EX3_LAG_02 - 1 Lagrange equations Example 3 Figure 1.1(a is...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
1 Lagrange equations - Example 3 Figure 1.1(a) is a schematic representation of an open kinematic chain (robot arm) consisting of three links 1, 2, 3, and a rigid body RB . Link 1 can be rotated at A in a “fixed” cartesian reference frame (0) of unit vectors [ ı 0 , 0 , k 0 ] about a vertical axis ı 0 . The unit vector ı 0 is fixed in link 1. Link 1 is connected to link 2 through pin joints B and B 0 . The link 2 rotates relative to 1 about an axis fixed in both 1 and 2, passing through B , and B 0 . The link 3 is connected to 2 by means of a slider joint 2’. The slider joint is rigidly attached to link 2. The last link 3 holds rigidly the rigid body RB . The mass centers of links 1, 2, 2’ and 3 are C 1 , C 2 = C 2 0 , and C 3 , respectively. The mass center of RB is C R . The mass if the link 1 is m 1 , the masses of the bars 2 and 3 are m 2 and m 3 , the mass of the slider 2’ is m 2 0 and the mass of RB is m R . The length of 2 is l and the length of 3 is L . Find the equations of motion for the robotic system. Solution A reference frame (1) of unit vectors [ ı 1 , 1 , k 1 ] is attached to body 1, with ı 1 = ı 0 . A reference frame (2) of unit vectors [ ı 2 , 2 , k 2 ] is attached to link 2, as it is shown in Fig. 1.1. The unit vector 2 is parallel to the axis of link 2, BB 0 , and 2 = 1 . The unit vector k 2 is parallel to the axis of link 3, C 2 C R . To characterize the instantaneous configuration of the arm, the general- ized coordinates q 1 ( t ) , q 2 ( t ) , q 3 ( t ) are employed. The generalized coordinates are quantities associated with the position of the system. The first generalized coordinate q 1 denotes the radian measure of the angle between the axes of (1) and (0) , Fig. 1.1(b). The second generalized coordinate q 2 designates also a radian measure of rotation angle between (1) and (2), Fig. 1.1(c). The last generalized coordinate q 3 is the distance from C 2 to C 3 . Angular velocities Next the angular velocities of the links and the rigid body will be ex- pressed in the fixed reference frame (0). One can express the angular velocity of link 1 in (0) as ω 10 = ˙ q 1 ı 1 = ˙ q 1 ı 0 . (1.1) The angular velocity of link 2 with respect to (1) is ω 21 = ˙ q 2 2 = ˙ q 2 1 , (1.2) 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
and the angular velocity of link 2 with respect to the fixed reference frame (0) is ω 20 = ω 10 + ω 21 = ˙ q 1 ı 1 + ˙ q 2 2 . (1.3) The unit vector ı 1
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 8

EX3_LAG_02 - 1 Lagrange equations Example 3 Figure 1.1(a is...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online