1. Standing Balance Control Using a Trajectory Library
Chenggang Liu and Chris Atkeson

2. Outline Introduction Robot Model Neighboring Optimal Control
Balance Controller
Simulation and Experiment Results

3. Introduction
Multiple Balance Strategies from One Optimization Criterion.

4. Robot Model

5. Trajectory Library Generation
Combined Method
Direct minimization with SNOPT (Sequential Quadratic Programming)
Differential Dynamic Programming
A library on a uniform grid of initial conditions

6. Neighboring Optimal Control
Given the discrete time dynamics of the robot:
and the optimal value function:
The neighboring optimal control is given by:
Where

7. Neighboring Optimal Control
Having the optimal trajectories of state over
time, , control over time, , and the
gain matrices over time, .
A local approximation to optimal control is:
where is the closest state to the current state on , and are those corresponding to on and .

8. Controller Architecture

9. Trajectory Library Generation
The library is refined to get a new library on an adaptive grid of initial conditions according to the proposed controller’s performance.

10. State/Push Estimation
State to be estimated:
Measurements:

11. State/Push Estimation
State transition and observation models

12. State/Push Estimation
To predict the next state before measurements are taken:
To update the state after measurements are taken:

13. Simulation Results Constant push of 42 Newtons at head
Short forward push at head of 50 Newtons, lasting 0.5 seconds
Random pushes sequence

17. Simulation Results
Comparison with the optimal controller.

18. Experiment Result
Push forward
Trajectory index
State index

19. Experiment Result
Push backward
Trajectory index
State index

20. Thanks!
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