ME451 Kinematics and Dynamics of Machine Systems

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ME451 Kinematics and Dynamics of Machine Systems. Driving Constraints 3.5 Start Position, Velocity, and Acc. Analysis 3.6 February 26, 2009. © Dan Negrut, 2009 ME451, UW-Madison. Before we get started…. Today: Continue with relative driving constraints

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### ME451 Kinematics and Dynamics of Machine Systems

Driving Constraints 3.5

Start Position, Velocity, and Acc. Analysis 3.6February 26, 2009

Before we get started…
• Today:
• Continue with relative driving constraints
• Proper Kinematic Analysis: Position, Velocity, and Acceleration Stages
• HW due next Th (Mar. 5): 3.5.1, 3.5.4, 3.5.5, 3.5.6, ADAMS
• 3.5.5: note that the angle 2 is not displayed correctly
• 3.5.6: get rid of vi , take it unit vector
• Last Time
• Finished kinematic constraints
• Point-Follower
• Started to talk about driving constraints
• A set of ndof independent drivers need to be specified to “occupy” all the degrees of freedom
• Driver constraints
• Absolute: x, y, 
• Relative: we’ve only talked about relative distance driver

2

Example: Specifying Relative Distance Drivers
• Generalized coordinates:
• Motions prescribed:
• Derive the constraints acting on system
• Derive linear system whose solution provides velocities

3

Revolute Rotational Driver
• The framework: at point P we have a revolute joint
• It boils down to this: you prescribe the time evolution of the angle in the revolute joint
• Driver constraint formulated as
• Note that i and j are attributes of the constraint

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This driver says that the distance between point Pi on “Body i” and point Pj on “Body j” measured along in the direction of changes in time according to a user prescribed function C(t):

Translational Distance Driver
• The framework: we have a translational joint between two bodies
• Direction of translational join on “Body i” is defined by the vector vi

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Translational Distance Driver (Cntd.)
• The book complicates the formulation with no good reason
• There is nothing to prevent me to specify the direction viby selecting this quantity to have magnitude 1
• Equivalently, the constraint then becomes
• Keep in mind that the direction of translation is indicated now through a unit vector (you are going to get the wrong motion if you work with a vi that is not unit length)
• This is the reason why I wanted to discuss this prior to assigning the problem 3.5.6

6

Driver Constraints, Departing Thoughts
• What is after all a driving constraint?
• You take your kinematic constraint, which indicates that a certain kinematic quantity should stay equal to zero
• Rather than equating this kinematic quantity to zero, you have it change with time…
• Or equivalently…

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Driver Constraints, Departing Thoughts(cntd.)
• Notation used:
• For Kinematic Constraints: K(q)
• For Driver Constraints: D(q,t)
• Note the arguments (for K, there is no time dependency)
• Correcting the RHS…
• Computing for Driver Constraints the right hand side of the velocity equation and acceleration equation is straightforward
• Once you know who to compute these quantities for K(q), when dealing with D(q,t) is just a matter of correcting…
• …  (RHS of velocity equation) with the first derivative of C(t)
• …  (RHS of acceleration equation) with second derivative of C(t)
• Section 3.5.3 discusses these issues

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Mechanism Analysis: Steps
• Step A: Identify *all* physical joints and drivers present in the system
• Step B: Identify the corresponding constraint equations (q,t)
• Step C: Solve for the Position as a function of time (q is needed)
• Step D: Solve for the Velocities as a function of time ( is needed)
• Step E: Solve for the Accelerations as a function of time ( is needed)

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Position, Velocity, and Acceleration Analysis (3.6)
• The position analysis [Step C]:
• It’s the tougher of the three
• Requires the solution of a system of nonlinear equations
• What you are after is determining at each time the location of every component (body) of the mechanism
• The velocity analysis [Step D]:
• Requires the solution of a linear system of equations
• Relatively simple
• Carried out after you are finished with the position analysis
• The velocity analysis [Step E]:
• Requires the solution of a linear system of equations
• What is challenging is generating the RHS of acceleration equation, 
• Carried out after you are finished with the position and velocity analyses

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Position Analysis
• Framework:
• Somebody presents you with a mechanism and you select the set of nc generalized coordinates to position and orient each body of the mechanism:
• You inspect the mechanism and identify a set of nk kinematic constraints that must be satisfied by your coordinates:
• Next, you identify the set of nd driver constraints that move the mechanism:

NOTE: YOU MUST HAVE nc = nk + nd

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Position Analysis
• We end up with this problem: given a time t, find that set of generalized coordinates q that satisfy the equations:
• What’s the idea here?
• Set time t=0, and find a solution q by solving above equations
• Next, set time t=0.001 and find a solution q by solving above equations
• Next, set time t=0.002 and find a solution q by solving above equations
• Next, set time t=0.003 and find a solution q by solving above equations
• Stop when you reach the end of the interval in which you are interested in the position
• What you do is find the time evolution on a time grid with step size t=0.001
• You can then plot the solution as a function of time and get the time evolution of your mechanism

13

Position Analysis
• Two issues with the described methodology for finding the time evolution of the mechanism
• The equations that we have to solve at each time t are nonlinear, so a first hurdle is being able to solve this nonlinear system
• Deal with this issue later (next week, Newton-Raphson method)
• The second issue comes when you start thinking about the solution that you’ve just got using a numerical algorithm
• How do you know that what you got is a meaningful thing?
• Remember, a nonlinear system can have an arbitrary number of solutions
• Deal with this issue now

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Position Analysis: Implicit Function Theorem
• Is the solution of our nonlinear system well behaved? A sufficient condition is provided by the Implicit Function Theorem
• In layman’s words, this is what the theorem says:
• Let’s say that we are at some time tk, and we just found the solution qk and we question the quality of this solution
• If the constraint Jacobian is nonsingular in this configuration, that is,
• … then, we can conclude that the solution is unique, and not only at tk, but in a small interval  about time tk.
• Additionally, in this small time interval, there is an explicit functional dependency of q on t, that is, there is a function f(t) such that:

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