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ME451 Kinematics and Dynamics of Machine Systems Relative Kinematic Constraints, Composite Joints – 3.3 October 4, 2011 © Dan Negrut, 2011 ME451, UW-Madison “The real problem is not whether machines think but whether men do.” B. F. Skinner TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A AAAA

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Before we get started… Last Time We looked at several absolute constraints x, y, relativ e constraints Started absolute distance constraint For each kinematic constraint, recall the drill that you have to go through in order to provide what it takes to carry out Kinematics Analysis Five step procedure: Identify and analyze the physical joint Derive the constraint equations associated with the joint Compute constraint Jacobian q Get (RHS of velocity equation) Get (RHS of acceleration equation, this is challenging in some cases) Today Covering relative constraints: Revolute, translational, and composite joints 2

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Absolute distance-constraint Step 1: the distance from a point P i to an absolute (or global) reference frame stays constant, and equal to some known value C 4 3 Step 2: Identify dx(i) =0 Step 3: dx(i) q = ? Step 4: dx(i) = ? Step 5: dx(i) = ?

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Example 3.2.1 An example where you’d have to use absolute constraints: the simple pendulum 4

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Example 3.1.3 [Tricky] A case when the algebraic constraints fail to imply (enforce) the actual kinematics of a mechanism: block sliding on incline, incline angle /4 Use the following set of generalized coordinates: Formulate constraints defining the kinematics (motion) of the mechanism 5 O 1 moves along ® = ¼ /4 User prescribed motion (given)

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Example 3.1.3 Note that when passing through the origin, the algebraic constraints fail to specify the actual kinematics of the mechanism An example when the translating plain English into equations is not straightforward 6 Unexpected problem when passing through origin… Translating plain English into the right equations:

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Example 3.2.2 An example where you’d have to use an absolute angle constraint: slider along x-axis 7

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Moving on to relative constraints (section 3.3) 8

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Loose Ends: Switching representation between two Reference Frames with rotation matrices A i and A j, respectively The problem at hand: 9

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Loose Ends, Continued: Notation (related to changing representation from A j to A i ) 10 Note that: Then, using for the angle between the two reference frames the notation Note that the order is important: it is “A ij ” and not “A ji ” A ij gets multiplied from the right by a vector represented in the “j” Reference Frame (RF) and produces a vector represented in the “i” RF Note that when you see A j in fact you should had A 0j, where “0” is used to symbolize the global reference frame … we get that

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Loose Ends, Final Slide (related to changing representation from A j to A i ) 11 For later reference, it is useful to recall that, Therefore

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Intro: Vector between P i and P j Something that we’ll use a lot: the expression of the vector from P i to P j in terms of the Cartesian generalized coordinates q 12

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Relative x-constraint Step 1: In layman’s words, the difference between the x coordinates of point P j and point P i should stay constant and equal to some known value C 1 Step 2: Identify rx(i,j) =0 Step 3: rx(i,j) q = ? Step 4: rx(i,j) = ? Step 5 rx(i,j) = ? 13

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Relative y-constraint Step 1: The difference between the y coordinates of point P j and point P i should stay constant and equal to some known value C 2 Step 2: Identify ry(i,j) =0 Step 3: ry(i,j) q = ? Step 4: ry(i,j) = ? Step 5: ry(i,j) = ? 14

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Relative angle-constraint Step 1: The difference between the orientation angles of the RFs associated with bodies i and j stays constant and equal to some known value C 3 Step 2: Identify r (i,j) =0 Step 3: r (i,j) q = ? Step 4: r (i,j) = ? Step 5: r (i,j) = ? 15

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Relative distance-constraint Step 1: The distance between two points P j and point P i should stay constant and equal to some known value C 4 Step 2: Identify rd (i,j) =0 Step 3: rd (i,j) q = ? Step 4: rd (i,j) = ? Step 5: rd (i,j) = ? 16

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