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Bipedal MotionJon HagerScopeDefining biped locomotionZMPDynamic ModelsTrajectory PlanningWABOT 11973, Ichiro Kato, Waseda University

Static gait

Terminology

TerminologyWalk & GaitTerminologyWalk & GaitPhasesTerminologyWalk & GaitPhasesStatic/DynamicTerminologyWalk & GaitPhasesStatic/DynamicFCoMTerminologyWalk & GaitPhasesStatic/DynamicFCoMSupport PolygonTerminologyWalk & GaitPhasesStatic/DynamicFCoMSupport PolygonZMPZero-Moment Point (ZMP)1984 WL-10RD of the robotic family WABOTDynamic gait

Zero-Moment Point (ZMP)Central factors:Simplified dynamic forces and moments from above the ankleContact of the foot with the ground, center of pressureReaction force and frictionNet moments about the x and y axesThe support polygon

MIOMIR VUKOBRATOVIC and BRANISLAV BOROVAC, ZERO-MOMENT POINT THIRTY FIVE YEARS OF ITS LIFERobot side: - consolidated force from body - consolidated moment from bodyGround side: - reaction force - reaction moment (friction)P - Reaction force acting point (CoP)

The Support PolygonExtends to edge of contact area of foot(single support phase)Rigid footProvides the real world bounds for point P

When is dynamic stability ensured?At point P:

Thus the name ZERO-MOMENT pointWhere can the ZMP occur?

Observe static equilibrium

Projected on horizontal plane

G foot mass centerA Ankle jointms foot massO Origin of coordinate system - Ground reaction force

This equation can be used to calculate the location of P in order to sustain equilibriumWhat happens if the calculated P is outside the bounds of the foot?

Recall : Acting point of is limited to the bounds of the foot.If the calculated P is within the bounds, it represents ZMPOtherwise, this point is known as a fictional ZMP point (FZMP)FZMPActing point of goes to the edge of footA perturbation moment about the x or y axis is induced.Intensity of this moment is directly proportional to distance between P and FZMP

MIOMIR VUKOBRATOVIC and BRANISLAV BOROVAC, ZERO-MOMENT POINT THIRTY FIVE YEARS OF ITS LIFEDistinction between CoP & ZMPCoP : Can always describe the point where the ground reaction force acts on the footZMP : Only applies when the CoP balances all of the forces and moments about the foot.

MIOMIR VUKOBRATOVIC and BRANISLAV BOROVAC, ZERO-MOMENT POINT THIRTY FIVE YEARS OF ITS LIFEAdjusting to FZMPWith FZMP, emergency steps must be take to regain dynamic stability Increase the size of the support polygonUse body/arms to adjust FA / MA

NEOTRAINFZMPUSE OF ARMS/BODY TO CREATECOUNTER FORCE ANDMOMENTINCREASE SUPPORTPOLYGON SIZE USINGTOESDynamic Modeling of the Robot3D Linear Inverted Pendulum Mode (3D-LIPM)

3D Linear Inverted Pendulum Mode

Movement of CoM constrained to an arbitrary plane

3D Linear Inverted Pendulum Mode

Normal vectorArbitrary planeMassless and extendable pendulum leg

3D Linear Inverted Pendulum Mode

Additional constraint:

3D Linear Inverted Pendulum Mode

Result from applying constraints:

3D Linear Inverted Pendulum Mode

If horizontal plane constraint is applied,

Where is the location of the ZMP

3D Linear Inverted Pendulum Mode

Therefore,

Cart-table model

Cart-table model

Trajectory PlanningApplying both feetManaging ZMP and CoMTrajectory PlanningApplying both feetManaging ZMP and CoM

Trajectory PlanningApplying both feetManaging ZMP and CoM

Trajectory PlanningGOAL: Calculate CoM given the ZMPTrajectory PlanningGOAL: Calculate CoM given the ZMPSolve with FFTPreview ControlFFT Method

Preview Control

Preview ControlServo controller1969, Hayase & IchikawaLinear Quadratic optimal servo controllerUses future values

Preview ControlTranslate ZMP equations to dynamical systemConsider the third derivative of position of CoM,

Preview ControlUse as input for ZMP equation

Discretize the system

Preview ControlPerformance index to be minimized

Servo errorIncremental stateIncremental input

Preview ControlController which minimizes given that the ZMP reference is previewed for step future:

G values are gains calculated by Q and R weights

Preview Control

Preview Control