From: Ilian Bonev Newsgroups: sci.math.research Subject: Overdetermined system of quadratics Date: Mon, 20 Apr 1998 16:38:20 +0900 A six-degree-of-freedom parallel robot consists (in its most common configuration) of a rigid body (the mobile platform) connected to a fixed base via six legs of variable length (linear actuators). The direct kinematics problem deals with finding the position and orientation of the mobile platform given the values of the six leg lengths. Or mathematically, we want to find the position and orientation in space of a body for which 6 distinct points are bound to lie on six spheres with known radii and centers. The direct kinematics problem leads to the solution of a system of nonlinear equations whose form depends on the method selected for representing the rigid body motion of the mobile platform. Let the 9 coordinates of 3 non-collinear points from the mobile platform be chosen to represent its position and orientation in space. Then, the direct kinematics problem reduces to the solution of 6 quadratic equations representing the variable leg lengths, plus 3 quadratic equations representing the distance constraints on three points of the mobile platform. In the general case no closed-form solution can be found for this system. Further, this system has many solutions (e.g. 40). By using three extra linear displacement sensors, one can measure the distances between the three points on the mobile platform used for representing the rigid body motion of the mobile platform and respectively three points fixed on the base. This data is represented by 3 new quadratic equations. They can be used to linearize the first 6 equations and to express 6 of the unknowns as linear functions of the other 3 (chosen as dependent) coordinates. When these expressions are substituted in the last 6 quadratic equations, an overdetermined system of 6 quadratic equations in three unknowns is obtained. Numerical analysis have shown for a particular configuration, that this system has only one solution. My question is whether there is any apparatus dealing with solving a system of 6 quadratic equations in three unknowns. The system may be decoupled in two subsystems, each of which has maximum 8 solutions. What is the maximum number of common solutions that these two systems of linearly independent quadratic equations can have? (4?) Under what conditions they could have always one and only one common solution? Ilian Bonev e-mail: ilian@geguri.kjist.ac.kr ============================================================================== From: Dave Rusin Date: Mon, 27 Apr 1998 16:18:59 -0500 (CDT) To: ilian@geguri.kjist.ac.kr Subject: Re: Overdetermined system of quadratics Newsgroups: sci.math.research In article <353AFB6C.716F36B0@geguri.kjist.ac.kr> you write: >A six-degree-of-freedom parallel robot consists (in its most common >configuration) of a rigid body (the mobile platform) connected to a >fixed base via six legs of variable length (linear actuators). The >direct kinematics problem deals with finding the position and >orientation of the mobile platform given the values of the six leg >lengths. Or mathematically, we want to find the position and orientation >in space of a body for which 6 distinct points are bound to lie on six >spheres with known radii and centers. I understand this is called a Stewart Platform. >Further, this system has many solutions (e.g. 40). Is this count a well-known result? I would like to see a pointer. >By using three extra linear displacement sensors, If you treat the new data as reliable, this should make the problem nearly linear, right? Given the distances from P to two fixed points we constrain P to lie in a certain plane perpendicular to the line between those fixed points; if we have the distance to a third fixed point, P is constrained to a line. Any one of the three distances may then be used to deduce P (with a solution set of cardinality at most 2). On the other hand, if you treat the data as noisy, there may be no solutions consistent with all data. I suppose one can estimate the location by using a least-squares criterion: select the point P=(P1,P2,P3) in R^9 so as to minimize sum-over-sensors-S_i sum_{j=1,2,3} (dist(S_i,P_j)-nominal_distance)^2 Rather than do this directly I suppose I would rely on existing numerical-optimization software to hunt for optimal coordinates for P. I have no real reason to think that this criterion for goodness-of-fit is any better than another criterion, however. I would be interest to hear of any other responses you get. dave ============================================================================== Date: Wed, 29 Apr 1998 20:20:31 +0900 To: Dave Rusin From: Ilian Bonev Subject: Re: Overdetermined system of quadratics Dear Prof. Rusin, Thank you for your reply. >I understand this is called a Stewart Platform. The general term is a parallel robot. Stewart Platform stands for the same thing though the "real" Stewart Platform is only a particular case. Other terms you may encounter for 6 degrees-of- freedom parallel robots are hexapods, platform manipulators, or closed-loop mechanisms. >Is this count a well-known result? I would like to see a pointer. There are several articles presenting this result. Personally, I like the one by Prof. J.C. Faug=E8re and D. Lazard, "The Combinatorial Classes of Parallel Manipulators", Mech. and Machine Theory, 30(6), pp. 765-776. It gives the upper bounds on the number of solutions in the cases of different leg connectivities. The results are obtained using Grobner basis in a package Gb developed by the first author. In fact the direct kinematic model of a general Stewart platform may be reduced to the solution of a system of 6 quadratics and 3 linear equations in 9 unknowns. According to Bezout's theorem, this system can have up to 64 solutions. A study of the solutions at infinity (i.e. the solutions obtained by homogenizing the equations and looking at the solutions for which the homogenizing variable is 0) shows that they are of dimension 0 and of degree 24... However, no more that 16 real solutions have been found up to now so the question for the exact bound still exists. As for your suggestions, you are right that my extra data is "noisy", or more precisely the accuracy of the extra sensors is worse than that of the sensors measuring the leg lengths. However, for the time being I assume that all data is reliable. >I would be interest to hear of any other responses you get. I will inform you when some results appear. If you are interested in other information concerning parallel manipulators, do not hesitate to ask me. Best regards Ilian Bonev assistant researcher K-JIST, Mechatronics Dpt., VDC1 lab 572 Sangam-dong, Kwangsan-ku, Kwangju, 506-712 Korea Tel: +82 62 970 2417 URL: ============================================================================== Date: Thu, 30 Apr 1998 13:01:13 +0900 To: Dave Rusin From: Ilian Bonev Subject: Re: Overdetermined system of quadratics Dear Prof. Rusin, There has been much development in area of parallel manipulators. There are about 500 to 1000 articles published in that field. Indeed, the kinematics of a parallel robot or more precisely its kinematic geometry is a pure mathematical problem. For example the direct kinematics problem can be regarded as the problem of finding all possible positions of a body in space for which 6 points are constrained to be on 6 given spheres. The workspace of a parallel manipulator with 6 legs can only be represented in a 6 dimensional space. For some 3D varieties of the whole workspace (e.g. the 3D workspace when the orientation is fixed) the fastest algorithms are the geometric ones. They proceed with finding the intersection of simple geometries like spheres, cylinders, cones etc., which calls for computational geometry algorithms. Finally, as in my problem, the solution of a system of three quadratics in three unknowns is the same as finding the intersection points of three quadratic surfaces. 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