Constraints: Connectivity and Mobility, Lecture 2 ME 3230 Kinematics and Mechatronics Dr. R. Lindeke, UMD–MIE

Points for Discussion Degrees of Freedom of a Mechanism or CONNECTIVITY MOBILITY of a Mechanism relative to a particular member chosen as Fixed Closure of a Mechanism Idle DOF’s and OverConstrained Linkages ME 3230 11/18/2018

Mobility Issues Mobility of a linkage defines the minimum number of coordinates needed to define all member positions A “Structure” has mobility of 0 (zero) and the structure is said to be “statically indeterminate” Linkages have mobility of 1 or more (robots up to 6 or even more if redundantly constrained) ME 3230 11/18/2018

Thinking about Mobility A single point in free space (unconstrained) has mobility 6 – 6 DOF Now supposing that we have a planer linkage (of n-members) and all are free to move, it has mobility of 3*n – in a planer sense If one link is chosen as base, it loses “all degrees of freedom” or mobility thus is also reduced: Linkage mobility is now: 3(n-1) – without any joints between links ME 3230 11/18/2018

Thinking about Mobility Now joining two links with a joint exhibiting some connectivity fi – connectivity of a joint equals its degrees of freedom – the mobility of the system is reduced to (3 – fi) When joints are made between all links, loss of system mobility is then: ME 3230 11/18/2018

Mobility Considerations: Thus, Total Mobility of a Linkage is given by: This Equation is called the “constraint criterion” of a Planer Linkage ME 3230 11/18/2018

Some Simple Examples: Here a simple 4-bar mechanism – we see that the mobility is 1 Here is a 6-bar mechanism with what looks like 6 joints, but it actually has 7 joints (2 where Links 3-4-5 meet). Thus M = 3(6-7-1) +7*1 = -6+7 = 1 ME 3230 11/18/2018

Let’s Try One: This is a Frame of the Water Pump seen in Problem 1.3 on page 54 in our text. Compute M for this device ME 3230 11/18/2018

Some More: How many links and joints? Determine Mobility? 11/18/2018

With Multi-Connected Links How many links and joints With Multi-Connected Links How many links and joints? Determine Mobility? ME 3230 11/18/2018

Special Case: The Diophantine Equation for planer mechanisms This occurs when we desire a mobility of 1 and all joints have connectivity of 1! A Diophantine equation admits only integral solutions ME 3230 11/18/2018

Diophantine Specifics Link count n and Joint count j must be integers n must be even j = (3n/2) – 2 In table 1.4 we see that the number of “potential configurations” grows rapidly as n exceeds 8 Most design requirements can be met by 4- or 6-link mechanisms! ME 3230 11/18/2018

Expansion to Spatial Linkages Points in free space have 6 dof Individual Links in free space now also have 6 dof! From here we develop the Kutzbach Criteria for spatial mechanisms: Planer links are a special case! ME 3230 11/18/2018

With only Lower Pair (connectivity of 1) Joints are used The Mobility equation is reduced to M = 6(n – j – 1) + j = 6n – 5j – 6 For this Diophantine equation (with Mobility of 1) the above further reduces to: 6n = 7 + 5j  text calls this eqn. 1.4 Where n = # Links; j = # Joints ME 3230 11/18/2018

Example of Spatial Linkage 4 Links; 4 Joints Connectivity of a Sph. Joint is 3 Connectivity of a Cyl. Joint is 2 fi = 2*3 + 1*2 + 1*1 = 9 M = 6(4-4-1) + 9 = 3 ME 3230 11/18/2018

Building Mechanisms with Closures – a different way to view the problem! Starting with a Base Link (fixed) Add (successively) links and joints When a joint connects another member to a linkage, mobility or #DOF’s is increased by fi (or the connectivity of the joint) At the same time, we add one link (nnew = norig +1) and one joint (jnew = jorig +1) These work to affect the Mobility of the linkage ME 3230 11/18/2018

On the other hand  Joining 2 links already present in a linkage reduces the total number of system DOF’s by the number of constraints imposed by the joint This number of constraints (lost) equals the number of DOF’s lost by the system when the joint is formed (in spatial mechanisms, it equals 6–fi for the joint type added) ME 3230 11/18/2018

Closure Analysis Cont. We refer to this process of adding joints to already present linkages as ‘forming a closure’ Thus: ME 3230 11/18/2018

What it Means: Planer Linkages do not follow, logically, the general Kutzbach criteria; M computed is always 3c less than the correct value Mathematically: ME 3230 11/18/2018

Why Worry: In Robotic structures, which are considered open kinematic chains, the number of links n = j+1 (the added ‘link member’ is the base) We find (generally): c = j + 1 – n Looking at it this way relates to positional analysis of Spatial Mechanisms: 6c = 6(j + 1 – n) gives the number of equations available for positional analysis of the mechanism The mobility of a system is the number of variables less the number of equations for the system ME 3230 11/18/2018

Some Random Effects: If M = -1; the number of equations for position problems exceeds the number of variables Therefore, there is no solution to the position problem. For a solution to exist, it is necessary for the equations to be dependent If mobility is greater than 1, the number of position variables is greater than the number of position equations. Solution to these systems exist but they are not unique ME 3230 11/18/2018

In Robotics: This is a Spherical Robot w/ Spherical Wrist It has 6 Joints and 7 links Mobility is: M = 6(7-6-1)+6 = 6 To position it we must set each joint to a specific value it will move robot end where we want it ME 3230 11/18/2018

Closing The Loop By grasping the block – connecting ‘link 7’ to the base – we create a closure Now M = 0 (it is a structure) and we are not free to set each joint to any value – only the unique ones needed to achieve this “pose” The is a part of Robot Mapping we will cover later in the course ME 3230 11/18/2018

Idle Degrees of Freedom An Idle degree of freedom is one that appears (and is) present but its value has no effect on the input – output relationships of interest To identify Idle degrees of freedom, first identify the input and output links Then we must determine if a single link or combinations of links can move without affecting the input/output link positions Like a connecting link rotating (about its axis) in a steering mechanism without changing the relationship between the steering wheel and the front tires in a vehicle ME 3230 11/18/2018

For Example: Here the Structure has two Idle DOF’s They are the roller and the cam rocker See solution – next page ME 3230 11/18/2018

Note: Pin-in-slot & Cam Contact have connectivity of 2: fi = 1*13 + 2*2 = 17 ME 3230 11/18/2018

Over Constrained Linkages This can occur Locally – indicating that a portion of the linkage is a “Structure” and we can replace this structure with a single rigid body Why a designer might build a (partially) over-constrained component is to greatly increase stiffness and rigidity of the over-constrained “member” ME 3230 11/18/2018