PRM Methods for Closed Kinematic Chains

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PRM Methods for Closed Kinematic Chains

• Chris Colasuonno

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Outline

• What are closed kinematic chains?
• Why are they more difficult than open kinematic
  chains?
• What methods have been developed to deal with
  these difficulties?
• How good are these methods?

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Closed Chains

• In class, we discussed open kinematic chains with
  one end completely free to move around and used
  Denavit-Hartenberg parameters to represent them
• Closed chains form a closed loop, from the base
  to the tool to the base
• Can be thought of as two or more open/serial
  chains that are linked at two points

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Examples

• Stewart platform, closed molecular chains,
  reconfigurable robots, two arms grasping the same
  object

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Difficulties for normal PRM

• Probabilistic Roadmap methods have been effective
  in solving high-dimensional problems, including
  open kinematic chains
• In closed chains, each serial arm is constrained
  by its connection to the other(s).
• This makes things hard, even for PRM

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Random Samples and Closure

• Let Ccons be the set of all configurations in C
  that satisfy the constraints of a closed chain
  system
• Let CSAT Ccons n Cfree
• Since CSAT is generally of lower dimension than
  C, the probability that a randomly chosen point
  lies in CSAT is zero.
• Blindly generating the nodes of the PRM will
  therefore result in very poor results.

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A little more background

• Forward Kinematics As seen in class, this
  problem solves for the location of the end
  effector from the known joint angles
• Inverse Kinematics Solves the inverse problem
  Knowing the location of the end effector, solve
  for all possible sets of angles yielding that
  location (sometimes multiple solutions, more
  difficult).

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Dealing with limitations of basic PRM

• Promising methods for motion planning of closed
  chain systems have been found
• Most are based on splitting the closed loop into
  an active and passive part
• The active part is solved using usual PRM
  methods, the passive part joint angles are found
  using inverse kinematics
• This maintains loop closure

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Han Amato Paper

• Basic principles
• Break chain into active and passive parts, using
  PRM to solve active and inverse kinematics to
  maintain closure
• Use two-stage approach
• 1st, generate kinematics-valid configurations
• 2nd, fill the space with copies of these
  configurations and remove parts that intersect
  obstacles

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Han Amato Two-stage process

• Kinematic Roadmap
• Ignore environment, fix a single rigid link, use
  PRM to generate random configurations
• Stage two
• Place copies of Kinematic Roadmap in environment
  by random configs of fixed link
• Remove parts that intersect obstacles
• Use local path planners to connect the roadmaps

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Algorithm

• Kinematic Roadmap Construction
• 1. Node Generation
• (find self-collision-free closure configurations)
• 2. Connection
• (connect nodes and save paths with edges)
• (repeat as desired)
• Prototype Kinematics-Based PRM
• I. Populate Environment with Kinematic Roadmap
• generate random base configurations and retain
  collision-free parts of kinematic roadmap in
  roadmap
• II. Additional Connection of Roadmap Nodes
• connect roadmap nodes with the same closure
  structure using rigid body planners

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Node Generation for Kinematic Roadmap

• 1. Randomly generate Ta
• 2. Use forward kinematics for active chains to
• compute end-frame configurations at the break
  point of each closed chain
• 3. Use inverse kinematics for passive chains to
  compute
• joint variables Tp to achieve the end-frame
• configurations computed in Step 2.
• 4. If a solution is found in Step 3 (closure
  exists)
• 5. If closure configuration T (Ta Tp )
• is self-collision free
• 6. Retain as a kinematic roadmap node

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Node Connection for Kinematic Roadmap

• 1. For any two nearby' closure configurations Ti
  and Tj
• 2. Use (simple) local planner to find path from
  Tia to
• Tja Ta(t) t from 0 1, where Ta(0) Tia ,
  a(1) Tja
• 3. For each intermediate point on the path Ta(t)
• 4. If inverse kinematics determines that no Tp(t)
  exists to satisfy the closure constraints
• 5. return no-edge
• 6. Choose the closure configuration T(t) so that
  is continuous from previous step
• 7. If T(t) involves self-collision, then return
  no-edge
• 8. endfor
• 9. save the edge (and with it, the path T(t) t
  from 0 1)
• 10. return edge-exist

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Populating Environment with copies

• 1. Choose random vertex T from the kinematic
  roadmap
• 2. Generate random base configuration gwb
• 3. If the configuration (gwb, T) is
  collision-free
• 4. Retain (gwb, T) as a roadmap vertex
• 5. For each neighbor of T, say T, in the
  kinematic map
• 6. If (gwb, T) is collision-free
• 7. Retain (gwb, T) as a roadmap vertex
• 8. Retrieve the path T(t) connecting T and T
  from the kinematic map
• 9. If (gwb, T(t)) is collision-free for all
  intermediate closure configurations along the
  path
• 10. Add an edge between (gwb, T) and (gwb, T)
• (repeat as desired)

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Connecting Nodes of Same Closure Type

• 1. For each closure configuration T in kinematic
  roadmap
• 2. Collect all roadmap nodes with this closure
  configuration in a set
• 3. Use rigid body PRM connection methods to
  connect configurations in the set
• 4. Add the edges generated in Step 3 to the
  roadmap
• 5. endfor

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Some advantages of 2-phase algorithm

• Generates achievable configurations for closed
  chain systems with reasonable speed
• Reusing the kinematic roadmaps greatly reduces
  collision checks since self-collisions are
  checked for once per reused kinematic
  configuration

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Results
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Results
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Comments

• Using kinematics to guide PRM certainly has made
  an improvement
• Overall, works decently well but seems to have
  trouble with larger numbers of links
• Authors plan to continue working on optimizing
  for higher number of links

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Cortés, Siméon, Laumond

• Another similar method
• Builds on Han Amato method
• Identifies major drawback of method and tries to
  solve it
• Random Loop Generator Algorithm

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Problems with 2-stage method

• Choice of where to break loop into active and
  passive parts is important
• A configuration is valid only when the end-frame
  of the active-chain is in the workspace of the
  passive chain
• The probability of a valid configuration
  therefore depends on the intersection of active
  workspace and passive workspace
• This probability can be judged by the size of the
  volume of the intersection of these spaces

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Problems with 2-stage method

• A closed-form inverse kinematics solution is
  required for an efficient roadmap method
• Therefore the passive chain must be short
• With long active chains and short passive chains,
  the volume of intersection (and mentioned
  probability) will be small

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Rand Loop Generator Algorithm

• Rather than simply randomly choosing a set of
  angles within each joint limit, choose angles one
  at a time and change the interval at each step
• Change in a way to guide resulting tool
  configuration toward a conservative approximation
  of the space reachable by the passive chain
• Shorter passive chains with efficient inverse
  solutions are OK

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Random Loop Generator

• Similar to collision detection methods, uses
  spherical shapes to represent reachable
  workspaces and determine intersections

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RLG Algorithm

• For each joint in the chain

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• RESULTS
• With passive reachable space as a sphere,
  generate 1000 active configurations for which the
  tool is inside the sphere
• N-Number of sampled configs
• T-computation time in seconds

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Random Loop Generator

• This approach was implemented into the generic
  motion planning software Move3D
• Used Visibility-PRM for an even more efficient
  implementation of the method
• Two examples
• Pipe held by a car-like robot and by a mobile
  articulated arm, with obstacles in the room (12
  joint closed loop)
• Two holonomic mobile manipulators carrying a
  plate object (18 dofs) with obstacles

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Results

• Pipe problem Only solved by car moving through
  wider passage while other robot passes through
  smaller. Took 5 seconds to solve on Sun Blade
  100.
• Plate problem solved in 25 seconds what took 10
  minutes by the standard random sampling

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Comments

• RLG shows good performance
• RLG not greatly affected by increased complexity,
  thus its comparative performance is even great
  for tough problems
• Used in A Manipulation Planner for Pick and Place
  Operations under Continuous Grasps and Placements
  by same authors for solving pick and place
  manipulation type problems.

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Comments

• Also used in developing an extremely general
  approach to solving parallel mechanisms in
  Probabilistic Motion Planning for Parallel
  Mechanisms by same authors
• Examples
• Stewart Platform, ring around snake-obstacle
  Graph computed in 60 s. Plans on this graph in
  hundredths of a second
• 4 Stewart Platform Puzzle Piece Problem 15
  seconds

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References

• A Kinematics-Based Probabilistic Roadmap Method
  for Closed Chain Systems, Han and Amato
• A Random Loop Generator for Planning the Motions
  of Closed Kinematic Chains using PRM Methods,
  Cortés, Siméon, Laumond
• Probabilistic Motion Planning for Parallel
  Mechanisms, Cortés, Siméon
• Also
• A Probabilistic Roadmap Approach for Systems with
  Closed Kinematic Chains, LaValle, Yakey, Kavraki