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ME451 Kinematics and Dynamics of Machine Systems Absolute Kinematic Constraints – 3.2 Relative Kinematic Constraints – 3.3 February 12, 2009 © Dan Negrut, 2009 ME451, UW-Madison Before we get started… Last Time Discussed stages of Kinematics Analysis Boiler plate approach: At each time step do Position Analysis (system of nonlinear equations) Velocity Analysis (system of linear equations, rhs denoted by ) Acceleration Analysis (system of linear equations, rhs denoted by ) Today Start discussion about geometric constraints Real life counterpart: joints between bodies Reminder: HW Assigned: 3.3.2, 3.3.4, 3.3.5, ADAMS problem Due on Tu, Feb. 17 ADAMS component available for download from class website 2 Focus of This Lecture: Geometric Constraints Learn how to write kinematic constraints that specify that the location and/or attitude of a body wrt the global (or absolute) RF is constrained in a certain way Sometimes called absolute constraints Learn how to write kinematic constraints that couple the relative motion of two bodies Sometimes called relative constraints 3 The Drill… Step 1: Identify a kinematic constraint (revolute, translational, relative distance, etc., i.e., the physical thing) acting between two components of a mechanism Step 2: Formulate the algebraic equations that capture that constraint, (q)=0 This is called “modeling” Step 3: Compute the Jacobian (or the sensitivity matrix) q Step 4: Compute , the right side of the velocity equation Step 5: Compute , the right side of the acceleration equation (ugly…) This is what we do almost exclusively in Chapter 3 (about two weeks) 4 Absolute Constraints Called “Absolute” since they express constraint between a body in a system and an absolute (ground) reference frame Types of Absolute Constraints Absolute position constraints Absolute orientation constraints Absolute distance constraints 5 Absolute Constraints (Cntd.) Absolute position constraints x-coordinate of Pi y-coordinate of Pi Absolute orientation constraint Orientation f of body 6 Absolute x-constraint Step 1: the absolute x component of the location of a point Pi in an absolute (or global) reference frame stays constant, and equal to some known value C1 Step 2: Identify ax(i)=0 Step 3: ax(i)q = ? Step 4: ax(i) = ? Step 5: ax(i) = ? NOTE: The same approach is used to get the y- and angle-constraints 7 Absolute distance-constraint Step 1: the distance from a point Pi to an absolute (or global) reference frame stays constant, and equal to some known value C4 Step 2: Identify dx(i)=0 Step 3: dx(i)q = ? Step 4: dx(i) = ? Step 5: dx(i) = ? 8