pendulum dynamics
DESCRIPTION
robotics lab dynamics notesTRANSCRIPT
![Page 1: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/1.jpg)
ESE 447
Dynamics of the Single-Link Inverted Pendulum.
![Page 2: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/2.jpg)
Right Handed Coordinate system
Counter-clockwise positive direction of rotation
Skeletel Diagram of theSingle-Inverted Pendulum.
![Page 3: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/3.jpg)
Generalized Coordinates
![Page 4: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/4.jpg)
Single Link Rotary Inverted Pendulum
q_1, q_2 – Joint Angles
![Page 5: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/5.jpg)
Lagrangian Dynamics
Link 1 Kinetic Energy:
Link 1 Potential Energy:q1
q2
![Page 6: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/6.jpg)
Lagrangian Dynamics
q1
q2
x
Velocity is obtained by taking the time derivative
![Page 7: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/7.jpg)
Lagrangian Dynamics
q1
q2
x
Total Kinetic Energy of link 2 is obtained by integratinginfinitesimal Kinetic energies along link2.
![Page 8: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/8.jpg)
Lagrangian Dynamics
Total Kinetic Energy of system:
Physical Parameters of the system.
![Page 9: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/9.jpg)
Lagrangian Dynamics
Potential Energy:
q1
q2
lc2
Lagrangian Dynamics:
![Page 10: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/10.jpg)
Lagrangian Dynamics
![Page 11: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/11.jpg)
Actuator Effects (DC Motor)
Gear Transfer:
Ideal PowerTransfer:
qm, tm q1, t1
![Page 12: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/12.jpg)
Friction Effects
Friction acts opposite to the direction of motion
Assume no Static FrictionDynamic Friction is proportional to velocity
joint 1:
joint 2:
![Page 13: Pendulum Dynamics](https://reader034.vdocuments.site/reader034/viewer/2022042507/563db92f550346aa9a9adaa3/html5/thumbnails/13.jpg)
Langrangian Dynamics with Friction and Actuator Effects