1. Optimization to solve Inverted Pendulum problem Zhili Chen

2. Problem Description Given a initial Phi>0, Apply F on cart to regain balance (Phi=0)

3. Problem Description Performance Evaluation ◦ Minimum time period before balance, or ◦ Minimum cart movement, or ◦ Minimum energy used by force F

4. Physical Equations => Approximate and linearize

5. Solution 1 Optimization-Based Interactive Motion Synthesis, SUMIT JAIN, YUTING YE, and C. KAREN LIU Use current joint configuration, get configuration of next time step.

6. Solution 1 Variables Constraints  Where  Upper and lower bounds for

7. Solution 1 Objective function  Want to decrease angle using smaller car movement Initial value ◦ Solution of the last time step is the initial value for current time step Use ‘fmincon’ in MATLAB ◦ Algorithm: active set (small scale)

8. Solution 1 Result ◦ Not good ◦ Constraints are too lax or too strict? ◦ Step by step optimization not suitable for this problem? (optimal solution of whole process does not imply optimal solution for every time step)

9. Solution 2 Spacetime Constraints, Andrew Witkin,Michael Kass Regard the process as a whole (from initial state to balance state)

10. Solution 2 Optimization variable  n = T/h (h, length of timestep) Constraints ◦ => Ay=b ◦ Upper and lower bounds for

11. Solution 2 Objective function Initial value (or boundary condition) Use ‘fmincon’ in MATLAB ◦ Algorithm: interior-point (large scale, sparse matrix, both inequality and equality)

12. Solution 2 Result ◦ Close to result from classic control theory

13. Thanks