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Generation and Control of Humanoid Locomotion Course Project for COMP768

Generation and Control of Humanoid Locomotion Course Project for COMP768. Yu Zheng Department of Computer Science University of North Carolina at Chapel Hill November 1, 2010. Outline. Lagrange’s equation — a restatement of Newton’s law in generalized coordinates

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Generation and Control of Humanoid Locomotion Course Project for COMP768

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  1. Generation and Control of Humanoid LocomotionCourse Project for COMP768 Yu Zheng Department of Computer Science University of North Carolina at Chapel Hill November 1, 2010

  2. Outline • Lagrange’s equation — a restatement of Newton’s law in generalized coordinates • Linear inverted pendulum — a simplified dynamics model of a humanoid character • Articulated rigid bodies — a brief introduction

  3. Newton’s law in cartesian coordinates Describe Newton’s law in terms of the Lagrangian Newton’s law: Lagrangian: Kinetic energy Potential energy where xi are the cartesian coordinates of a particle

  4. Newton’s law in generalized coordinates Cartesian coordinates Generalized coordinates . Lagrangian L as a function of qj and qj ? Verify

  5. Newton’s law in generalized coordinates Chain rule:

  6. Newton’s law in generalized coordinates Chain rule:

  7. Newton’s law in generalized coordinates Newton’s law in cartesian coordinates Lagrange’s equation

  8. Overview of the simulation system Simplified dynamics model Linear inverted pendulum (LIP) Articulated dynamics

  9. Overview of motion generation Choose footsteps Choose reference trajectory of Zero Moment Point Compute all joint values Compute CoM motion Inverse kinematics

  10. Linear inverted pendulum (LIP) x represents the position of ZMP q indicates the position of CoM Lagrangian:

  11. Linear inverted pendulum (LIP) x represents the position of ZMP q indicates the position of CoM Lagrange’s equations:

  12. Linear inverted pendulum (LIP) x represents the position of ZMP q indicates the position of CoM State-space equation of motion:

  13. Linear inverted pendulum (LIP) x represents the position of ZMP q indicates the position of CoM Choice of the input u: For example: Output: Foot location IK Joint values of legs Body position

  14. Extension of (LIP) q1 denotes the rolling angle of the ball q2 indicates the relative position of the wheel on the ball q3 indicates the rotation of the wheel q4 indicates the relative position of the CoM to the wheel LIP on a rolling ball to the motion of a ball walker: Lagrange’s equation: State-space equation:

  15. Extension of (LIP) q1 denotes the rolling angle of the ball q2 indicates the relative position of the wheel on the ball q3 indicates the rotation of the wheel q4 indicates the relative position of the CoM to the wheel Examples of the input u: where Choice 1: is a constant value show demos Choice 2:

  16. Controller & Simulator Controller: where Connection between controller and simulator: x, y y * where

  17. Articulated body dynamics Lagrange’s equation: Canonical form of the equation of motion: Equation of motion for a N rigid-body system: (H is positive-definite) Considering the motion constraint: where is the constraint force

  18. Articulated body dynamics Motion constraint (holonomic): where consists of generalized coordinates of each body and consists of free variables where is the Jacobian Relation between and : Then where and (HG is positive-definite)

  19. Articulated body dynamics We have where What is ? where are the joint torques and are the contact forces Then and are variables

  20. Optimization (sum of errors) Minimize where and are variables of interest (reference and actual joint accelerations) (reference and actual contact accelerations) (reference and actual joint torques) (reference and actual contact forces)

  21. References • A Mathematical Introduction to Robotic ManipulationR. N. Murray, Z. X. Li, S. S. Sastry, CRC Press • Rigid Body Dynamics AlgorithmsR. Featherstone, Springer

  22. Thank You! & Question?

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