Modeling Robot with n total limbs n-1 limbs on the ground World , Abdomen Contacts arbitrary, Coulomb Friction Shoulders , End-Effector ,Feet Each link’s COM* is at its coordinate origin Quasi-static Joint Angles: Velocity: is ‘the velocity of w.r.t as seen by ‘ Lets start obtaining our tools by obtaining a description of abdomen motion. 1st limb n th limb Center Of Mass* (Support Region)

Abdomen Motions Stance Map: Contact Forces Abdomen Forces Stance Map Transposed: Contact Velocity Abdomen Velocity Analogous to Grasp Map Enforce contact constraint Feet cannot penetrate the ground Legs must comply with body Conventions follow Murray, Li, Sastry 1994 This allows us to impose a fundamental constraint on stance, and obtain the stance jacobian.

Abdomen Velocity Stance Jacobian: Joint Velocities Abdomen Velocities Analogous to Hand Jacobian From here, describing the way to reach out with a particular velocity is straightforward (contact information)

Free Limb Motions = (Abdomen Motion w.r.t world) Reach Jacobian: Joint velocities end-effector velocities 2 Pieces: Free Limb w.rt world Lets step aside now and make some interesting observations. = (Abdomen Motion w.r.t world) + (Free Limb Motion w.r.t Abdomen)

Analogies to Manipulation Similarity previously noted by many, e.g. [Waldron 1986], [Kumar and Waldron 1987], [Hauser et al 2008], [Bretl and Lall 2008], [Johnson and Koditschek 2012]. Stance Kinematics make analogy explicit. Now we consider a very important piece of the puzzle -- how to deal with gravity Multi-Fingered Hand Kinematics* Multi-Legged Robot Kinematics Stance Jacobian Hand Jacobian Stance Map Grasp Map Manipulable Grasp Force Closure *Introduced by [Kerr, 1984] and [Salisbury, 1982]

Center of Mass Velocity Differentiate mass weighted average of link positions to get Stance Constrained Center of Mass Jacobian: Where Center of Mass Velocity (Weighted Jacobians)

Center of Mass Velocity Center of Mass Jacobian: Joint Velocities Center of Mass Velocity Used to encourage ‘Without Falling’ Relationships expressed in contact frames Contact properties are explicit Lots of structure to be exploited Center of Mass Velocity It seems that these kinematic building blocks might be enough to try to do something.

Kinematics of Stance/Reach Stance Jacobian Stance MapT   Reach Jacobian     Stance/Reach Kinematics   Center-of-Mass Kinematics Center-of-Mass Jacobian   Weighted Link Jacobian Shankar & Burdick ‘ICRA 14

“Balanced Priority” Planning “Balanced Priority” Solution “Balanced Priority” Planning Existence & Uniqueness results [ICRA’14] Closed form solution [IJRR’15] Extends to these geometries Shankar & Burdick ‘ICRA 14, IJRR’15

Efficient QP Formulation Quadratic Programming (QP)   Sub. to (t = 1 if soln. exists, t = 0 otherwise) Linear feasibility Prog.     Subject to A LOT of constraints can be immediately incorporated Self-Collision Constraints (Bullet) Point link in constant direction (gaze constraint) Static Equilibrium Constraint Hard Boundaries on Link motion Configuration Biasing (preferred walking pose) L1 norm—release min number of joint breaks CVXGEN (Mattingley and Boyd) --(http://www.cvxgen.com) Custom solver runs extremely fast 160-320 microseconds for SURROGATE (21 DOF) 220-480 microseconds for RoboSimian (28 DOF)

Efficient QP Formulation Quadratic Programming (QP)   Sub. to Linear Constraint Relax.     Subject to A LOT of constraints can be immediately incorporated Self-Collision Constraints (Bullet) Point link in constant direction (gaze constraint) Static Equilibrium Constraint Hard Boundaries on Link motion Configuration Biasing (preferred walking pose) L1 norm—release min number of joint breaks CVXGEN (Mattingley and Boyd) --(http://www.cvxgen.com) Custom solver runs extremely fast 160-320 microseconds for SURROGATE (21 DOF) 220-480 microseconds for RoboSimian (28 DOF)

Stance Jacobian

Reach Jacobian

Center of Mass Jacobian