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

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

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

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.

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

Stages Kim, Jung-Yup (2006)

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)

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

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

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

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

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”

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

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

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

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

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

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)

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)

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)

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

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

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

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

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

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

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

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

Trajectory Parameters

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