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Title: Joint Advanced Student School 2006


1
Joint Advanced Student School 2006
  • Binary Manipulator Motion Planning

by Vasiliy Chernonozhkin CSRI RTC,
SPbSPU Saint-Petersburg 2006
2
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.

3
Binary manipulator concept
  • The binary manipulator concept is influenced by
    several successive concepts
  • discrete actuation
  • hyper-redundant structure
  • sensorless systems.

4
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.

5
Related literature Hyper-redundancy
  • Fig. 1. Three major types of hyper-redundant
    manipulators
  • (a) continuous, (b) serial, and (c) cascaded
    platforms

6
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.

7
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.

8
Variable Geometry Truss manipulator
  • Fig. 2. All possible configurations of a 3-bit
    planar binary platform manipulator one VGT
    module

9
Variable Geometry Truss manipulator
  • Fig. 3. Configurations 110001110001110 and
    001110001110001 of 15-DOF planar VGT manipulator

10
Variable Geometry Truss manipulator
  • Fig. 4. Planar 30-DOF binary robot manipulator by
    Chirikjian and Burdick

11
Variable Geometry Truss manipulator
  • Fig. 5. Ebert-Uphoffs binary robot manipulator

12
Variable Geometry Truss manipulator
  • Fig. 6. Suthakorns discretely-actuated
    hyper-redundant robotic manipulator

13
Binary Robotic Articulated Intelligent Device
(BRAID)
  • Fig. 7. BRAIDs i-th parallel link stage
  • (a) physical parallel link stage (b)
    diagrammatic representation.

14
Binary Robotic Articulated Intelligent Device
(BRAID)
  • Fig. 8. Detent based binary joint

15
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.

16
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
17
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,
18
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

19
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

20
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
21
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
22
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).

23
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

24
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.

25
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.

26
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.

27
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.

28
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.

29
Efficient workspace generation
  • Fig. 11. Representation of a workspace as a
    density set

30
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

31
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

32
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
33
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.

34
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

35
HRM Modeler
  • Fig. 12. The main window of the modeling program

36
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.

37
HRM Modeler
  • Fig. 13. The outcome of a combinatorial algorithm
    for the inverse kinematics of 30-DOF VGT
    manipulator

38
HRM Modeler
  • Fig. 13. The outcome of a combinatorial algorithm
    for the inverse kinematics of 30-DOF VGT
    manipulator

39
HRM Modeler
  • Fig. 13. The outcome of a combinatorial algorithm
    for the inverse kinematics of 30-DOF VGT
    manipulator

40
HRM Modeler
  • Fig. 13. The outcome of a combinatorial algorithm
    for the inverse kinematics of 30-DOF VGT
    manipulator

41
HRM Modeler
  • Fig. 14. The 15-DOF VGT manipulator workspace

42
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.
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