1. 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)

2. 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.

3. 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)

4. 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)

5. 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]

6. 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)

7. 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.

8. 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

9. “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

10. 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) --(
Custom solver runs extremely fast
microseconds for SURROGATE (21 DOF)
microseconds for RoboSimian (28 DOF)

11. 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) --(
Custom solver runs extremely fast
microseconds for SURROGATE (21 DOF)
microseconds for RoboSimian (28 DOF)

12. Stance Jacobian

13. Reach Jacobian

14. Center of Mass Jacobian