1. Benjamin Stephens Carnegie Mellon University Monday June 29, 2009 The Linear Biped Model and Application to Humanoid Estimation and Control

2. Introduction 2

3. Motivation 3 Robotics Find simple models for complex systems Develop algorithms that use simple models to make humanoid control simpler Better way to understand and explain dynamic balance and locomotion Human Physiology Evaluating biomechanical models Understand and prevent falls, which can lead to hip/wrist fractures.

4. Take-Home Message 4 “The Linear Biped Model is a simple model of balance that can describe a wide range of behaviors and be directly applied to humanoid robot estimation and control”

5. Outline Modeling Balance Overview Linear Biped Model Orbital Energy Control Lateral Foot Placement Control Humanoid Robot Center of Mass Estimation Feed-forward Control Future Work Conclusion 5

6. Modeling Balance 6

7. Sum of forces Center of pressure Base of support Intro to Modeling Balance 7 center of pressure

8. Linear Inverted Pendulum Model Features: All mass concentrated at CoM Massless legs Does not move vertically Linear Kajita, S.; Tani, K., "Study of dynamic biped locomotion on rugged terrain-derivation and application of the linear inverted pendulum mode," IEEE International Conference on Robotics and Automation, vol.2, pp.1405-1411, 1991. (Linearize) 8

9. Stability of Linear Inverted Pendulum What’s the best we can do? Apply maximum allowable force to the ground Move center of pressure to edge of base of support 9 Benjamin Stephens, "Humanoid Push Recovery," The IEEE-RAS 2007 International Conference on Humanoid Robots, Pittsburgh, PA, November 2007

10. The Linear Biped Model Weighted sum of the dynamics due to two linear inverted pendulum models (rooted at the feet) 10 Benjamin Stephens, " Energy and Stepping Control of Linear Biped Model in the Coronal Plane," Submitted to The IEEE-RAS 2009 International Conference on Humanoid Robots.

11. The Double Support Region 11 We define the “Double Support Region” as a fixed fraction of the stance width.

12. Dynamics of Double Support 12 The dynamics during double support simplify to a simple harmonic oscillator LIPM Dynamics

13. Stability of the Linear Biped Model 13 What’s the best we can do? Apply maximum allowable force to the ground Move center of pressure to edge of base of support

14. Phase Space of LiBM Location of feet Double Support Region 14

15. Controlling Balance 15

16. Static Balance Control 16 Goal: Return to a state of static balance (zero velocity) Strategies:

17. Periodic Balance Goal: Balance while moving in a cyclic motion, returning to the cycle if perturbed. 17 Slow Swaying Fast Swaying Marching in Place or Walking

18. Orbital Energy Control Orbital Energy: Solution is a simple harmonic oscillator: We control the energy: 18

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20. Energy Control Trajectories 20

21. Stepping Control Because we define double support region, when to step is pre-determined, we only have to decide how far to step DSP region moves 21

22. N-Step Controller Because DSP region is fixed, we know when to take a step, only need to decide where N-Step lookahead over a set foot step distances Benefits: Very fast Works for any desired energy Recovers from Pushes Stabilizes position 22

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25. Application to Humanoid Balance 25

26. Humanoid Applications 26 Linear Biped Model predicts gross body motion and determines a set of forces that can produce that motion State Estimation Combine sensors to predict important features, like center of mass motion. Feed-Forward Control Perform force control to generate the desired ground contact forces.

27. Robot Joint Level Controller Potentiometers Force/Torque Sensors IMU Joint Torques Kinematics Model Flatness Calculation Position Measurement High Level Controller Acceleration Estimate Estimate Fusion & Filter Robot Model Acceleration Measurement Robot Sensing Overview State Estimate PROCESS NOISE MEASUREMENT NOISE Force Measurement

28. Center of Mass Filtering 28 A (linear) Kalman Filter can combine multiple measurements to give improved position and velocity center of mass estimates. NOTE: Because we measure force, we should also be able to estimate push/disturbance magnitudes Joint Kinematics Hip Accelerometer Feet Force Sensors Kalman Filter Periodic Humanoid Balance CoM State

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30. Feed-Forward Force Control 30 LiBM can be used for feedforward control of a complex biped system. Torques can be generated by force control of the CoM with respect to each foot Additional controls are applied to bias towards a home pose and to keep the torso vertical.

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32. Movie Summary 32

33. Conclusion 33 “The Linear Biped Model is a simple model of balance that can describe a wide range of behaviors and be directly applied to humanoid robot estimation and control” Slow SwayingFast Swaying Marching in Place or Walking Joint Kinematics Hip Accel Force Sensors Kalman Filter Periodic Humanoid Balance CoM State

34. Future Work 34 3D Linear Biped Model Refine Robot Behaviors Foot Placement Push Recovery Walking Robust Control/Estimation Sliding Mode Control of LiBM Push Force Estimation Online LiBM Parameter Estimation/Adaptation

35. The End 35 Thanks to Research Committee Members: Chris Atkeson Jessica Hodgins Martial Herbert Stuart Anderson Questions?

36. A Note on Generality 36 The derivation of the Linear Biped Model is easily extended to include more contacts: Grasping for support >2-Legged Locomotion As the number of contacts increases, the difficulties are: How to distribute the desired force on the CoM How to coordinate/plan contact reconfiguration

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38. Dynamic Constraints DSPRSPLSP 38

39. Friction Constraints on LiBM

40. Double Support Right Support Left Support

41. Hybrid Orbital Energy DSP region! In DSP region, we use the same energy equation as before, x is relative to half way between feet In SSP region, we use the orbital energy, x is relative to stance foot Energy at middle of SSP determines curve