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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; sensorless systems. hyper-redundant structure;

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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-Uphoff’s binary robot manipulator

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Variable Geometry Truss manipulator Fig. 6. Suthakorn’s discretely- actuated hyper-redundant robotic manipulator

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Binary Robotic Articulated Intelligent Device (BRAID) Fig. 7. BRAID’s 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 Fig. 9. Serial-revolute manipulator For a serial-revolute manipulator (Fig. 9) the kinematics of an individual module in a particular state are described by:

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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: Fig. 10. VGT module where: So,

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An efficient algorithm for computing the forward kinematics Pre-computation algorithm Inputs to the algorithm: q j min, q j max, the joint limits for each actuator in the manipulator in states 0 and 1 respectively, for j = 1,…,J. w i, the width of the top of module i for a truss-type manipulator, or l i, 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 C i for state j. 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. C i, for i = 1,…,B, the configuration sets for the modules in the manipulator. Outputs from the algorithm: EE pos, the position of the manipulator’s end-effector. s ee, c ee, the sin and cos of the end-effector orientation angle for the 2D case, or R ee, the rotation matrix describing end-effector orientation, in the 3D case. EE pos = 0; R ee = I; for i = 1 to B select C i EE pos = EE pos + R si b si ; R ee = R si R ee ; 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: EE des, the desired position of the manipulator’s end-effector. EE now, the current position of the manipulator’s end-effector. p now, the manipulator’s current state vector. n max, the maximum number of bits allowed to change in the manipulator’s 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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Algorithm description d min = cost(EE des, EE now ); p min = p now ; b min = 0; for i = 1 to n max for j = 1 to c = combo(B, i, j); p test = c p now ; d test = cost(EE des,fwdKin(p test )); if d test < d min then d min = d test ; p min = p test ; b min = i; end return d min, p min, b min A combinatorial method for computing the inverse kinematics

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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(2 n ), 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 ρ 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 i th 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 R N is a transformation of the form y = Ax + b, where x, y, and b are vectors in R N, and A is an arbitrary matrix in R NxN. 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 2 k 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, i L, j L, k L, 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 indicescan be find as follows: First are chosen to be the indices corresponding to the middle point x 0 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 indicescan 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 indicescan 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-s+1: 1. Estimate size and location of intermediate workspace W m-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-s+1: 2. For all configurations, (), apply the corresponding homogeneous transformation to the density array D m : For all indices for which the entry of the density array D m 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.

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