Motion Planning of Multi-Limbed Robots Subject to Equilibrium Constraints. Timothy Bretl Presented...
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![Page 1: Motion Planning of Multi-Limbed Robots Subject to Equilibrium Constraints. Timothy Bretl Presented by Patrick Mihelich and Salik Syed.](https://reader030.fdocuments.net/reader030/viewer/2022032800/56649d385503460f94a11b1a/html5/thumbnails/1.jpg)
Motion Planning of Multi-Limbed Robots Subject to Equilibrium Constraints.
Timothy Bretl
Presented by Patrick Mihelich and Salik Syed
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Introduction● Free-Climbing robots must coordinate limb motions to satisfy a set of complex constraints.
●Decision made early on affect ability to complete task.
● Two primary constraints Contact (keep hands at a carefully chosen set of hands)
Equilibrium (Apply forces exactly to compensate for gravity without slip)
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Basic Approach
● First plan for which set of holds to use
●Search if a path exists between this set of holds.●Use sampling based techniques to compute both holds, and connectivity between holds.
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Configuration Space of a 4 Limb climbing robot.
●Workspace is a plane●Configuration space is parametrized by (x,y), angle of body and 2 angles per arm. (in empty space)
(x,y,Θ)
Configuration in empty space:q=(x,y,Θ,β1.1,β1.2....β4.1,β4.2)
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Better characterizing the C-Space● A climbing robot must maintain contact with at least some
number of holds
● Because certain limbs are constrained when holding a given hold. The configurations of those limbs can be parametrized. The configuration of the robot under constraint is thus a submanifold of the C-Space. This is a “Stance Manifold”
●The feasible space is part of this lower dimension “stance manifold” which avoids collision or dis-equillibrium. (analagous to freespace)
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3 or 4 Hold stances:Stance Manifold is a 5D manifold in 11D C-space.
Stance Manifold is a 3D manifold in 11D C-space.
Due to constraints body configuration can be defined by 3 parameter
Support seg. Support seg.
Inverse kinematics would yield a definite solution given 5 parameters + constrainsts
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Transitions●Transitions are are configurations which belong to more than one state manifold. i.e they are configurations which could be used to transition from one state to another.
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Representing connectivity of sub-manifolds
●Stance Graph: Simply represent connectivity of stance manifolds with graph. (relatively) Easy to compute (requires only finding intersecting configurations)
A B
C
D
E
Each node in the graph representsa seperate stance manifold.
If there is a overlap betweenA,B add an edge
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Problem with stance graph●Path in stance graph is a necessary but not sufficient condition for changing stances..●Stance graph does not take into account obstacle collisions. Since a stance manifold represents all configurations. (not to be confused with feasible region)●Just because your two stances are connected does not mean you can use those stances to reach the goal w/o collision (animation)●It is difficult to compute the stance graphs explicitly ... (sampling based techniques are used)
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1
3
4
2
1 23
4
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Component Graphs
●Instead look at the connected components of the stance manifolds.●If two components A,B have the property A∩B ≠{}then add an edge between the components.●This provides both a necessary and sufficient condition.
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S1
S2
S4
S3
Two different connected components
S1..S4 are stance manifolds.
Note dimensionality of manifolds may not be the same,intersections illustrate that both S1,S2contain the same configuration
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Transition graph●Can also have “transition graph” two points q1 of A,B and q2 of B,C are connected if there exists a collision free path between q1,q2 inside B●Similar to component graph, but easier to compute (don't need to compute connected components)
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S1
S2
S4
S3
S1..S4 are stance manifolds.
q1 q2
q1q2
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Two-stage search strategy
● Explore stance graph● Edges are transitions between stances● Generated quickly● No guarantee of feasible path
● Explore transition graph● Edges are paths between transitions● Expensive, but can use sample-based techniques
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Exploring the stance graph
● Maintains a priority queue of nodes to explore, ordered according to a heuristic.
● Computes candidate sequence of nodes, edges
A B
C
D
E
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Exploring the transition graph● Explores transition for each edge in candidate
sequence● Explores path for each node● Find furthest reachable stance
● If not the final stance, delete edge and re-search stance graph
A B
C
D
E
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Exploring the transition graph● Explores transition for each edge in candidate
sequence● Explores path for each node● Find furthest reachable stance
● If not the final stance, delete edge and re-search stance graph
A B
C
D
E
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Justification
● Stance graph cheaper than transition graph● Finding transition cheaper than finding path to reach
it
● Stance graph is good approximation of transition graph
● Can prune large portions of transition graph
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Ordering the search
● “Lazy strategy”● Order priority queue by likeliness that candidate
sequence will actually be feasible● Estimate probability that each edge in stance
graph will be path-connected● Time spent exploring● Learned classifier
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Finding transitions
● Transition lies in intersection of feasibility spaces at adjacent stances
● Sample configurations from this intersection
S2
S3
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Finding paths
● Path lies in stance manifold between transitions● Use PRM approach
● Retain samples where robot is at equilibrium
S4
q1q2
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Non-uniform sampling strategy
● C-space can be decomposed into subsets by elbow bends
● Configurations with different elbow bends may not be connected by straight line
● Need to explicity sample singular configurations
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Conclusions
● “Stance before motion” approach is effective
● Satisfaction of complex conditions well approximated by simple ones
● Non-uniform PRM sampling strategy● Explicitly sample singular configurations
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Future work
● Dynamics
● Autonomous hold detection
● Identifying “crux” steps