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

• More 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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Papers

• LaValle, Yakey, Kavraki (1999,2001)
• Works but very slow performance
• Han and Amato (2000)
• Better performance
• Cortes, Simeon, Laumond (2002)
• Best performance

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LaValle, Yakey, Kavraki

• Uses the standard method of generating roadmap
• Randomly samples c-space
• Attempts to connect nodes with local planner
• Admits most points are not in Csat so uses
  steepest descent type of method to move samples
  into Csat

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LaValle, Yakey, Kavraki

• Worked but resulting performance not so great
• Initial results took hours to compute
• Some improvements in 2001 paper but still takes
  15 to 30 minutes for 9 and 11 link examples.

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Han Amato Paper

• Basic principles
• Break chain into active and passive parts, using
  PRM (OBPRM) 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 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
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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 Probabilistic Roadmap Approach for Systems with
  Closed Kinematic Chains by Steven M. LaValle,
  Jeffrey Yakey, and Lydia Kavraki, IEEE
  International Conference on Robotics and
  Automation, 1999.
• A Kinematics-Based Probabilistic Roadmap Method
  for Closed Chain Systems, Li Han and Nancy M.
  Amato, Proceedings of the Workshop on Algorithmic
  Foundations of Robotics (WAFR'00), March 2000
• A random loop generator for planning the motions
  of closed kinematic chains using PRM
  methodsCortes, J. Simeon, T. Laumond,
  J.P.Robotics and Automation, 2002. Proceedings.
  ICRA '02. IEEE International Conference onVolume
  2,  11-15 May 2002 Page(s)2141 - 2146 vol.2
• Probabilistic motion planning for parallel
  mechanismsCortes, J. Simeon, T.Robotics and
  Automation, 2003. Proceedings. ICRA '03. IEEE
  International Conference onVolume 3,  14-19
  Sept. 2003 Page(s)4354 - 4359 vol.3 Also