Cosine function with horizontal and vertical offset 5

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Cosine function with horizontal and vertical offset 5. DYNAMIC ANALYSIS The dynamic behavior of manipulator is described in terms of the time rate of change of the robot configuration in relation to the joint torques exerted by the actuators. Hence, the resulting equation of motions represents such relationship in form of set of differential equations that govern the dynamic response of the robot linkage to input joint torques. Dynamic model for two DOF robot using Newton’s Euler approach, the dynamic equations are written separately for each link. Equations are evaluated in numeric or recursive manner. In this case, the sum of forces is equal to variation of linear momentum. The joint torques τ ´ and τ are coupling moments. M±q². q L µ C±q, q L². q ¹ µ g(q) µf±q ¹² ³ τ M Where, q ³ Joint displacements q ¹ ³ Joint velocities q L ³ Joint accelerations M±q² ³ Manipulator inertia matrix C±q, q L² ³ Matrix of centripetal & coriolis torques g±q² ³ Gravitational torques f±q ¹² ³ Friction torques τ M ³ Applied torque inputs For a two degree of freedom manipulator, / τ ´ τ 1 ³ ! i ´´ i ´¶ i ¶´ i ¶¶ " / x ° x V 1 µ / c ´´ c ´¶ c ¶´ c ¶¶ 1 / x x W 1 µ / g ´´ g ¶´ 1 Xx ´ Y Where, q ´ L ³ x ° q ´ ¹ ³ x q ´ ³ x ´ q L ³ x V q ¹ ³ x W q ³ x Z The above equation can be rearranged in the form of, / τ ´ τ 1 ³ ! g ´´ c ´´ i ´´ 0 c ´¶ i ´¶ g ¶´ c ¶´ i ¶´ 0 c ¶¶ i ¶¶ " [ \ \ \ \ ] x ´ x x ° x Z x W x V ^ _ _ _ _ The inertia matrix of 2 DOF robot arm is symmetric and positive definite and its elements are, ! i ´´ i ´¶ i ¶´ i ¶¶ " Where, i ´´ ³ m ´ a µ m ±a ´ µ a µ 2a ´ a C ² µ I ´ µ I i ´¶ ³ m ±a µ a ´ a C ² µ I i ¶´ ³ m ±a µ a ´ a C ² µ I i ¶¶ ³ m a µ I The inertia has maximum value for fully extended arm. The elements of centripetal and coriolis matrix, / c ´´ c ´¶ c ¶´ c ¶¶ 1 ³ ! ¸2m a ´ a S x W ¸m a ´ a S x W m a ´ a S x 0 " Entries of gravitational terms are, / g ´´ g ¶´ 1 ³ ! m ´ ga C ´ µ m g±a ´ C ´ µ a C ´¶ ² m ga C ´¶ " When arm is fully extended in the x-direction, the effect of gravitational moment becomes maximums. The closed form dynamic equations of n-degree-of-freedom robot in the form of, τ - ³ c M -d q L d e df´ µ c c C -dg q ¹ d q ¹ g µ G - ±q - ² µ f - ±q ¹ - ² e g;´ e d;´ , i ³ 1, … . . , n Lagrangian Euler approach is based on differentiation of energy terms with respect to the systems variable and time. Internal reaction forces are automatically eliminated using this method. Closed form equations are directly obtained. Here, the sum of torques is equal to variation of angular momentum. The total energy stored in each link is represented as, τ - ³ c i 1 2 m - |V a- | µ 1 2 I - ω - l -;´ The resulting expression as same as obtained using Newton’s Euler approach. Lagrangian formulation is simpler than Newton’s Euler approach. -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 0 100 200 300 400 Amplitude Angle (deg) T=2 cos(360*x/T)-1 cos(360*x/T)+10 cos(360*x/T)
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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 @ 81 5.1 Mass Properties of Both Links
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