CHAPTER 3-2. Planar Cartesian Kinematics

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

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

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

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

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

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

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

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

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

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

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

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.

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

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.

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

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

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

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

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

Constraints between Pairs of Bodies(2)

Relative Position of Two Bodies

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

Translational Joint Constraint equation

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

Translational Joint Right side of acceleration equation where,