Mechanics Assignment 3 Solved

For each parallel mechanism, PMi, shown below, do the following:

 

PMi.1 
Identify the system of wrench constraints (of the platform) for each leg.  
PMi.2 
Identify the total wrench constraint system of the platform.
PMi.3 
Identify the system of platform freedoms.
PMi.4 
Answer additional questions, if any.
PMi.5 
Answer additional questions, if any.
PMi.6 
…
 

When describing each wrench or twist system: (a) provide a basis giving the vector components of the basis twist/wrenches in a chosen reference frame; (b) give an adequate geometric description of the system by specifying screws of what pitches underlie the system and what are the locations of their axes (for finite-pitch screws) or their directions (for infinite-pitch screws).  

 

 

 

  

 

PM1. In The 5R Leg 1, joint axes 1 and 2 are parallel, and so are 4, 5, and 6, while 2 and 3 are skew. In the PRRR Leg 2, the axes of joints 2, 3, and 4 are parallel, and the direction of joint 1 (the slider) is not perpendicular to the revolute-joint axes. In the configuration shown: all joint axes in Leg 1 are horizontal, while those in Leg 2 are not.   PM1.4. Repeat steps PM1.1-3 for the mechanism obtained by removing the first two joints in leg 1 and the P joint in leg 2. Can these three joints be used to actuate the original mechanism?

 

 

  

PM2. In each 5R leg, joints 1, 2, and 3 intersect in a point in the base, common for the four legs. Joint axes 4 and 5 are parallel to each other and to a plane in the platform common for the four legs.

 PM2.4. Describe and draw a serial chain with the same full-cycle mobility of the end-effector as the PM.  

 PM2.5. Repeat steps PM2.1-3 for the mechanism obtained from PM2 by removing (blocking) joint 4 in one of the legs. Would the result change if a joint 5 is removed instead? What if all 4th and 5th joints are removed?  

 

  

  

PM3. In each leg: joint axes 1 and 2 are parallel to a common direction fixed in the base. Joint axes 3, 4, 5 intersect in a point in the platform, common for all legs. Assume that the five joint twists in a leg are independent.  

            PM3.4. Consider one of the legs as a separate serial chain. Can its joint twists become dependent? If yes, describe and illustrate all such leg postures. How would the wrench and twist systems of the PM change if one of its legs is in such a singular posture?  

 

 

  

PM4. In each 5R leg, joint axes 2, 3, 4, are parallel (but not coplanar) and perpendicular to both joint axes 1 and 5. The three base-joint axes, as well as the three platform-joint axes, are concurrent and coplanar. In this configuration, joints axes 1 and 5 coincide for each leg and all joint 3 axes are horizontal (i.e., all joint axes are horizontal).  

 PM4.4. Repeat steps PM1.1-3 for the mechanism obtained from PM1 by removing (i.e. blocking) the first joints in all legs.