Kinematics of pantograph masts

Abstract

This paper deals with the kinematics of pantographs masts, which have widespread use as deployable structures in space. They are overconstrained mechanisms with degree-of-freedom (d.o.f), evaluated by the Grubler-Kutzbach formula, as less than one. In this paper, we present a numerical algorithm to evaluate the d.o.f of pantograph masts by obtaining the null-space of a constraint Jacobian matrix. In the process we obtain redundant joints and links in the masts. We also present a method based on symbolic computation to obtain the closed-form kinematic equations of triangular and box shaped pantograph masts and obtain the various configurations such masts can attain during deployment. IntroductionDeployable masts used in space are prefabricated structures that can be transformed from a closed compact configuration to a predetermined expanded form in which they are stable and can carry loads.Deployable / foldable mast have one or more internal mechanisms [1-2] and their d.o.f as evaluated by the Grubler-Kutzbach criterion often turns out to be less than 1 [3]. In this paper we study the kinematics of deployable masts made up of pantograph mechanisms or scissor like element (SLE). An SLE in two dimensional form has straight rods of equal length connected by pivots in the middle. The assembly has one d.o.f and the basic model can be folded and deployed freely. Three dimensional masts are created with SLE in such a way that they form a structural unit which in plan view is a normal polygon with each side being an SLE. The polygon can be equilateral triangle, square or normal n-sided polygon. By combining several of these normal polygon shaped units, structures of various geometric configurations can be created [4]. Active cables control the deployment and prestress the pantograph and passive cables are pre-tensioned in the fully deployed configuration. These cables have the function of increasing the stiffness when fully deployed. The whole system deploys synchronously.The kinematics of pantograph masts can be studied by use of relative coordinates [5], reference point coordinates (as in the software package ADAMS) or Cartesian coordinates [6]. In this paper Cartesian coordinates, also called natural / basic coordinates, have been used. This method uses the constant distance condition for two or more basic points of the same link. Using unitary vectors the method can be extended to spatial mechanisms. The main advantage of using Cartesian coordinates is that the constraint equations are quadratic as opposed to transcendental equations, and the number of variables tends to be (on average) in between relative coordinates and reference point coordinates. In an earlier study, the foldability equations were formulated for SLEs based on geometric approach [7].

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