Computing stable equilibrium stances of a legged robot in frictional environments

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Computing stable equilibrium stances of a legged robot in frictional environments. Yizhar Or Dept. of ME, Technion – Israel Institute of Technology Ph.D. Advisor: Prof. Elon Rimon. g. x 3. x 2. x 1. Outline. 2D:. Computation of frictional equilibrium stances

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Computing stable equilibrium stances of a legged robot in frictional environments

Yizhar Or

Dept. of ME, Technion – Israel Institute of Technology

g

x3

x2

x1

Outline

2D:

• Computation of frictional equilibrium stances
• Robustness w.r.t disturbance forces and torques
• Dynamics – contact modes and strong stability

3D:

• The support polygon principle for flat terrains
• Geometric Parametrization of equilibrium forces in 3D
• Exact Computation of frictional equilibrium stances
• Polyhedral approximation of equilibrium stances

g

Problem Statement

Setup: mechanism modeled as a variable c.o.m. body

Given a 2D (3D) multi-limbed mechanism standing on a terrain

with k frictional contacts, where should the center-of-mass be for:

• Static equilibrium?
• Robustness w.r.t disturbance forces?
• Dynamic stability?
Applications
• Quasistatic legged locomotion on rough terrain
• (spider robots, snake robots, climbing, search-and-rescue robots)
• Graspless manipulation (part feeding, assemblies)
• Motion planning for Hybrid wheeled-legged robots
• Semi-dynamic locomotion

Application - Three-Legged Locomotion

2-3-2 gait pattern:

• Select 2-contact postures that share a common contact point
• 3-contact stage connecting two consequent 2-contact postures
• At the 3-contact stage: straight line motion of center-of-mass
Related Work – 2D
• Mason, Rimon, Burdick (1995):

Frictionless postures under gravity

• Greenfield, Choset and Rizzi (2005):

planning quasistatic climbing via bracing

• Mason (1991): Graphical methods for frictional equilibrium in 2D
• Erdmann (1998): Two-palm manipulation with friction in 2D
• Lotstedt (1982); Erdmann (1984); Mason & Wang (1988);

Rajan, Burridge and Schwartz (1987); Dupont (1992):

Contact modes and frictional dynamic ambiguity

• Trinkle & Pang (1998):

Strong stability, LCP formulation of frictional dynamics

Related Work – 3D
• McGhee and Frank, 1968:

The support polygon principle for legged locomotion

• Bretl, Latombe (2003): PRM-based motion planning

algorithm for climbing on vertical walls with discrete supports

• Mason, Rimon and Burdick , 1997:

Computing stable equilibrium frictionless stances in 3D

• Han, Trinkle and Li, 2000:

Feasibility test of frictional postures in 3D as LMI problem

• Bretl and Lall, 2006:

Adaptive polyhedral approximation of 3D equilibrium stances

Equilibrium condition:

• where t(x)= xJfext+text

x

Statics in 2D - LP Formulation
• Center of mass:x2
• External wrench:w=(fext,text) 2
• Contact forces:f2k
• Friction Cones Bounds: Bf ≥ 0

tmin = min{-Gtf}

tmax = min{-Gtf}

s.t.

s.t.

Gff=-fext

Bf ≥ 0

Gff=-fext

Bf ≥ 0

Statics - LP Formulation (cont’d)

Theorem:

The feasible k-contact equilibrium region:

R(w) = {x: tmin xJfext + texttmax}

where

Infinite strip parallel to fext

S++ = Strip (C1+,C2+)

S--= Strip (C1-,C2-)

+

-

+

+

S

S

S-+ = Strip (C1-,C2+)

S+-= Strip (C1+,C2-)

P

Theorem:

R(wo)= [(S++ S--)  P]

 [(S+- S-+)  P ]

R(wo)

_

2 Contacts - Graphical Characterization

x

2

P = Strip (x1, x2)

x

1

R(wo)

R

R

g

56

13

Algorithm:

k-Contacts - Graphical Characterization

R(w) =

=conv{Rij (w) }

x

x

1

x

x

x

4

6

x

5

3

2

g

b

fext

c.o.m.

dx

• Robust Equilibrium Region:R(N) = R(w)

wN

External Wrench Neighborhood

• Wrench magnitude scales static response
• Parametrize wext=(fx,fy,text):
• External wrench neighborhood:
• N = {(p,q): -k≤p≤k , -n≤q≤n}

R(N) = R(w)

fext

wN

fext

n

Then R(N) =R(wi)

fg

i

x

2

x

1

Robust Equilibrium Region – Example

Recipe:

If N = conv {wi}

R(N)

3 equations

3+2k unknowns

Dynamic Contact Modes Theory

ma = fext + f1 + f2 + …+ fk

Ica = text + (x1-x)×f1 +… (xk-x) ×fk

Contact modes (F, R, U, W)

(or S)

• Contact modes add 2k equations

 a unique dynamic solution as a function of x

• Contact Mode’s inequalities  Feasibility Region of x

g

Example of Dynamic Ambiguity

Contact mode UF:

The Strong Stability Criterion

Strong Stability (Trinkle and Pang, 1998):

S (w) = RSS(w) - RFF (w) RUF (w) ...  RWW(w)

• Eliminates ambiguity – only static solution is feasible
• Any roll/slide/break motion cannot evolve (at zero velocity)
• Yet, not formally related to classical dynamic stability
• (bounded response to bounded position/velocity perturbations)
• Does not always imply bounds on c.o.m. height

 Must be augmented with robustness

Robust Stability:S(N) = S(w)

wN

Define:

Robust Equilibrium Region:

wN

RXY(N) =  RXY(w)

Non-Static Modes’ N-Feasible Region:

wN

Robust Stability Region:

S(N) = RSS(N) -RFF(N)  RUF(N) ...  RWW(N)

Robust Stability - Definitions

Strong Stability:

S (w) = RSS(w) - RFF (w) RUF (w) ...  RWW(w)

Definition:RXY(N) =  RXY(w)

wN

Non-Static Modes N-Feasible Region

• Express RXY as an intersection of halfspaces
• in a four-dimensional space: Fi(x,y,p,q) ≥ 0
• RXY(N) is the projection of RXY onto xy plane

The Silhouette Theorem:

The Silhouette curves of the projection are critical values of the projection function, on which the generalized normal of RXY is parallel to xy plane.

N

N-Feasible Region of UF Mode

• Critical curves fi(x,y,p,q) are
• linear in p,q and quadratic in x,y
• Critical curves generate
• cell arrangement in xy plane
• Line-Sweep Algorithm:
• identifies the cells and generates sample points
• Checking cell membership: LP problem in p,q

N

N

N

S(N)

N

S(N)

S(N)

S(N)

Example - Robust Stability Region

S(N) = RSS(N) - RFF(N)  RUF(N) ...  RWW(N)

r=0.05

r=0.1

r=0.25

Strong Stability and Dynamic stability

Force Closure asymp. stability under keep-contact perturbations

Here: no force closure, passive contacts, arbitrary perturbations

Two contacts - neutralstability under keep-contact perturbations

Strong Stability non-static modedecays until collision

• How to model collisions? treat sequence of collisions?
• Does strong stability really leads to dynamic stability?
• How to design stabilizing joints’ control laws
• for a legged robot?
Frictional Equilibrium Stances in 3D
• Analyze 3D equilibrium stances of legged mechanisms in frictional environments
• Support Polygon criterion does not apply for non-flat terrains
• Exact formulation of equilibrium region
• Efficient conservative approximation by projection of convex polytopes

g

Problem Statement
• Characterize feasible equilibrium postures of a multi-limbed mechanism supported against frictional environment in 3D.
• Given k frictional contacts, find the feasible region R

of center-of-mass locations achieving frictional equilibrium.

• Assumption: point contacts, uniform friction coefficient m.
• Friction Cones in 3D:

Ci = {fi : (fi⋅ni)≥0and(fi⋅si)2 + (fi⋅ti)2≤m2(fi⋅ni)2}

• Feasible equilibrium region in 3D:

R is a vertical prism with horizontal cross-section .

Focus on computing the boundary of for 3-contact stances.

Basic Properties of R
• R is a convex and connected set.
• The dimension of R is generically min{k,3}.

Assumption: upward pointing contacts: fie > 0 for all fi  Ci,

where e is the upward direction

Motivational Example

m = 0.5

R

g

z

x

y

The Support Polygon Principle:

x must lie in the vertical prism spanned by the contacts:

X3

y

???

X2

X1

x

x

Motivational Example (cont’d)

top view

m = 0.2

g

z

x3

x2

y

x1

Support Polygon Principle is unsafe!!!

Parametrizing Equilibrium Forces
• Horizontal and vertical components:
• fi must intersect a common vertical line lr

Permissible Polygonal Region of r
• Projected frictional constraints:
• rmust lie in the polygonal regionP = P+ P- ,where

p2

p3

p1

where

Complete Graphical Parametrization

• Action line of fi intersects the common vertical line lr at pi
• Define zi – height of pi about xi

zi = e∙(pi –xi)

• Parametrize contact forces by (r,z) 2 3, where z=(z1,z2,z3):

P

and

Q = Q1 Q2 Q3

Qi

where

Ci

P

Permissible Region in (r,z) space
• The permissible region: (r,z)  Q = Q1 Q2 Q3 , where
• for fi lying on the boundary of Ci ,zi=zi*(r)

Computing the Boundary of

• Torque balance implies a map from (r,z) to :

Horizontal cross section is the image of Q under

• Formulate the restriction of to all possible manifolds of Q
• Compute critical curves of on each manifold of Q
• Candidate boundary curves of are -image of critical curves

type-2 boundary

fiCi,fjCj, r=r*

type-3 boundary

fi Ci, i=1..3

type-1 boundary

fi,fj≠0 ; fk=0

Graphical Example of

y

x

Conservative Polyhedral Approximation
• Replacing exact friction cones with inscribed pyramids.
• Reduces to projection of a convex polytope onto a plane
• Approximate outer bound by taking circumscribing pyramids
• Graphical example – with 6-sided pyramids

x

3

y

x

x

2

1

x

Polyhedral Approximation of R - Example

top view

In progress

Future Research
• Physical geometric intuition of boundary curves, effect of m
• Relation to line geometry and parallel robots’ singularities
• Generalization to multiple contact points
• Robustness with respect to disturbance forces and torques
• Elimination of non-static contact modes
• (Complementarity formulation, Pang and Trinkle, 2000)
• Application to legged locomotion on rough terrain in 3D

g

x3

x2

x1

Computing stable equilibrium stances of a legged robot in frictional environments

Yizhar Or

Dept. of ME, Technion – Israel Institute of Technology