1. Lecture 3 Intro to Posture Control Working with Dynamic Models

2. The need for Posture Controllers Joint Controllers (PD) Generate voltage to produce desired torque/speed Move links to follow generated path Posture Controllers Compensates for non- idealities:  Vibration  Compliance in joints and limbs  Changes in mass distribution  Force/torque disturbances  Changes to the mathematical model

3. Landing Controllers What must be controlled? Angular momentum Landing Impact Force Landing timing Uneven foot placement How it’s done Apply torque at ankle during landing Deflect knee joint to create virtual shock absorber Extend or retract landing leg to change cycle timing Add delay to walking cycle until landing is achieved,

4. Keeping the Torso Vertical Tilted floors introduce pitch/roll error Need independent orientation data Correct with simple PI control to ankles/knees

5. Impact Force Reduction Prescribe knee deflection as ideal mass-spring-damper Determine maximum allowable force Determine total landing time dt Apply 2 constraints to determine ideal k, c

6. Angular Momentum compensation Desired angular momentum Compensate by applying angular impulse

7. Dynamic Modeling

8. Types of Modeling Tools AutoLev Script-based Can produce Inverse Kinematics, Equations of motion, and PID control Relies on external solver (Matlab or C) ADAMS GUI based dynamics simulation Uses predefined joints and rigid bodies to build system Simulates system based on a specified input Difficult to interface with external software packages

9. What do we model? Should be included Large and heavy bodies Joints with motors Compliant joints  virtual spring and damper Can be sometimes be ignored Flexure of structure under low stress / low frequency excitement Lump small masses Ignore small inertia contributions

10. Simplification of Hubo Actual HuboSimplified model (10 DOF) LLL LUL TSO RLL RUL LFT RFT

11. Joint design for Inverse Kinematics Need to be able to describe limbs in terms of end position Ball joints at hip and ankle require Euler angles Use gimbal joints to simplify description

12. Modeling Procedure Define rigid bodies (geometry, mass, inertia) Define any extra reference frames Describe geometric relationships between bodies/frames Define p, v, a for each mass center Define θ, ω, α for each body Define applied forces/torques Use Kane’s method to produce equations of motion

13. Validate the model Zero Input Behavior Intuitive behavior:  Collapse due to gravity  Swing infinitely with no applied damping  Settle with damping at joints Static Equilibrium Multiple inverted pendulum has equilibria at:  q i =0, all links vertical  q1=pi, qi=0, (i=2..n)  (All links hanging) Equilibria w/ static Torque Apply torque at each joint to balance weight T 1 (t) T 2 (t) T 3 (t)

14. Add PD Control to joints Proportional Control Define error qerr i Compare to reference r i Increase proportional gain to improve trajectory tracking Derivative Control Define error qerr i Compare to reference r i Increase derivative gain to damp overshoot

15. Inputs and Initial conditions Initial Conditions Initial angles and velocities must be valid AL: Solve initial conditions using nonlinear eq. solver (See IC example) Input from Inverse Kinematics # ref inputs = # Independent system DOF Use IK to translate trajectories into r i (see IK example )

16. Boundaries Work envelope limits possible trajectories Choose desired trajectory within work envelope

17. Model Update: Taking a step Double Support (3D) Necessary conditions at switch time T P D (T)=P S (T) V D (T)=V S (T) Very simple - No collision dynamics Single support (3D)

18. Model Update: Foot Landing Single Support (3D) Necessary conditions @ T P S (T)=P D (T) V S (T)≠V D (T) H S (T)=H D (T) V i,n =0 Double support (3D)