charts, showing the optimum kinematic design of the 4 link planer manipulator are obtained. It is clear from this result, that the performance of the manipulator is maximum for a position interval in addition to a certain position. Also, at some positions better relationship between force and motion is observed where local mobility index is not exactly unit.

Keywords Optimal kinematic design; Local mobility index; Force manipulability

1. INTRODUCTION

   To design a kinematic optimum manipulator is the central focus of researchers. Some criteria are there to design robotic manipulators. Yoshikawa [1&2] introduced the concept of end-effector manipulability as a measure of the kinematic transmission characteristics of manipulator. One of the most important criteria in the optimal robot design is that the robot can achieve isotropic configurations. The equal forces may be exerted in all directions. Salisbury and Craig [3] proposed to use the ratio of the largest and smallest singular values of Jacobian matrix. The most manipulability measures have been derived from the condition number of the Jacobian matrix, and have been used by Asada, Chiu and Park [4, 5 & 6] for analyzing and optimizing kinematic performance of the manipulators based on the motion and force manipulability concepts. Klein and Blaho [7] use the minimum singular value

   Roth [14] investigated the effects of actuation schemes on the minimum singular value, the manipulability and the condition number measures of kinematic performance. A new definition of the force ellipsoid was proposed in the study of Chiacchio et al. [15] and an algorithm to correctly evaluate the task space force polytope was given.

   This paper presents how the link lengths are optimized for the closed-loop, four-bar planer mechanism as a manipulator with the help of local mobility index. The performance index is defined as the ratio between the minimum and maximum joint torque vector norms. For the investigated manipulator, optimum link measurements that maximize the performance index have been achieved based on the position of the input link. Comparing these results with those of the parallel manipulator, it is seen that better conditions are reached in terms of force-motion relationship with the planar manipulator.

   Nomenclature

   2 length of the input link 2

   2 length ratio of the input link 2

   3 length of the coupler link

   3 length ratio of the coupler link

   4 length of the output link

   4 length ratio of the output link

   1 length of the input link 1

   derivative of 92 with respect to q

   Jacobian matrix

   generalized Jacobian matrix

   as a measure. Gosselin and Angeles [8] applied a global

   conditioning index to four different cases, which is based on

   1

   ,

   2, 3

   coefficients of Freudenstein's equation

   the distribution of the condition number of the Jacobian matrix over the entire manipulator workspace. Park and Brockett [9] introduced the kinematic distortion index, which is a measure of the distortion associated with the mapping from configuration to work space. Singh and Rastegar [10] developed global velocity ellipsoid concept, which represents the velocity transmission characteristics of a manipulator. Doel and Pai [11] introduced formalism for the systematic construction of performance measures of robot manipulators in a unified framework based on differential geometry. Kircanski [12] determined the isotropic configurations of planar and spatial manipulators in the form of polynominal. Zanganeh and Angeles [13] introduced a set of conditions for the submatrices of the Jacobians under which a parallel manipulator can attain an isotropic configuration. Matone and

   rotation of the input link 2

   { } vector of cartesian velocities end-effector link length ratio

   rotation of the output link

   { } vector of joints rates

   1 rotation of the input link 1

   2 rotation of the output link (end-effector)

   local mobility index

   maximum local mobility index with i, j, k design

   variables

   , minimum and maximum eigenvalues of the

   generalized Jacobian matrix

2. MOTION MANIPUBILITY CHARACTERISTICS

   =

   The planar manipulator with a four-bar mechanism under study is shown in Fig. 1. Planar manipulator with a four-bar mechanism has two actuators that are fixed to the base link and drive the two input links. One of the links in a four-bar mechanism connected to the input link 1 oscillates while the other has a full rotation. Manipulator with a parallelogram

   2 1, 3 = 3 1, 4 = 4 1

   (4)

   The angle is found explicitly as a function of p and the parameters L2, L3, L4. Such a solution is obtained by expressing sin and cos in terms of tan(/2),

   mechanism is a special case of the four-bar mechanism consisting of equal two opposite links.

   2 /2

   1+ 2 /2

   = ,

   (5)

   1 2 /2

   1+ 2 /2

   When the end-effector position is taken as the task domain position vector, the Jacobian matrix is defined as the matrix representing the transformation mapping the joint rates into the cartesian velocities. This transformation is written as

   and substituting those values in Eq. (3), the angle 2 is found as shown below

   = 2 2+ 2 2

   J = r (1)

   where = 1 is the vector of joints rates and

   is the vector of cartesian velocities, the Jacobian matrix is given by

   =

   (6)

   (7)

   2 = +

   Using the above equations, derivative in the Jacobian

   1 1

   1 1

   +

   2

   1

   matrix is expressed as

   1 + 2

   1 + 2

   =

   = ( ) 1

   1 1 + 1 1 + 2 1

   (2)

   (8)

   where = 2

   is the derivative of the auxiliary or

   coordinates

   2with respect to and is the link length ratio.

   (9)

   =

   =

   (

   ) 1

   The generalized Jacobian matrix, , defined as the quadratic form of the Jacobian matrix, , is used to characterize the force manipulability of the mechanism. can be written as the product of two matrices:

   =

   Fig. 1. The planar manipulator with a four-bar mechanism.

   As seen in Fig. 1, 2 = + expression can be written. In this case, = 2 = is

   obtained. The derivative can be determined by

   Freudenstein's equation. Freudenstein's equation is a displacement equation for the four-bar linkage, which holds true for each position of the linkage. In a compact form, it

   (10)

   The propagation from the input joint torques to the output end effector force is directly proportonal to the eigenvalues of . If the eigenvalues of the generalized Jacobian

   matrix are max and min, one can conclude that when the

   endpoint force is in the direction of eigenvector emax related to the maximum eigenvalue of , the largest joint torque is required to exert a unit endpoint force. Thus, the force

   manipulability at the mechanism's endpoint is at its worst. On the other hand, the force manipulability is at its best when a force is exerted in the direction of eigenvector emin. The local mobility index (LMI) is denned as the ratio between the minimum and maximum eigenvalues of the generalized Jacobian matrix , Lee (1993):

   =

   writes

   (3)

   +

   + = 0

   (11)

   The local mobility index is bounded as 0 1.

   Where, = , = 2 + ,

   = 1 + 3, 1 = 1 4,

   2 = 1 2,

   3 = ( 32 22 42 1) 2 2 4, 2 =

   The best-conditioned point is identical to an isotropic point where local mobility index equals one. This indicates that the joint torque vector norms are equal for endpoint force exerted in any direction. The isotropic point is a special condition

   where the mechanism has a uniform mobility in all motion directions.

   The local mobility index of the planar manipulator shown in Fig. 1 can be derived from the generalized Jacobian matrix

   as

   =

   1+ 2+2 2+2

   1+ 2+2 2+2

   (12)

   1+4 2+ 2+22 4+4 4

   2 +4 + 1+2 3 2+2 1+2 2

   + 1+4 2+ 2+22 4+4 4

   2 +4 + 1+2 3 2+2 1+2 2

   2 2

   2 2

   Fig. 2. The local mobility index versus joint angle and the link

   The expression LMI in Eq. (12) defines a five-variable function depending on the input link position as seen in Eqs. (4)-(9) and dimensionless link lengths L2, L3, L4 and . For

   the subject of optimal kinematic design of planar manipulator with four-bar mechanism, variables L2, L3, L4, which are the characteristic dimensions of this mechanism and variable ,

   which determines the position of the end-effector on output link have been used. In order to generalize the results, using the length of input link 1, link lengths have been made dimensionless. The design variables , L2 and L4 which

   maximize the objective function LMI have been searched in the study. In the derived numerical results, first and later

   , L2; ., L2 and L4 variables left free, respectively and the optimization problem is solved giving constant values to the other variables. The unconstrained optimal design problem is solved by employing Newton's method iteratively, Papalambros (2000).

3. NUMERICAL EXAMPLES

   The performance index of the manipulator for a parallel planar manipulator with equal opposite link lengths is drawn in Fig. 2 using L2 = 0.3, L3 = 1 and L4 = 0.3 numerical values and depending on input link and end-effector position. As seen in

   this figure, the = /4 position of the input link and =

   0.707107 value for the end-effector position equal local mobility index to one in such a parallel manipulator. These values obtained for the manipulator with parallelogram mechanism are the same with results for the planar two link manipulator obtained in the studies of optimal kinematic design by Gosselin and Angeles [8], Singh and Rastegar (1995), Kircanski (1996) and Lee et al. (1993).

   length ratio for parallel manipulator.

   Fig. 3 displays the maximum local mobility index values obtained by the optimization of different link lengths of the manipulator depending on the position . As seen in

   graph (full line), if only the value is optimized, local

   mobility index equals 1 at = /4, = 0.707107 point. With L3 = 1, L4 = 0.3 constant values and and L2 as design variables graph 2 (dotted line), with L3 = 1 and , L2 and L4 as desig-

   Fig. 3. The maximum local mobility index v/s joint angle ( , L2 = 0.3, L3 = 1, L4 = 0.3)

   gn variables graph 2 4 (dashed line) are derived. It is seen in these graphs that with the variable-length manipulator,

   the performance index can be kept at its maximum value in a position interval. As the number of free design variable increase, force-motion quality rises. As seen in the 2

   graph, it is possible to keep performance index constant at its

   maximum value from = 0 to approximately 1.2 rad. position by the optimization of l2 and variables. This position can be

   increased to approximately 1.4 rad. by the optimization of ,

   L2 and L4. Such a manipulator which has a higher kinematic performance than that of a parallel manipulator can be realized through hydraulically actuated linkage.

   In Tables 1-3, maximum local mobility index obtained by the chosen 8 input link positions and the optimum design

   variables giving these values are shown with numerical values.

   In these tables, it is seen that at = /4 position, = 1, =

   0.707107, L2 = 0.3, L3 = 1, L4 = 0.3 are obtained independent from the number of design variables of the local mobility index that are found optimum. A similar condition appears at

   = 3 /4 position. At this position = 0.029437, =

   0.707107, L2 = 0.3, L3 = 1 and L4 = 0.3 values are attained independent from the number of design variables in the optimization problem. Comparisons on Fig. 3 and tables reveal that better conditions for the force-motion relationship at the other points besides the maximum of the performance index can be obtained.

4. CONCLUSIONS

Optimal kinematic design of planar manipulator with a four-bar mechanism has been presented. The performance index is defined as the ratio called local mobility index between the minimum and maximum eigenvalues of the generalized Jacobian matrix. For the manipulator, the link measurements that maximize the performance index depending on the input link position have been found. The attained results have been compared to those of the parallel manipulator and it is seen that better results have been achieved in terms of force- motion relationship. The force-motion performance of the manipulator can be elevated by the optimization of two design variables with setting the input link long and end-effector

position short formerly, from the beginning of the motion and

TABLE I. THE MAXIMUM LMI VERSES JOINT ANGLE OPTIMUM LINK RATIO (L2 = 0.3, L3 = 1, L4 = 0.3)

AND

vice versa latter. In this case, the performance index can be kept at its highest value from = 0 to approximately 1.2 rad.

position. If the optimization problem is solved with three variables as the output link length is added among the design variables, it is possible to raise this interval to 1.4 rad. As a result of the comparisons, better conditions have been obtained in terms of force-motion relationship also at the other points in addition to this maximum value of the performance index. As the number of free design variables increases, quality of the force-motion is possible to be raised.

TABLE II. THE MAXIMUM LMI VERSES JOINT ANGLE AND OPTIMUM LINK RATIO , L2 ( L3 = 1, L4 = 0.3)

TABLE III. THE MAXIMUM LMI VERSES JOINT ANGLE AND OPTIMUM LINK RATIO , L2 , L4 ( L3 = 1.0)

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