1. Introduction to Biped Walking
Lecture 1
Background, simple dynamics, and control

2. Some Sample Videos
Human Walk.avi
Hubo straight leg.avi

3. Human Leg Anatomy
Torso
Hip, 3DOF
Knee, 1DOF
Ankle, 2DOF
Toes, ~2 DOF

4. Building Blocks of Biped Walking
Dynamic modeling
Trajectory generation
Inverse kinematic model
Trajectory error controllers
Additional failure mode controllers
Mechatronics
Programming
Provides virtual experimentation platform
The ideal path that the hips and feet follow.
Specifies the joint movements to make feet and hips follow the trajectory
Specify how the joints should move to compensate for trajectory error.
Adjusts the trajectory to compensate for nonidealities.
The structure and implementation and the limitations thereof
Reading sensors, processing and filtering their data, sending joint position commands.

5. Walking Cycle (2D)
Kim, Jung-Yup (2006)

6. Stages
Kim, Jung-Yup (2006)

7. Controllers
Damping Controller reduces reactive oscillations to swinging legs
ZMP controller minimizes ankle torque and optimizes hip trajectory
Landing controller limits impact forces at foot, controls foot angle
Torso/pelvis controllers follow prescribed trajectory
Tilt-over controller adjusts foot placement if ZMP becomes unstable
Landing position controller adjusts foot landing to compensate for excess angular velocity
Kim, Jung-Yup (2006)

8. Block Diagram of KHR-2
Kim, Jung-Yup (2006)

9. Balance Control Controls Center of mass location Consists of:
Prevents tiltover
Controls foot placement during landings
Consists of:
Torso sway damping controller
ZMP controller
Foot placement controller
Foot Landing Controller

10. Single Support Vibration Modeling
Compliance between ankle and torso
Model robot body as lumped mass
Model flexible parts and joints as spring
Use Torque along X axis of ankle to counteract motion
Linearize with small angle

11. Vibration Damping Control
Apply Laplace Transform
Factor out Θ(s) and U(s) to form transfer function
Substitute to find TF of Torque wrt input angle

12. Damping Controller Substitute Apply derivative feedback of error
β= K/ml2−g/l
α=K/ml2
Apply derivative feedback of error
Simulation shows effect of damping on vibrations
(See )“vibdamp.mdl”

13. Joint Motor Controller Basics
DC brush motors
Harmonic drive gear reduction
Simple governing equations
Inefficient at low speeds

14. Joint Motor Controller
Motor Voltage/Speed constant (V-s/rad)
Rotor Inductance (Henry)
Rotor Resistance (Ω)
Input Voltage (V)
Current (Amp)
Output Torque (N-m)
Motor equivalent viscous friction (N-m-s)
Block Diagram of System

15. Effects of Motor on Control
Torque limit due to R
torque inversely proportional to speed
High current (and heat) at zero speed

16. Ankle model with motor Assume simple inverted pendulum
Combine electrical and mechancal ODE’s

17. Zero Moment Point
Point about which sum of inertia and gravitational forces = 0
Requires no applied moment to attain instantaneous equilibrium
Control objective: minimize horizontal distance between COM and ZMP

18. Single Support Model Divide ZMP control into 2 planes
Track hip center to ZMP
Requires dynamic model or experiment to determine model parameters
Pole placement compensator
(See “ZMP.mdl”)
Investigate model
Double inverted pendulum
Kim, Jung-Yup (2006)

19. Foot Landing Placement
IMU measures X and Y angular velocity
Hip sway monitored by trajectory controllers
Excess angular velocity reduced by widening landing stance
Reduced angular velocity maintains hip trajectory
Kim, Jung-Yup (2006)

20. Landing Problem Foot landing causes impact and shock to system
Dynamics of shock are difficult to model
Large reaction forces
Angular momentum controlled with 1 ankle
Before After v2
v’1=0
v’2 v1
Fz(t)
M(t)

21. Simplified Collision Dynamics
Governing Formulas
Impact Energy Losses
Power Input
Impact
Before After v2 v1

22. Deriving the ideal model
Ideal mass-spring-damper
mT≈53kg (hubo’s mass)
c, k = model constants
Form transfer function
Solve numerically mT

23. Dynamic Model of knee mT Lump mass of torso at hip
Lagrange method to derive dynamics
Add artificial damping to reduce simulation noise
Use PID control to stabilize

24. Knee Inverse Kinematics
Need to solve θi(x,t) (i=1,2)
Desired path along y axis (x=0)
Setup constraint equations & solve
Apply as input to model

25. Trajectory Generation
“Goal” Control
Trajectory Feedforward
Needs no knowledge of model
Low computation overhead
Non-optimal path
Requires mathematical model
Input conditioned for system
Requires online computation
Allows path optimization

26. Hubo’s Hip Trajectory Y=A*sin(ωt) Simplifies frequency domain design
A=sway amplitude
Ω= stride frequency (rad/s)
Simplifies frequency domain design
X=c*A1cos (ωt)+(1-c)A2*t
A2=A1*π/(2 ω)
c controls start/end velocity
Amplitude A1 controls step length

27. Basic foot trajectory Continuous function of t
Cycloid function
Continuous function of t
0 velocity at each full cycle
Velocity adjustable by linear component

28. Timing of walking cycle
Short double support phase (<10% of half cycle)
Knee compression and extension
Short landing phase
Kim, Jung-Yup (2006)

29. Trajectory Parameters

30. What’s Next Biped Design Procedure Next Lecture: Concepts
Dynamic modeling
Simulations
Trajectory generation
Fundamentals of dynamics
Fundamentals of controls
2d dynamic modeling
Implementing posture control systems
Basic X and Z axis trajectories