Joint Advanced Student School 2006

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Joint Advanced Student School 2006

• Binary Manipulator Motion Planning

by Vasiliy Chernonozhkin CSRI RTC,
SPbSPU Saint-Petersburg 2006
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Introduction

• Most robots available today are powered by
  continuous actuators such as DC motors or
  hydraulic cylinders.
• Continuously actuated robots can be built to be
  precise and to carry large pay loads, but they
  usually have high price/performance ratios, as
  evidenced by the high cost of industrial robots
  available today.
• Thus, there is a need for a new paradigm in
  robotics which will lead to lower cost and higher
  reliability.

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Binary manipulator concept

• The binary manipulator concept is influenced by
  several successive concepts

• discrete actuation

• hyper-redundant structure

• sensorless systems.

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Related literature

• Discrete actuation
• 1 Pieper D.L. The Kinematics of Manipulators
  under Computer Control, 1968.
• 2 Roth B., Rastegar J., Scheinman V. On the
  Design of Computer Controlled Manipulators, 1973.
• 3 Koliskor A. The 1-Coordinate Approach to the
  Industrial Robots Design, 1986.
• 4 Kumar A., Waldron K.J. Numerical Plotting of
  Surfaces of Positioning Accuracy of manipulators,
  1980.
• 5 Sen D., Mruthyunjaya T.S. A Discrete State
  Perspective of Manipulator Workspaces, 1994.
• Hyper-redundancy
• 6 Chirikjian G.S. Theory and Applications of
  Hyper-Redundant Robotic Manipulators, 1992.
• 7 Chirikjian G.S., Burdick J.W. An Obstacle
  Avoidance Algorithm for Hyper-Redundant
  Manipulators, 1990.
• 8 Anderson V.V., Horn R.C. Tensor-Arm
  Manipulator Design, 1967.
• 9 Hirose S., Umetani Y. Kinematic Control of
  Active Cord Mechanism with Tactile Sensors, 1976.
• 10 Hirose S., Yokoshima K., Ma S. 2 DOF Moray
  Drive for Hyper-Redundant Manipulator, 1992.

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Related literature Hyper-redundancy

• Fig. 1. Three major types of hyper-redundant
  manipulators
• (a) continuous, (b) serial, and (c) cascaded
  platforms

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Related literature

• Sensorless systems
• 14 Canny J., Goldberg K. A Rise Paradigm for
  Industrial Robotics, 1993.
• 15 Mason M.T. Kicking the Sensing Habit, 1993.
• 16 Goldberg K. Orienting Polygonal Parts
  Without Sensors, 1992.
• 17 Erdmann M.A., Mason M.T. Exploration of
  Sensorless Manipulation, 1988.
• Binary manipulators
• 18 Bergstrom P.L., Tamagawa T., Polla D.L.
  Design and Fabrication of Micromechanical Logic
  Elements, 1990.
• 19 Ebert-Uphoff I., Chirikjian G.S. Efficient
  Workspace Generation for Binary Manipulators with
  Many Actuators, 1995.
• 20 Chirikjian G.S. Kinematic Synthesis of
  Mechanisms and Robotic Manipulators with Binary
  Actuators, 1995.
• 21 Chirikjian G.S., Lees D. Inverse Kinematics
  of Binary Manipulators with Applications to
  Service Robotics, 1995.

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Binary manipulator concept

• Binary manipulators are a particular kind of
  discrete device in which actuators have two
  stable states.
• Major benefits of binary manipulators are
• they can be operated without extensive feedback
  control
• they are relatively inexpensive
• they are relatively lightweight and have a high
  payload to arm weight ratio
• their task repeatability is very high.

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Variable Geometry Truss manipulator

• Fig. 2. All possible configurations of a 3-bit
  planar binary platform manipulator one VGT
  module

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Variable Geometry Truss manipulator

• Fig. 3. Configurations 110001110001110 and
  001110001110001 of 15-DOF planar VGT manipulator

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Variable Geometry Truss manipulator

• Fig. 4. Planar 30-DOF binary robot manipulator by
  Chirikjian and Burdick

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Variable Geometry Truss manipulator

• Fig. 5. Ebert-Uphoffs binary robot manipulator

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Variable Geometry Truss manipulator

• Fig. 6. Suthakorns discretely-actuated
  hyper-redundant robotic manipulator

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Binary Robotic Articulated Intelligent Device
(BRAID)

• Fig. 7. BRAIDs i-th parallel link stage
• (a) physical parallel link stage (b)
  diagrammatic representation.

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Binary Robotic Articulated Intelligent Device
(BRAID)

• Fig. 8. Detent based binary joint

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Binary manipulator motion planning

• While the hardware costs of a binary manipulator
  are lower than of a continuously actuated
  manipulator, there is a tradeoff in the
  complexity of the trajectory planning software.
• The large number of possible states of a binary
  manipulator makes it highly desirable to have
  efficient algorithms for searching through some
  (potentially large) subset of manipulator
  configurations that satisfy a particular
  constraint, so that an optimal configuration
  can be chosen.
• Since the mid 1990s Chirikjian and co-workers
  have developed a variety of efficient algorithms
  for highly actuated discrete-state robots and
  mechanisms. These include approaches to the
  kinematic synthesis of such mechanisms, the
  generation of workspaces and inverse kinematics.

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An efficient algorithm for computing the forward
kinematics
For a serial-revolute manipulator (Fig. 9) the
kinematics of an individual module in a
particular state are described by
Fig. 9. Serial-revolute manipulator
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An efficient algorithm for computing the forward
kinematics
The forward kinematics for a single VGT module
(Fig. 10) obey the following geometric
constraints
These equations are solved simultaneously for the
coordinates of the top plate of the truss
where
Fig. 10. VGT module
So,
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An efficient algorithm for computing the forward
kinematics

• Pre-computation algorithm
• Inputs to the algorithm
• qjmin, qjmax, the joint limits for each actuator
  in the manipulator in states 0 and 1
  respectively, for j 1,,J.
• wi, the width of the top of module i for a
  truss-type manipulator, or li, the length of link
  i in a serial-revolute manipulator, for i
  1,,B.
• for i 1 to B
• for j 1 to J
• Use the kinematic parameters of module i
• to compute Ci for state j.
• end
• end

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An efficient algorithm for computing the forward
kinematics

• Main algorithm
• Inputs to the algorithm
• S, a J-bit binary number representing the current
  state of the manipulator.
• Ci, for i 1,,B, the configuration sets for the
  modules in the manipulator.
• Outputs from the algorithm
• EEpos, the position of the manipulators
  end-effector.
• see, cee, the sin and cos of the end-effector
  orientation angle for the 2D case, or Ree, the
  rotation matrix describing end-effector
  orientation, in the 3D case.
• EEpos 0 Ree I
• for i 1 to B
• select Ci
• EEpos EEpos Rsibsi
• Ree Rsi Ree
• end
• end

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A combinatorial method for computing the inverse
kinematics
The standard definition of the binomial theorem
If we let x 1 and let y 1, we get the
following result
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A combinatorial method for computing the inverse
kinematics
Consider a VGT robot with J actuators, which we
move toward its target location by changing no
more than k of its actuators at a time. To do
this we must search through
candidate states to find the one that best moves
the robot toward its target position.
Operation is defined as follows
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A combinatorial method for computing the inverse
kinematics

• Algorithm description
• Inputs to the algorithm
• EEdes, the desired position of the manipulators
  end-effector.
• EEnow, the current position of the manipulators
  end-effector.
• pnow, the manipulators current state vector.
• nmax, the maximum number of bits allowed to
  change in the manipulators state vector.
• B, the number of degrees of freedom (same as
  number of bits) of the manipulator.
• The geometry of the manipulator modules for
  computing the forward kinematics (using, for
  example, the method described earlier).

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A combinatorial method for computing the inverse
kinematics

• Algorithm description
• dmin cost(EEdes, EEnow)
• pmin pnow
• bmin 0
• for i 1 to nmax
• for j 1 to
• c combo(B, i, j)
• ptest c pnow
• dtest cost(EEdes, fwdKin(ptest))
• if dtest lt dmin then
• dmin dtest
• pmin ptest
• bmin i
• end
• end
• end
• return dmin, pmin, bmin

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Efficient workspace generation

• Determining the workspace of a binary manipulator
  is of great practical importance for a variety of
  applications.
• A representation of the workspace is essential
  for
• trajectory tracking,
• motion planning,
• the optimal design
• of binary manipulators.
• Given that the number of configurations
  attainable by binary manipulators grows
  exponentially in the number of actuated DOF,
  O(2n), brute force representation of binary
  manipulator workspaces is not feasible in the
  highly actuated case.

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Efficient workspace generation

• Concepts for discrete workspaces
• The point density ? assigns each block of
  the number of binary manipulator states
  resulting in an end-effector position within the
  block, normalized by the volume of the block
• The point density array, or density array for
  short, is an N-dimensional array of integers
  (D(i, j) for N 2 or D(i, j, k) for N 3) in
  which each field/element corresponds to one block
  of the workspace and contains the number of
  binary manipulator states causing the
  end-effector to be in this block.

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Efficient workspace generation

• Concepts for discrete workspaces
• The ith intermediate workspace of a
  macroscopically serial manipulator composed of B
  modules is the workspace of the partial
  manipulator from module i 1 to the
  end-effector.
• An affine transformation in RN is a
  transformation of the form
• y Ax b, where x, y, and b are vectors in RN,
  and A is an arbitrary matrix in RNxN.
• A homogenous transformation is a special case of
  an affine transformation y Rx b, where R is
  a special orthogonal matrix, i.e., an orthogonal
  matrix with determinant 1.

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Efficient workspace generation

• Efficient representation of workspaces
• Requirements for potential workspace
  representations
• 1. The amount of data stored at any time must be
  far less than the explicit storage of an
  intermediate workspace, which would require 2k
  N-dimensional vectors for k n.
• 2. The positional error caused by the
  representation of the workspace has to be small.
  In the ideal case it must stay below a given
  bound.
• 3. It is crucial that the workspace
  representation used supports efficient
  computation of affine transformations.
• 4. It is desirable to be able to quickly test
  whether a particular vector lies in an
  intermediate workspace.

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Efficient workspace generation

• Efficient representation of workspaces
• A density set is a computational
  structure containing the following information
• A reference point that defines a
  point of the workspace in real coordinates.
• The resolution of the discretization, i.e. block
  dimensions given by .
• The dimensions/length of the array in each
  direction, either in real (workspace)
  coordinates, , or as integers, iL,
  jL, kL, giving the numbers of pixels/voxels for
  the particular resolution.
• The density array, D, of the workspace, which is
  an N-dimensional array of integers representing
  the point density of the workspace multiplied by
  block volume.

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Efficient workspace generation

• Fig. 11. Representation of a workspace as a
  density set

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Efficient workspace generation

• Efficient representation of workspaces
• For given workspace coordinates , the
  corresponding array indices can be find as
  follows
• First are chosen to be the indices
  corresponding to the middle point x0 of the
  workspace, such that the range of possible
  indices of the array is simply

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Efficient workspace generation

• Efficient representation of workspaces
• For given workspace coordinates , the
  corresponding array indices can be find as
  follows
• The rule to calculate workspace coordinates from
  array indices is

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Efficient workspace generation

• Efficient representation of workspaces
• For given workspace coordinates , the
  corresponding array indices can be find as
  follows
• The inverse problem is solved as follows

is the floor operation
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Efficient workspace generation

• The workspace mapping algorithm
• Iteration s of the algorithm, which deals with
  module m B-s1
• 1. Estimate size and location of intermediate
  workspace Wm-1. Based on this information
• Choose the dimensions of a block in the new
  density array
• .
• Based on these dimensions determine the number of
  fields of the density array in each direction
  .
• Allocate sufficient memory for this density array
  and initialize it with zeros.
• Determine the coordinates of the middle point of
  the new workspace .
• Determine the array indices, , of the
  middle point of the new array.

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Efficient workspace generation

• The workspace mapping algorithm
• Iteration s of the algorithm, which deals with
  module m B-s1
• 2. For all configurations , ( ), apply
  the corresponding homogeneous transformation to
  the density array Dm
• For all indices for which the entry
  of the density array Dm is not zero, the
  following steps are applied
• Calculate the vector from the array
  indices .
• Calculate the coordinate vector .
• Find the array indices of x' in the
  new array.
• Increment entries in the block of the new array
  by the corresponding entry of the old array

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HRM Modeler

• Fig. 12. The main window of the modeling program

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HRM Modeler

• Main features of the program are
• VGT structure graphic modeling
• end-effector coordinates and orientation
  calculation
• manipulator configuration, allowing to reach any
  given point, computing
• position error minimization by reducing the
  distance between end-effector and given point
• manipulator workspace computing and graphic
  representation.

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HRM Modeler

• Fig. 13. The outcome of a combinatorial algorithm
  for the inverse kinematics of 30-DOF VGT
  manipulator

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HRM Modeler

• Fig. 13. The outcome of a combinatorial algorithm
  for the inverse kinematics of 30-DOF VGT
  manipulator

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HRM Modeler

• Fig. 13. The outcome of a combinatorial algorithm
  for the inverse kinematics of 30-DOF VGT
  manipulator

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HRM Modeler

• Fig. 13. The outcome of a combinatorial algorithm
  for the inverse kinematics of 30-DOF VGT
  manipulator

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HRM Modeler

• Fig. 14. The 15-DOF VGT manipulator workspace

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Summary

• Binary manipulators are promising alternative to
  traditional continuously actuated robots.
• Such advantages of binary robots, like low cost,
  light weight, high task repeatability and other,
  makes efforts in studying of binary manipulators
  very perspective and practical.
• While the latest algorithm, described in this
  report, is extremely fast and efficient, there is
  some limitation in number of binary manipulator
  modules. When this limitation is exceeded high
  computational costs become an insoluble problem
  again.
• Thus, there is still a problem of finding new
  decisions and making new algorithms in binary
  manipulator motion planning.