ESE 447

Dynamics of the Single-Link Inverted Pendulum.

Right Handed Coordinate system

Counter-clockwise positive direction of rotation

Skeletel Diagram of theSingle-Inverted Pendulum.

Generalized Coordinates

Single Link Rotary Inverted Pendulum

q_1, q_2 – Joint Angles

Lagrangian Dynamics

Link 1 Kinetic Energy:

Link 1 Potential Energy:q1

q2

Lagrangian Dynamics

q1

q2

x

Velocity is obtained by taking the time derivative

Lagrangian Dynamics

q1

q2

x

Total Kinetic Energy of link 2 is obtained by integratinginfinitesimal Kinetic energies along link2.

Lagrangian Dynamics

Total Kinetic Energy of system:

Physical Parameters of the system.

Lagrangian Dynamics

Potential Energy:

q1

q2

lc2

Lagrangian Dynamics:

Lagrangian Dynamics

Actuator Effects (DC Motor)

Gear Transfer:

Ideal PowerTransfer:

qm, tm q1, t1

Friction Effects

Friction acts opposite to the direction of motion

Assume no Static FrictionDynamic Friction is proportional to velocity

joint 1:

joint 2:

Langrangian Dynamics with Friction and Actuator Effects