Finding Reachable Workspace of a Robotic Manipulator by Edge Detection Algorithm
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1 International Journal of Advanced Mechatronics and Robotics (IJAMR) Vol. 3, No. 2, July-December 2011; pp ; International Science Press, ISSN: Finding Reachable Workspace of a Robotic Manipulator by Edge Detection Algorithm Khushdeep Goyal 1* & Roshan Lal Virdi 1 1 Yadavindra College of Engineering, Punjabi University Guru Kashi Campus, Talwandi Sabo , India ( * Corresponding Author: khushgoyal@yahoo.com) ABSTRACT An effective algorithm for finding the full three dimensional reachable workspace for a robotic manipulator is presented in this paper. This algorithm is developed by using Denavit-Hartnberg representation and then finding Jacobian of the manipulator by row rank deficiency condition. At the edge of the workspace, the velocity vector normal to that edge becomes zero. By equating the Jacobian of this vector to zero, the singularities of that edge are detected. And these singularities are further processed to find the singular surfaces of the workspace. In the end all these singular surfaces are combined to generate the full three dimensional workspace of the robotic manipulator. In this work, MATLAB is used to find and plot all the singularities of the manipulator, which gives a full three dimensional view of the workspace. The practical examples of PUMA, and other robotic manipulator are treated with this algorithm in this paper. Keywords: Robot workspace, kinematics, Jacobian, edge surfaces, singularities 1. INTRODUCTION In recent years, some authors have developed the reachable workspace boundaries of the robotic manipulator. Exact computation of the workspace and its boundary is of significant importance because of its impact on manipulator design, manipulator placement in an environment, and manipulator dexterity. Goyal and Sethi [1] explained one method to determine the Workspace of the Robotic Manipulators, which is applicable to kinematic chains that can be modeled using the Denavit-Hartenberg representation for serial kinematic chains. This method is based upon analytical criteria for determining singular behavior of the mechanism. Bi and Lang6 have proposed a forward kinematic model for determining the workspace of tripod machine tool. The joint motions are used to calculate the workspace. Malek and Yeh [2] explained a broadly applicable formulation for representing the boundary of swept geometric entities using Jacobian rank deficiency conditions. A constraint function is defined as one entity is swept along another. Boundaries in terms of inequality constraints imposed on each entity are considered which gives rise to an ability of modeling complex solids. The intersection curves between two parametric surfaces are determined by the method explained by Malek and Yeh [3, 4]. This paper presents a method for determining the intersection curves of two intersecting parametric
2 44 International Journal of Advanced Mechatronics and Robotics surfaces using continuation methods. Cao et al [5] used a numerical method based on random probatility to generate the planar boundary curves of spatial robot in its main working plane. Then 3D shape and volume of robot workspace are generated by commercial software unigraphics. A synthesis algorithm for the workpace of three revolute robotic manipulators is proposed by Ceccarelli, with the help of formulating a set of non linear linear algebraic equations whose unknown are workspace structural coefficients. The authors Snyman and Plessis [6] gave an optimization approach to the determination of the boundaries of manipulator workspace. This numerical method consists of finding a suitable radiating point in the output coordinate space and then determining the points of intersection of a representative pencil of rays, emanating from the radiating point, with the boundary of the accessible set. This is done by application of a novel constrained optimization approach that has the considerable advantage that it may easily be automated. Kumar and Waldron [7] presented another algorithm to compute the manipulator s workspace. The intersection curves between two parametric surfaces are determined by the method explained by Malek and Yeh [8]. This paper presents a method for determining the intersection curves of two intersecting parametric surfaces using continuation methods. Malek and Yeh [9] explained a broadly applicable formulation for representing the boundary of swept geometric entities using Jacobian rank deficiency conditions. A constraint function is defined as one entity is swept along another. Boundaries in terms of inequality constraints imposed on each entity are considered which gives rise to an ability of modeling complex solids. Determination of workspace boundaries is also referred by Malek [10], Ricard et al [11], Tsai and Soni [12], and Haug et al [13]. Malek and Othmen [14] demonstrated a mathematical formulation for creation of solid models. It is shown that Denavit-Hartenberg representation method adopted from kinematics is well suited for the representation of soid models that are created as a result of multiple sweeps. 2. METHODOLOGY This algorithm is developed by using Denavit-Hartnberg representation and then finding Jacobian of the manipulator by row rank deficiency condition. At the edge of the workspace, the velocity vector normal to that edge becomes zero. By equating the Jacobian of this vector to zero, the singularities of that edge are detected. And these singularities are further processed to find the singular surfaces of the workspace. In the end all these singular surfaces are combined to generate the full three dimensional workspace of the robotic manipulator. 3. EDGE DETECTION ALGORITHM The mathematics of positioning of the wrist point for edge is readily available by the use of the Denavit-Hartenberg (D-H) representation. The D-H representation provides a systematic method for describing the relationship between adjacent links. The 4 x 4 transformation matrix describing a transformation from link (i-1) to link i for a revolute joint is :
3 Finding Reachable Workspace of a Robotic Manipulator by Edge Detection Algorithm 45 i 1 T = i cosθi cosαi sin θi sin αisin θi ai cosθi sin θi cosαi cosθi sin αi cosθi ai sin θi 0 sin α cosα i i i d where θ i, depicted in Figure 1, is the joint angle from x i-1 to the x i axis, d i is the distance from the origin of the (i-1)th coordinate frame to the intersection of the z i-1 axis with the x i, a i is the offset distance from the intersection of the z i-1 axis with the x i axis, and α i is the offset angle from the z i-1 axis to the z i axis. Figure 1: D-H Representation The homogeneous transformation matrix 0 T i that specifies the configuration of the ith frame with respect to the base coordinate system is the product of successive transformation matrices of i-1 T i, 0 T i = 0 T 1 1. T 2.. i-1 T i = i j-1 T j (1) where i is the number of degree of freedom and i-1 T i is of the form j=1 R P i 1 i 1 i i 1 i Ti where i-1 R i is the rotation matrix between frame i-1 and frame i and i-1 p i is the position vector from the origin of the i-1 frame to the ith frame. For a six axis manipulator with a spherical (2)
4 46 International Journal of Advanced Mechatronics and Robotics wrist, the homogeneous transformation matrix relating the end-effector and the wrist to the reference frame is 0 T 6 = 0 T 3. 3 T 6 (3) This is applied to determine the wrist points of robotic manipulator. One reference frame is developed at the base of the robot and the other one at the reference frame of link 3 (shown in the Figure 2). Figure 2: Notation Used in Obtaining the Accessible Output Set 0 x q = 0 R 3 3 x w + 3 p 3 (4) where the vector 0 x q describes the accessible output set of the wrist point and 3 x w is the vector describing the wrist point, resolved in the reference frame of link 3. In order to determine the edge of the wrist-accessible output set for a mechanism, it has been shown that singularities (both internal and boundary) can be computed by proper manipulation of the Jacobian of the mechanism. These singularities are substituted into the constraint equation to parameterize boundaries of the wrist accessible output set. At a certain position in space, the generalized coordinates satisfy independent holonomic kinematics constraint equations of the form ξ (θ) = 0 X q 0 R 3 3 X w 0 p 3 = 0 In addition, the generalized co-ordinates θ are subject to inequality constraints representing joint limits. θ min θ θ 1 1 1max θ min θ θ 2 2 2max (4a)
5 Finding Reachable Workspace of a Robotic Manipulator by Edge Detection Algorithm 47 θ min θ θ 3 3 3max The constraint Jacobian of the constraint function ξ(θ) of Eq.(4a) for a certain configuration is the 3x3 matrix. J () X = ξi() θ θj Equating the determinant of the Jacobian to zero, the singularities of the system are determined. To impose the inequality constraints, it is convenient to parameterize Eq. (2.8) by introducing new generalized coordinates λ i such that for an inequality constraint of the form can be parameterized as θ i min θ i θ i max (5) (6) θ i = a i + b i sin θ i (7) where a i = ( θ i max + θ imin )/2 and b i = ( θ i max -θ i min )/2 are the mid point and half range of the inequality constraint. The Jacobian with respect to the new coordinates can be written as: ξi θ j = ξ θ θλ θj λj (8) Singularities can be determined by equating the determinant of the Jacobian to zero such that J(X) = ξ θ θ λ (9) Solving J(x) and substituting the results into Eq. (7), a set of singularities µ i (i=1,...,m) is generated, where m is the total number of singularities. Substituting each singularity into the accessible output set, a set of surfaces Ψ 1 (µ i ) are parameterized such that Ψ i (µ i ) = [Ψ 1 (µ i ), Ψ 2 (µ 2 ),,Ψ m (µ m )] (10) where i = 1,...m. These surfaces determines the boundary of the dexterous workspace Robot PUMA EXPERIMENT RESULTS The Puma 560 is a six degree-of-freedom (DOF) elbow manipulator; it consists of three large joints that constitute the arm and three small joints that form the wrist. Joint rotations and their limits are shown in Figure 3.
6 48 International Journal of Advanced Mechatronics and Robotics Figure 3: PUMA DENAVIT-HARTENBERG Representation of PUMA 560 The D-H parameters of PUMA 560 are given is Table 1. Table 1 D-H Parameters Joint i a i a i d i 1 θ 1 π/2 0 d i 2 θ 2 0 a 2 d 2 3 θ 3 π/3 a θ 4 π/2 0 d 4 5 θ 5 π/ θ d 6 After applying the above described algorithm to the PUMA, the following factors and boundary edges are found out: π π The first factor, cos(λ 1 ) = 0, indicates that λ 1 =,. Thes result in two singularities 2 2 π π θ 1 = 160 0, Similarly for, cos(λ 2 ) = 0, λ 2 =,. i.e. two singularities θ2 = 125 0, π π and for, cos(λ 3 ) = 0, λ 3 =,. i.e. θ3 = 135 0,
7 Finding Reachable Workspace of a Robotic Manipulator by Edge Detection Algorithm 49 These singularities are the joint angle limits of the respective boundary edges, and from these the boundary edges are plotted in MATLAB, these edges are shown in Figure 4 to Figure 6. Figure 4: Boundary Edge Due to Singularity at θ 1 = Figure 5: Boundary Edge Due to Singularity at θ 2 =
8 50 International Journal of Advanced Mechatronics and Robotics The forth surface Ψ 4 due to singularity at θ 2 = is shown in Figure 6. The range of θ 1 and θ 3 are θ , θ Combining all Boundary Edges Figure 6: Boundary Edge Due to Singularity at θ 2 = After determining all these boundary edges surfaces, they are united together to develop the complete workspace. The complete workspace is shown in the Figure 7. Figure 7: Complete View of the Workspace
9 Finding Reachable Workspace of a Robotic Manipulator by Edge Detection Algorithm CONCLUSIONS An edge detection algorithm for determining the reachable workspace is presented in this paper. This algorithm is used to find the wrist vector of robot wrist with the help of robotics kinematics. This is processed to find the Jacobian of manipulator. At the extreme edge of the workspace, this Jacobian becomes equal to zero, because the normal velocity vector vanishes. The singularities then used to plot the extreme edges of the robot by applying constraints on the angular limits of the joints, and these edges are combined to develop the final workspace of the robot manipulator. In this work, the experimental example of PUMA robotic manipulator is treated with this edge detection algorithm. This edge detection algorithm can be extended to design manipulator workable area, manipulator placement in an environment, and manipulator dexterity. REFERENCES [1] Khushdeep Goyal, Davinder Sethi (2010), An Analytical Method to Find Workspace of a Robotic Manipulator, Journal of Mechanical Engineering, Transaction of the Mech. Eng. Div., The Institution of Engineers, Bangladesh, Vol. 41(1), pp ISSN: [2] Bi, Z. M., Lang, S.Y.T. (2009), Joint Workspace of Parallel Kinematic Machines, Journal of Robotics and Computer Integrated Manufacturing, 25(1), pp [3] Malek and Yeh (2000), On the Placement of Serial Manipulators, Proceedings of DETC00, 2000, ASME Design Engineering Technical Conferences. [4] Malek and Yeh (2000), Interior and Exterior Boundaries to the Workspace of Mechanical Manipulator, Journal of Robotics and Computer Integrated Manufacturing, Vol.16, pp [5] Cao, Y., Qi, S., Lu, K., Zang, Y. and Yang, G. (2009), An Integrated Method for Workspace Computation of Robot Manipulator, Proceedings of the International Joint conference on Computational Sciences and Optimization, pp [6] Snyman, J. A. and Plessis, L. (2000), An Optimization Approach to the Determination of the Boundaries of Manipulator Workspaces, ASME Journal of Mechanical Design, Vol.122, pp [7] Kumar, A. and Waldron, K. J. (1981), The Workspace of a Mechanical Manipulator, ASME Journal of Mechanical Design, Vol. 103, pp [8] Malek and Yeh (1997), Analytical Boundary of the Workspace for General 3-DOF Mechanisms, The International Journal of Robotics Research, Vol.16, No. 2, pp [9] Malek and Yeh (2000), Interior and Exterior Boundaries to the Workspace of Mechanical Manipulator, Journal of Robotics and Computer Integrated Manufacturing, Vol. 16, pp [10] Malek, K. (1997), On Determination of the Boundaries to the Workspace of the Manipulator, Journal of Robotics and Computer Integrated Manufacturing, Vol. 13, No.1, pp [11] Ricard, R. and Gosellin, C. M. (1998), On the Determination of the Workspace of Complex Planar Robotic Manipulators, ASME Journal of Mechanical Design, Vol. 120, pp [12] Tsai, T. and Soni, A. (1983), An Algorithm for the Workspace of a General n-r Robot, ASME Journal of Mechanical Design, Vol.105, pp [13] Haug, E. J., Luh, Chi-Mei, Adkins, F. A., and Wang, J. Y. (1996), Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces, Transactions of the ASME, Vol. 118, pp [14] Abdelmalek, K. (1999), Multiple Sweeping Using the Denavit-Hartenberg Representation Method, Journal of Computer-Aided Design, Vol. 31, pp
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