Parameters are analyzed using annova to imitate the

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parameters are analyzed using ANNOVA to imitate the real time performance measurement of 2-DOF planar manipulator [5]. Optimal dynamic balancing is formulated by minimization of the root-mean-square value of the input torque of 2DOF serial manipulator and results are simulated using ADAMS software [6]. The dynamic parameters of 2 DOF are estimated and results are validated through simulation [7]. Non-linear control law for serially arranged n-link is derived using Lyapunov-based theory by M W Sponge [8]. Recently, position control using neuro-fuzzy controller is proposed for 2-DOF serial manipulator [9]. Forward and inverse kinematics, DH parameters formulations and dynamic analysis for 2 DOF robot arm is as under, Fig -1 : Two DOF planar manipulator
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IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308 __________________________________________________________________________________________ Volume: 02 Issue: 09 | Sep-2013, Available @ 78 2. FORWARD KINEMATICS The position and orientation of end effectors is a non linear function of joint variables P ° ±x, y² ³ f±q² . x ³ L ´ cosq ´ µ L cos±q ´ µ q ² (1) y ³ L ´ sinq ´ µ L sin±q ´ µ q ² (2) Joint ranges are constrained for the manipulator analysis in the present case as, µ90 ° · q ´ ≪ ¸90 ° and µ 45 ° · q ≪ ¸45 ° By differentiating the above two expressions, x ¹ ³ ¸L ´ sinq ´ ∙ q ¹ ´ ¸ L sin±q ´ µ q ²±q ¹ ´ µ q ¹ ² (3) y ¹ ³ L ´ cosq ´ ∙ q ¹ ´ µ L cos±q ´ µ q ²±q ¹ ´ µ q ¹ ² (4) In matrix form, ! x ¹ y ¹ " ³ ! ¸L ´ sinq ´ ¸ L sin±q ´ µ q ² ¸L sin±q ´ µ q ² L ´ cosq ´ µ L cos±q ´ µ q ² L cos±q ´ µ q ² " ! q ¹ ´ q ¹ " X ¹ ³ J ∙ q ¹ Where, X ¹ is the velocity of end effector, J is jacobian matrix and q ¹ represents joint rates. For a rotational joint the analytic a geometric jacobian are different. Jacobian matrix represents the relationship between rates of change of pose with respect to joint rates. Rank deficiency of jacobian represents singularity. 2.1 DH parameters of 2DOF arm Fig -2: Frame attachment for Two DOF planar manipulator Table -1: DH Parameters Link α a d θ 1 0 L ´ 0 θ ´ 2 0 L 0 θ T ³ ’ C ´ ¸S ´ 0 L ´ C ´ S ´ C ´ 0 L ´ S ´ 0 0 1 0 0 0 0 1 + ´ , T ³ ’ C ¸S 0 L C S C 0 L S 0 0 1 0 0 0 0 1 + ´ T ³ ’ C ´¶ ¸S ´¶ 0 L ´ C ´ µ L C ´¶ S ´¶ C ´¶ 0 L ´ S ´ µ L S ´¶ 0 0 1 0 0 0 0 1 + , Where, L - ³ Link length of ith link θ ¹ - ³ Joint rates of ith actuator C ´ ³ cosθ ´ S ´ ³ sinθ ´ C ´¶ ³ cos ±θ ´ µ θ ² S ´¶ ³ sin ±θ ´ µ θ ² Fig -3: Kinematic Mapping 3. INVERSE KINEMATICS The pose of the robot manipulator H ³ / R P 0 1 1 is known and the joint variables need to be identified for a pose. The geometric solution of the inverse kinematics are summarized in the form of, cosq ³ x µ y ¸ L ´ ¸ L 2L ´ L tanq ´ ³ y±L cosq µ L ´ ² ¸ x ´ L sinq x±L ´ µ L cosq ² µ yL sinq Joint velocity and the end-effectors velocity has a velocity constraint and is expressed as, q ¹ ³ J X ¹ J ³ ! C ´¶ /±L ´ S ² S ´¶ /±L ´ S ² ¸±L C ´¶ µ L ´ C ´ ²/L ´ L S ¸±L S ´¶ µ L ´ S ´ ²/L ´ L S " Task Space Forward Kinematics Inverse Kinematics Joint Space Rate of change of Cartesian pose Jacobian Rate of change of joint position
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