1. CHAPTER 3-2. Planar Cartesian Kinematics

2. Basic Concepts in Planar Kinematics(1)
Kinematically driven system
If number of independent driving constraints is specified as many as numer of DOF in the system, such a system is called kinematically driven.
nc : number of coordinates
nh : number of holonomic constraints
ndof : number of degrees of freedom
Jacobian matrix of constraint equation in the kinematically driven system has dimension nc × nc

3. Basic Concepts in Planar Kinematics(2)
Constraint equation
Solve for q in the kinematically driven system
Since is highly nonlinear equation, it is difficult to obtain analytical from of solution q.
If a numerical method is applied to solve constraint equation for q at discrete instants in time, q is not known as an explicit function of time, i.e.,
Thus, can not be differentiated to obtain or

4. Basic Concepts in Planar Kinematics(3)
One way to compute and Interpolation method
Numerical
Method such as Newton Raphson method
Cubic spline interpolation
method
Differentiating interpolation function

5. Basic Concepts in Planar Kinematics(5)
The other way to compute and Chain rule of differentiation
Velocity Equation
Acceleration Equation

6. Example : Simple Pendulum(1)
A Physical pendulum system
Assign triads for analysis
Select
generalized
coordinates
Mathematical Model (Formulation)
Compute system DOF
Assign driving constraint

7. Example : Simple Pendulum(2)
Solve
Compute
for using numerical methods for at any instant in time.

8. Example : Simple Pendulum(3)
Compute
At any instant in time, velocity equations are as follows.

9. Example : Simple Pendulum(4)
Compute
Validate (9),(10) by directly differentiate constraint equations and velocity equations.
Thus, (9), (10) have been validated.

10. Example : Slider Crank Mechanism(1)
Geometric conditions to construct kinematic constraints.
Point on the crank(body1) = Point in the Global frame.
Point on the coupler(body2) must lie on the axis.
Point on the crank = Point on the coupler

11. Example : Slider Crank Mechanism(2)
Kinematic constraints
DOF
Driving constraint equation
Kinematic constraint equations for kinematically driven system.

12. Example : Slider Crank Mechanism(3)
Velocity equations can be obtained by differentiating
at any instant in time

13. Example : Slider Crank Mechanism(4)
Acceleration equations can be obtained by twice differentiating
at any instant in time
HW 3.1.3, 3.2.3
Introducing a systematic approach for formulating and solving kinematic equations are the objectives in this course.

14. Constraints between a Body and Grand(1) (Absolute Constraints)
Absolute distance constraint
Constraint equation
where, and
If , Constraint equation requires that both and

15. Constraints between a Body and Grand(1) (Absolute Constraints)
Jacobian matrix
Right side of the velocity equation
Right side of the acceleration equation
If , the Jacobian of equation is zero.

16. Constraints between a Body and Ground(2)
Absolute position constraint
Constraint equation
Jacobian matrix
Right side of velocity equation
Right side of acceleration equation

17. Constraints between Pairs of Bodies
Relative coordinate constraints
A relative constraint
Jacobian matrix
where,
Right side of velocity equation
Right side of acceleration equation

18. Constraints between Pairs of Bodies
A relative constraint
Jacobian matrix
where,
Right side of velocity equation
Right side of acceleration equation

19. Constraints between Pairs of Bodies
A relative constraint
Jacobian matrix
where,
Right side of velocity equation
Right side of acceleration equation

20. Constraints between Pairs of Bodies(1)
A relative distance constraint
Jacobian matrix
where,
Right side of velocity and equations acceleration

21. Constraints between Pairs of Bodies(2)

22. Relative Position of Two Bodies

23. Revolute Joint Right side of velocity equation
Constraint equation
Jacobian matrix
where,
Right side of velocity equation
Right side of acceleration equation

24. Translational Joint
Constraint equation

25. Translational Joint Jacobian matrix Right side of velocity equations
where,
Right side of velocity equations

26. Translational Joint
Right side of acceleration equation
where,