Haug, E.J., Luh, C.M., Adkins, F.A., and Wang, 1996, "Numerical algorithms for mapping boundaries of manipulator workspaces", Transactions of the ASME Journal of Mechanical Design, 118, No. 1, pp. 228-234.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....algorithm for identifying and analyzing barriers to output control of manipulators using first and second order Taylor approximations of the output in selected directions. Haug and colleagues showed that the output velocity in the direction normal to such curves and surfaces must be zero [14] and manipulator boundaries were consequently mapped. The method presented in this paper is based on recent results by this group where singular surfaces in manipulator workspaces were delineated and acceleration based crossability criteria were defined [15 19] PROBLEM DEFINITION The objective ....

Haug, E.J., Luh, C.M., Adkins, F.A., and Wang, 1996, "Numerical algorithms for mapping boundaries of manipulator workspaces", Transactions of the ASME Journal of Mechanical Design, 118, No. 1, pp. 228-234.

....( st are the independent parametric coordinates. To impose the inequality constraints in numerical form, it is convenient to parametrize the above constraints by introducing new generalized coordinates l i such that an inequality constraint of the form qqq iii min max (7) can be parametrized [20] as qab iiii = sinl (8) where ( aqq ii i = max min 2 (9) and ( bq q iii = max min 2 (10) are the mid point and half range of the inequality constraint. In this paper, the computation scheme of the branches of the intersection curve is divided into two phases: 1) finding the starting ....

....quadratic convergence properties [19] The constraint equation (equation 11) has more rows than columns and the constraint sub Jacobian H q has more columns than rows, thus equation (13) has multiple solutions. One solution to this type of problems is to find the solution Dqwith minimum norm [20] and [21] i.e. to solve the minimization problem min 1 2 DDqq T (14) Hq H q D= 15) Using the Lagrange multiplier approach that appends a multiplier vector times the equations to be satisfied to the function that is to be minimized, define YD D 1 2 qq HzH q TT l) 16) As a ....

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Haug, E.J.; Luh, C.M.; Adkins, F.; Wang, J.Y. (1994) Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces. Proceedings of the 24th ASME Mechansims Conference.

....an improved understanding of the functionality of robotic arms. In the past, researchers in the field of robotics have studied functionality in terms of spaces. Reachable workspaces have been addressed by Roth (1975) Tsai and Soni (1981) Yang and Lee (1983) Gosselin (1990) Emiris (1993) and Haug et al. 1994). Dexterous workspaces have been studied by Kumar and Waldron (1980) Yang and Lai (1985) and Wang and Wu (1993) Workspaces, however, do not provide adequate information about the functionality of the robot at specific targets. For example, given a required dexterous workspace using the current ....

....to assemble a product in one cell while performing another task in another cell. The questions of where to locate the two cells with respect to the robot for maximum functionality may arise. To set the background for this paper, definitions of relevant terms are stated. Accessible Output Set (Haug et al. 1994) The region of space that can be reached by a point on the manipulator, for all combinations of joint coordinates. Dexterous Workspace (Kumar and Waldron, 1981) A subspace of the accessible output set within which a vector on the end effector may assume all orientations. Because of joint limits ....

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Haug, E.J., et al., 1994, "Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces," Proceedings of the 23rd ASME Mechanisms Conference, Minneapolis, MN.

.... 4 4 matrix M is called the blending function, and the parameter matrices are [ Uuuu= 1 23 , and [ Vvvv= 1 23 Parametrization of Inequality Constraints: The parametric form of an inequality given by uuu iii min max (6) is obtained by introducing a new variable l and the transformation [14] uab iii = sinl (7) where ( auu iii = max min 2 and ( buu iii = max min 2 are the mid point and half range of (6) For further discussion, entities are classified into two types: Type I: Ruled, curved, and surface patches Type II: Surfaces having closed boundary curves. Curves ....

Haug, E.J., Luh, C.M., Adkins, F., and Wang, J.Y., 1994, "Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces," Proceedings of the 24th ASME Mechansims Conference.

.... min max , are parameterized into an equality using the following transformation such that new generalized coordinates l i are introduced qab iiii = sinl (19) 6 where ( aqq ii i = max min 2 and ( bq q iii = max min 2 are the mid point and half range of the inequality constraint (Haug, et al. 1994). The constraint function is then written in terms of the extended vector s = ll l 12 . n T such that xsq= ##( l i (20) Differentiating with respect to time and using the chain rule, the velocity of the end effector is ## xqs qs = # (21) where q s = l q ij and # ss= ddt. On a ....

....either in the positive or negative direction of n, will be in the same sense of signs of s and K . 10 Example 1: A Planar 3 bar Linkage This example is presented to demonstrate the implementation of the presented formulation and to validate the results in comparison with those reported by Haug, et al. 1994) for the planar three bar linkage shown in Fig. 2 with the following imposed limits p p 33 q i ; i = 13. x y 4 2 1 q 1 q 2 q 3 Figure 2. A planar three DOF manipulator The position vector of a point on the end effector is defined by #( cos cos( cos( sin sin( sin( q = ....

Haug, E. J. Luh, C. M. Adkins, F. A. and Wang, J. Y. 1994. Numerical algorithms for mapping boundaries of manipulator workspaces. Advances in Design Automation, ASME DE 69(2):447-459.

....to determining the workspace was formulated and solved by Kumar and Waldron [5] by tracing boundary surfaces of a workspace. Tsai and Soni [6] studied accessible regions of planar manipulators, while Gupta and Roth [7] studied the effect of hand size on workspace analysis. Recently, Haug et al. [8] formulated numerical criteria to find the workspace (so called accessible output set) of a general multi degree of freedom system via the study of a row rank deficiency of its Jacobian. The algorithm computes tangent vectors at bifurcation points of continuation curves that define the boundary of ....

Haug, E.J., Adkins, R., and Luh, C.M., 1994, "Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces," Proceedings of the 23rd ASME Mechanisms Conference, Minneapolis, MN.

....in the ability to determine holes and voids in the accessible output set. The same example treated by Cecarelli will be addressed here for validation purposes. Other works that have dealt with manipulator workspace are reported by Kumar (1985) and Emiris (1993) and Zhang et al. 1996) Recently, Haug et al. 1996) formulated numerical criteria to find the workspace (called the accessible output set) of a general multi degree of freedom system using a continuation method to trace boundary curves suitable for the study of both open and closed loop manipulators. The initial criteria for this computational ....

....and to analytically represent it. At a specified position in space, Eq. 1 can be written as a constraint function #( qqqq0= fxgyhz T (4) Joint limits imposed in terms of inequality constraints in the form of qqq i L ii U , where in= 1, are transformed into equations (Haug et al. 1996) by introducing a new set of generalized coordinates # = ll 1 n T such that qqq qq ii L i U i U i L i = sin22 PUPU l in= 1, 5) These generalized coordinates l i are called slack variables in the field of optimization. In order to include the effect of joint limits, it ....

[Article contains additional citation context not shown here]

Haug, E.J., Luh, C.M., Adkins, F.A., and Wang, J.Y., 1996, "Numerical algorithms for mapping boundaries of manipulator workspaces," ASME Journal of Mechanical Design, 118, pp. 228-234.

....T ; w T ; v T i T , the boundary of the solution set in the u space is sought. Such collections of equations arise in analysis of the kinematics of mechanical systems, and computation of the u space boundary of the solution set arises in workspace analysis. In Haug, Luh, Adkins, and Wang [6], an analytic necessary condition for being on the u space boundary of the solution set of Phi(q) 0 is formulated, and when dim u 3 a practical approach to mapping the boundary is to define a family of planes (or hyperplanes if dim u 3 ) that intersect the boundary in one dimensional sets ....

....coordinates necessary to describe the mechanism, called the intermediate coordinates. The solution set of Eq. 1 in the u space is A = fu 2 nu : Phi(u; w; v) 0 for some w and vg (2) A necessary condition for u 2 A to be on the boundary A is that Phi w;v (q) be row rank deficient [6]. A necessary and sufficient condition for Phi w;v (q) to be row rank deficient is that there is some 2 m , 6= 0, such that Phi T w;v (q) 0 (3) Thus a necessary condition for describing the boundary of the solution to Eq. 1 in the u space is A ae S = fu 2 A : Phi T w;v = 0; ....

E.J. Haug, C.-M. Luh, F.A. Adkins, and J.Y Wang. Numerical algorithms for mapping boundaries of manipulator workspaces. Journal of Mechanical Design, 118(2):228--234, 1996.

....program that uses automatic differentiation to evaluate derivatives needed (Haug, Luh, Adkins, and Wang, 1994; Griewank, et al., 1994) Two planar examples are analyzed in this section to illustrate use of the method. A spatial Stewart platform is analyzed in detail in a companion paper (Luh, Haug, Qiu, 1994). In the figures presented here, numbered circles are bifurcation points that identify distinct segments of A that they separate. Solid curves are segments of A across which output is restricted when the manipulator is in the associated critical configuration. Dotted curves are segments of A ....

Haug, E.J., Luh, C.M. Adkins, F.A. and Wang, J.Y., 1994, Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces, Advances in Design Automation, ASME DE 69(2):447-459.

....between input and output coordinates has likewise been used to characterize singular surfaces of manipulators (see [5] and [9] Numerical methods for mapping boundaries of workspaces of 2 E.J. HAUG, F.A. ADKINS, AND C.M. LUH mechanisms and manipulators have recently been presented (see [3], 4] and [11] These methods are summarized here, with accompanying numerical methods. 2. Analytical Conditions For Working Domains And Boundaries In order to define working domains in bodies that move with an underlying mechanism, whose configuration is defined by a generalized coordinate ....

....domain D is, therefore, D ae n u 2 D : Psi T z (u; z) 0; T = 1; Psi (u; z) 0; for some z and o (4) DOMAINS OF OPERATION AND INTERFERENCE 3 3. Numerical Methods For Mapping Boundaries A brief summary of calculations involved in mapping one dimensional generators for D [3] is given here. Finding an Interior Point of D: Simulations can often be carried out to find a configuration of the mechanism and points on working bodies satisfying criteria of Eq. 1, yielding a point u 0 interior to D. Alternatively, an iterative method can be used to find such a point, as ....

[Article contains additional citation context not shown here]

Haug, E.J., C.M. Luh, F.A. Adkins, and J.Y Wang. 1994. Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces. Advances in Design Automation ASME DE-69:2:447-459.

....( u q u f r q A q f r q A q f q 0 , 2.5) Thus, the domain of interference I D 1 2 for Problem 1 is = I D 1 2 1 1 1 u u q 0 q : YY for some and (2. 6) If a point u 1 in I D 1 2 is on its boundary, denoted I D 1 2 , then it is necessary (Haug, Luh, Adkins, and Wang, 1994) that the sub Jacobian matrix YY FF FF FF 1 1 1 q q u q 0 f 0 r q A q f r q A q f A q f A q f q q q , 1 2 2 2 2 1 1 1 1 1 1 1 2 2 2 1 1 2 1 2 q ....

...., YY xx xx xx YY xx (2. 13) 3 NUMERICAL METHODS FOR MAPPING BOUNDARIES OF DOMAINS OF INTERFERENCE While analytical criteria for points on the boundary of a domain I of interference, denoted I , differ in detail from that for points on the boundary of an accessible output set (Haug, Luh, Adkins, and Wang, 1994), the nature of the conditions is identical. Therefore, numerical methods developed for mapping boundaries of accessible output sets can be used to map boundaries of domains of interference. A brief summary of calculations involved, following the development of Haug, Luh, Adkins, and Wang (1994) ....

[Article contains additional citation context not shown here]

Haug, E.J., Luh, C.M., Adkins, F.A., and Wang, J.Y., 1994, Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces, Advances in Design Automation ASME DE-69:2:447-459.

No context found.

Haug, E.J., Luh, C.M., Adkins, F.A., and Wang, J.Y., 1996, "Numerical algorithms for mapping boundaries of manipulator workspaces," ASME J. Mech. Design, 118, pp. 228-234.

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Haug, E.J., et al. Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces Proceedings of the 23rd ASME Mechanisms Conference, 1994, Minneapolis, MN.

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Haug, E.J., Luh, C.M., Adkins, F.A., and Wang, J.Y., 1996, "Numerical algorithms for mapping boundaries of manipulator workspaces," ASME Journal of Mechanical Design,

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