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Whole-body Planning and Stabilization of Dynamic Motion with ContactMichael Posa*, Scott Kuindersma, and Russ Tedrake**MIT and HarvardDynamic Walking 2015TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA1

Whole-body Planning and ControlGoal: efficient motions, periodic and aperiodicDRC demonstrated capability of QP controllersLargely ZMP/COM-basedFlat foot walkingBuild on classical optimal controland stabilization algorithmsIntegrate QP controllers with state of the art trajectory optimization# of 13DRC: QP control by IHMC, CMU-WPI, MIT! Want to combine QP with optimal control for more natural, efficient motionsEmphasize traj+contorl respecting constraints2Contact Implicit Trajectory OptimizationFind dynamically feasible trajectoryMinimizes cost (e.g. effort or cost of transport)Without a priori specification of contactsJointly optimize over state, input, and contact forces

Floating base Positions and VelocitiesGap function[Dyn. Walking 2012, WAFR 2013, IJRR 2014]

What about executing the trajectory?# of 13introduce framework3Trajectory AccuracyContact implicit method is O(h)Higher accuracy methods for smooth systemsDirect collocation O(h3) [Hargraves and Paris,1987]Extend direct collocation to handle contact constraintsFloating base coordinatesClosed kinematic chains (e.g. double support)Friction limitsTrajectories constrained to contact manifold

# of 13However, contact implicit gives a mode sequence (and initial seed trajectory).E.g. foot on ground mode4

Direct Collocation O(h3)Optimize over ComputeConstruct cubic splines between Match dynamics with spline derivative at midpoints

# of 13Take a step back and review H+P. Constraint defect.5Nave ApproachCompute constrained dynamicsAdd constraints for state to be on manifold

Consider the unactuated case: given find Overconstrained!

# of 13Go quicker here. Either relax constraints or add variables.6

Constrained Direct CollocationOptimize over ComputeConstraints

Spline matches dynamics at collocation point up to a projection onto contact manifold

Constrained DynamicsKnot points on manifold# of 13Explicit lambda allows for friction constraints.7Stabilization: Constrained LQRLinearize deviations from nominal

Dynamics satisfy (linearized) manifold constraint

Can only control in the kernel ofSuppose an orthonormal basis of kernelThen is controllable with

# of 13Not controllable8Stabilization: Quadratic ProgrammingDescend LQR cost-to-goOptimize over accelerations, control input, and contact forcesSubject to linearized constraints:Whole-body dynamicsContact manifoldCoulomb friction limitsInput saturations[Kuindersma et al., ICRA 2014]# of 13Leverage control from DRC9Example: Biped Walking

# of 13Prelim results10Example: Monopod Hopper

# of 13Example: Underactuated Biped

# of 13ConclusionExplicitly treating contact forces a natural extension to traditional algorithmsFuture workLateral stabilization and 3DAtlas experimentationContact implicit direct collocationHigher order accuracy without a mode sequence

Scott is looking for postdocs and students to join the Harvard Agile Robotics lab, talk to him or email scottk@seas.harvard.edu # of 13