Toward More Efficient Motion Planning with Differential ...mac/pubs/final-exam-slides.pdf · Final Oral Exam December 14th, 2007 Maciej Kalisiak Toward More Eﬃcient MP w/Diﬀerential

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Motion PlanningViability

Contributions

Toward More Efficient Motion Planning withDifferential Constraints

Maciej Kalisiak

Final Oral ExamDecember 14th, 2007

Maciej Kalisiak Toward More Efficient MP w/Differential Constraints

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Contributions

Outline

1 Motion Planning (MP)What is MP?Types of MP problemsMP is hard

2 Viability

3 ContributionsMP in highly constrained problemsMP w/viability filteringViability-based safety enforcement

4 Conclusion


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What is MP?Types of MP problemsMP is hard

Outline


2 Viability


4 Conclusion


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What is Motion Planning (MP)?

in a nutshell:“how to get from A to B?”

sometimes also:“... optimally?”

example problems


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example problems


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example problems


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example problems


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example problems


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Approach

solved by converting to dual problem (agent → point)complication: often cannot manipulate agent directly

X = x × y × θx ∈ X

x = (x, y, θ)


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Approach



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Approach



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Types of MP problems

common types:kinematicnonholonomickinodynamic

e.g., “Piano Mover’s Problem”


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e.g., agents w/rolling contacts


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e.g., inertia & balance play big role


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Differential Constraints (DC)DC: constraints on q′( d

dt of agent configuration)

nonholonomic kinodynamic

DC

DCs very common,but make MP more difficult


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MP is hard

hardnessPiano Mover’s Problem:→ PSPACE-completeMP problems w/DC: at least as hard

why?“curse of dimensionality”real world problems often high-DDCs complicate search space further

300 S. M. LaValle: Planning Algorithms

P NP PSPACE EXPTIME

Figure 6.40: It is known that P ⊂ EXPTIME is a strict subset; however, it is notknown precisely how large NP and PSPACE are.

or EXPTIME. A language A is called X-hard if every language B in class X ispolynomial-time reducible to A. In short, this means that in polynomial time,any language in B can be translated into instances for language A, and then thedecisions for A can be correctly translated back in polynomial time to correctlydecide B. Thus, if A can be decided, then within a polynomial-time factor, everylanguage in X can be decided. The hardness concept can even be applied toa language (problem) that does not belong to the class. For example, we candeclare that a language A is NP-hard even if A 6∈NP (it could be harder and lie inEXPTIME, for example). If it is known that the language is both hard for someclass X and is also a member of X, then it is called X-complete (i.e., NP-complete,PSPACE-complete, etc.).8 Note that because of this uncertainty regarding P, NP,and PSPACE, one cannot say that a problem is intractable if it is NP-hard orPSPACE-hard, but one can, however, if the problem is EXPTIME-hard. Oneadditional remark: it is useful to remember that PSPACE-hard implies NP-hard.

Lower bounds for motion planning The general motion planning problem,Formulation 4.1, was shown in 1979 to be PSPACE-hard by Reif [819]. In fact, theproblem was restricted to polyhedral obstacles and a finite number of polyhedralrobot bodies attached by spherical joints. The coordinates of all polyhedra areassumed to be in Q (this enables a finite-length string encoding of the probleminstance). The proof introduces a fascinating motion planning instance that in-volves many attached, dangling robot parts that must work their way through acomplicated system of tunnels, which together simulates the operation of a sym-metric Turing machine. Canny later established that the problem in Formulation4.1 (expressed using polynomials that have rational coefficients) lies in PSPACE[175]. Therefore, the general motion planning problem is PSPACE-complete.

Many other lower bounds have been shown for a variety of planning problems.One famous example is the Warehouseman’s problem shown in Figure 6.41. This

8If you remember hearing that a planning problem is NP-something, but cannot rememberwhether it was NP-hard or NP-complete, then it is safe to say NP-hard because NP-completeimplies NP-hard. This can similarly be said for other classes, such as PSPACE-complete vs.PSPACE-hard.


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MP is hard

hardnessPiano Mover’s Problem:→ PSPACE-completeMP problems w/DC: at least as hard

why?“curse of dimensionality”real world problems often high-DDCs complicate search space further

300 S. M. LaValle: Planning Algorithms

P NP PSPACE EXPTIME

Figure 6.40: It is known that P ⊂ EXPTIME is a strict subset; however, it is notknown precisely how large NP and PSPACE are.

or EXPTIME. A language A is called X-hard if every language B in class X ispolynomial-time reducible to A. In short, this means that in polynomial time,any language in B can be translated into instances for language A, and then thedecisions for A can be correctly translated back in polynomial time to correctlydecide B. Thus, if A can be decided, then within a polynomial-time factor, everylanguage in X can be decided. The hardness concept can even be applied toa language (problem) that does not belong to the class. For example, we candeclare that a language A is NP-hard even if A 6∈NP (it could be harder and lie inEXPTIME, for example). If it is known that the language is both hard for someclass X and is also a member of X, then it is called X-complete (i.e., NP-complete,PSPACE-complete, etc.).8 Note that because of this uncertainty regarding P, NP,and PSPACE, one cannot say that a problem is intractable if it is NP-hard orPSPACE-hard, but one can, however, if the problem is EXPTIME-hard. Oneadditional remark: it is useful to remember that PSPACE-hard implies NP-hard.

Lower bounds for motion planning The general motion planning problem,Formulation 4.1, was shown in 1979 to be PSPACE-hard by Reif [819]. In fact, theproblem was restricted to polyhedral obstacles and a finite number of polyhedralrobot bodies attached by spherical joints. The coordinates of all polyhedra areassumed to be in Q (this enables a finite-length string encoding of the probleminstance). The proof introduces a fascinating motion planning instance that in-volves many attached, dangling robot parts that must work their way through acomplicated system of tunnels, which together simulates the operation of a sym-metric Turing machine. Canny later established that the problem in Formulation4.1 (expressed using polynomials that have rational coefficients) lies in PSPACE[175]. Therefore, the general motion planning problem is PSPACE-complete.

Many other lower bounds have been shown for a variety of planning problems.One famous example is the Warehouseman’s problem shown in Figure 6.41. This

8If you remember hearing that a planning problem is NP-something, but cannot rememberwhether it was NP-hard or NP-complete, then it is safe to say NP-hard because NP-completeimplies NP-hard. This can similarly be said for other classes, such as PSPACE-complete vs.PSPACE-hard.


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Outline


2 Viability


4 Conclusion


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What is Viability?

“definition”viable state: ∃ an evasive actionnonviable state: constraint violation unavoidable


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What is Viability?



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What is Viability?


why of interest?crops up in many contexts, usefulexploited throughout thesis:

to expedite MPto aid in user-control


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MP in highly constrained problemsMP w/viability filteringViability-based safety enforcement

Outline


2 Viability


4 Conclusion


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Overall goal of thesis

aim: explore some novel ideas in MPfocus: improving MP speedgrand vision: MP with motion “macro-primitives”


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Outline


2 Viability


4 Conclusion


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MP in highly constrained problems

key idea: “any progress” is better than “no progress”


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improvement to RRT algorithmhighly-constrained problems:poor performanceproposed: RRT-Blossomresult: big speed ups (>10x)


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amount of constraint

runt

ime

ideal

RRT


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amount of constraint

runt

ime

ideal

RRT


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RRT operation review

grows two trees(from qinit and qgoal)

each tree grows toward qtgt


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RRT-Blossom

allow receding edges...but not if regressingfilter with regression testbottlenecks



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RRT-Blossom


regression if:∃other | ρ(parent, leaf ) > ρ(other , leaf )



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RRT-Blossom


21



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Outline


2 Viability


4 Conclusion


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MP w/viability filtering

drawbacks of tree-based MP:tactile-only sensingsearch ignores prior attempts

general idea:“work smarter, not harder”add “sight” + “learning” → faster MP


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MP w/viability filtering

drawbacks of tree-based MP:tactile-only sensingsearch ignores prior attempts

general idea:“work smarter, not harder”add “sight” + “learning” → faster MP


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Key extensions

“sight”virtual sensors: distance along pathyield “locally situated” state

“learning”prior trajectories → viability modelsmodels parametrized using sensors→ local models→ transferrable

ideally: bootstrapping


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Key extensions

“sight”virtual sensors: distance along pathyield “locally situated” state

“learning”prior trajectories → viability modelsmodels parametrized using sensors→ local models→ transferrable

ideally: bootstrapping


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Exploiting viability

observationscurrently: search in all of Xfreebut Xfree includes Xricxgoal usually unreachable from x ∈ Xric

⇒ avoid futile searching!model agent viabilitykeep MP search within Viab(Xfree)observed: speed-up of up to 10x

Xobst

Xfree

Xric

Viab(Xfree)


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Xobst

Xfree

Xric

Viab(Xfree)


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Xobst

Xfree

Xric

Viab(Xfree)


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Xobst

Xfree

Xric

Viab(Xfree)


Results: model transfer

agent problem posed model trained on

problem algo. runtimes

RRT-CT 371.5sRRT-Blossom 21.0sRRT-Blossom-VF 5.6s



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Results: model transfer

agent problem posed model trained on

problem algo. runtimes




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Results: tree structure

RRT-CT RRT-Blossom RRT-Blossom w/VF

no filtering viability filtering

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Outline


2 Viability


4 Conclusion


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Viability-based safety enforcement

⇒ assisted control:inherently usefulfacilitates obtaining user-demonstrated training datahelpful in user-assisted MP (future work)

key idea: viability more reliable for detecting imminent danger


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Viability-based safety enforcement

⇒ assisted control:inherently usefulfacilitates obtaining user-demonstrated training datahelpful in user-assisted MP (future work)

key idea: viability more reliable for detecting imminent danger


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Collision avoidance

typical (collision-based)based on predictive lookahead (Th seconds)weakness: Th is finite

Th may be too smallsafety↑ as Th →∞

better: viability-based safety enforcementonly a minimal lookahead neededlonger lookaheads: milder corrections


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Collision avoidance





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Collision avoidance





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Collision avoidance





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Collision avoidance





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Collision avoidance





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Operation

xk

xk

xkxk

F i(xk, vk)

L3

F i(xk, uj)

L2L1L0


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Viability of control actions

U

ThTebTh

Teb

U


Experimentsag

ents

envi

ronm

ents

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Results

environment

Viab(Xfree) model

enforcement

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Conclusion

contributionsbetter handling of constrained environments in RRTmore efficient MP by narrowing search to Viab(Xfree)more robust threat avoidance in computer-assisted control

future work:learning appropriate actions from motion dataMP w/motion “macro primitives”evaluate viability filtering with other MPslocal viability models for safety enforcement(near-)optimal solutions for MP w/DChuman-derived motion data (e.g., style content)human-guided MP: selection of style or topology


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Conclusion

contributionsbetter handling of constrained environments in RRTmore efficient MP by narrowing search to Viab(Xfree)more robust threat avoidance in computer-assisted control

future work:learning appropriate actions from motion dataMP w/motion “macro primitives”evaluate viability filtering with other MPslocal viability models for safety enforcement(near-)optimal solutions for MP w/DChuman-derived motion data (e.g., style content)human-guided MP: selection of style or topology


Documents

Toward More Efficient Motion Planning with Differential ...mac/pubs/final-exam-slides.pdf · Final Oral Exam December 14th, 2007 Maciej Kalisiak Toward More Eﬃcient MP w/Diﬀerential