Hindawi Modelling and Simulation in Engineering Volume 2020, Article ID 8523514, 13 pages https://doi.org/10.1155/2020/8523514 Research Article Modeling of a Tethered Testbed for a VTVL Vehicle Marco Sagliano ,1 Stephan Theil,1 Johanna Schramm,2 and Matthias Schwarzwald2 1German Aerospace Center, Robert Hooke Straße 7, 28359 Bremen, Germany 2University of Bremen, Bibliothekstraße 1, 28359 Bremen, Germany Correspondence should be addressed to Marco Sagliano; [email protected] Received 8 January 2020; Revised 20 May 2020; Accepted 3 September 2020; Published 1 October 2020 Academic Editor: Mohamed B. Trabia Copyright © 2020 Marco Sagliano et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper proposes an algorithm for modeling a three-dimensional tethered environment for testing vertical-take off, vertical landing vehicles. The method is able to take several geometrical configurations into account and combines the classical catenary model with the elasticity theory to predict the forces acting on the lander in quasistatic conditions, i.e., in conditions of hovering, where the motion of the vehicle is reduced. Numerical results confirm that the method is potentially able to provide real-time solutions, which can be included as feedforward contributions in the design of tethered experiments. 1. Introduction test the hovering capability, one of the first milestones to be achieved towards the development of the full free-flight capa- Recent and future missions involve a precise descent and bility. On the other side, a tethered configuration provides a landing in addition to the ascent phase to reach the target safe environment for the vehicle. It is mitigating the effect orbit. This can be on the one hand the powered descent of failures until the technology under development is mature and landing of a reusable first stage of a launch vehicle as it enough to allow a reliable free flight. Moreover, the use of was demonstrated several times by SpaceX with its Falcon 9 tethered configurations allows for making the test facility a launch system. On the other hand, precise descent and land- protective area for the team of engineers and researchers. ing has been applied to planetary missions and is foreseen for Thus, they can safely and closely track the progress in the many more future missions to Mars and to the Moon. The development of the vehicle. development of guidance, navigation, and control (GNC) This paper addresses the problem of modeling the teth- techniques for these applications remains a challenging task ered testbed during the hovering experiments of the VTVL although several missions have been already successfully vehicle EAGLE (Environment for Autonomous GNC Land- completed. Several ideas to support and accelerate the ing Experiments) developed by the German Aerospace Cen- GNC development using demonstrators have been con- ter (DLR) [9–11]. The modeling focuses on the tethering ceived in the past, for example NASA’s Morpheus lander effects on the vehicle in hovering conditions. It can poten- [1, 2] or the HOMER demonstrator of Airbus Defense and tially be extended to any vehicle where it is needed to include Space [3, 4]. In these and other developments of space sys- the forces generated by the tethers in the design of the tems, a wide variety of tethered experimental setups has been controller. created [1, 2, 5–8]. The modeling starts with the use of the catenary, a well- For vertical-take off, vertical landing (VTVL) vehicles, known concept in the field of structures and mechanics the use of tethered solutions has pros and cons. One of the [12], which physical meaning is the ideal shape that a hang- main drawbacks is the impact of the tethers on the flight ing rope or chain has when subject to its own weight while characteristics of the vehicle. They introduce effects which having its endpoints constrained in two points in the space. do not exist in the final free-flight scenario. A further down- The catenary is slightly different from the parabolic profile, side is that the allowed flight envelope is usually quite small which is the shape that intuition would (erroneously) sug- due to limited tether length. Nevertheless, it is sufficient to gest. The catenary concept is widely used and can be 2 Modelling and Simulation in Engineering �(x , y ) b x y extended to multibody net structures [13]. For instance, it � � ( b, b) can be used for the modeling of railway overheads or power lines as in [14] or [15]. In these applications, however, the shape is the most important aspect and is therefore the main factor to be studied. As a consequence, the forces at the sus- pension points are not so emphasized. In [16], the catenary curve and the corresponding equi- librium of forces are discussed in the frame of the develop- ment of a controller for an unmanned vehicle. However, this work focuses on the design of a controller which mini- mizes the tension in the ropes. Richardson focused instead y on the entanglement detection of swarms of robot in urban environments [17]. This paper focuses on the modeling and the analysis of x complex tethered configurations including hanging ropes o x y attached to the vehicle while hovering. In addition, elasticity ( 0, 0) is included in the model. The purpose is to know the forces which act on the VTVL vehicle and can be included as Figure 1: Example of catenary curve. Point a and b are the feed-forward contribution in the design of the EAGLE con- suspension points. Point o is the vertex. troller. We propose an algorithm implementing the afore- mentioned theories of catenary and elasticity that computes the forces acting on EAGLE in an iterative way. y, with a horizontal x-coordinate and a vertical y-coordi- The paper is structured as follows. In Section 2.1, the nate. An example is depicted in Figure 1. modeling of a hanging rope and the related concept of cate- We can derive three equations for three unknown param- r x y nary are introduced. The corresponding forces at the suspen- eters , 0, and 0. We know the position of the two suspen- sion points with and without elasticity are computed. sion points, defined as a = ðxa, yaÞ and b = ðxb, ybÞ. Numerical validations of the proposed modeling are carried Moreover, the length of the rope l is known. out in Section 2.2. In Section 3.1, the specific geometrical test x − x setup for EAGLE is described. The setup includes different a 0 hxðÞa = ya = r cosh + y rope materials and a configuration of three ropes which are r 0, x − x linked to the VTVL vehicle in a triangular configuration. b 0 hxðÞb = y = r cosh + y , In Section 3.2, the iterative solving procedure is illus- b r 0 ð2Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi trated. It is based on a MATLAB implementation, but it ðx ðx b dh 2 b x − x can be transferred to any other software. Section 4 shows l dx 0 dx: = 1+ dx = cosh r some numerical results obtained with the proposed algo- xa xa rithm. In general, every single point within the test area can be tested. However, for a better characterization of the sce- With this equations, we get the following system: nario involving the EAGLE motion, two specific hovering x − x paths are analyzed, and the accuracy of the computed solu- y y − r a 0 ð Þ tions is discussed. Finally, in Section 5, we draw some conclu- 0 = a cosh r , 3 sions on the work done. xb − x xa − x y = r cosh 0 − r cosh 0 + y ð4Þ 2. Rope Modeling b r r a, xb − x xa − x 2.1. A Brief Review. In this section, a brief introduction about l = r sinh 0 − r sinh 0 : ð5Þ the modeling of a static rope is given, and the computation r r procedure of the forces at the suspension points is explained. For a simplification of the system of equations, the cate- The function describing the shape of a hanging rope sub- nary curve is shifted. The suspension points are moved, such ject to a constant gravity is called catenary [18]. The catenary fi fi that the rst suspension point is on the origin. Thus, a local curve is de ned as coordinate system positioned at the first suspension point is used. x − x Now, Equations (4) and (5) can be used to obtain a for- hxðÞ r 0 y ð Þ = cosh + 0, 1 x r mula for 0: y − y x x x −r b a b + a : ð Þ r x x 0 = artanh + 6 where is the radius in the vertex, 0 is the -coordinate of l 2 y y ff the vertex and 0 is the -o set of the vertex. x y y A two-dimensional - reference frame is used for the Here, 0 is eliminated. Therefore, Equations (5) and (6) – x description of the motion. The gravity is directed towards become a system of two equations for two unknowns 0 Modelling and Simulation in Engineering 3 r y F and , whereas 0 can be computed by evaluating Equation b (3) at point a. x − x y y − r a 0 : ð Þ 0 = a cosh r 7 We can rewrite the condition on the length of the rope represented by Equation (6), which gives hi x − x 2 F b a 2 2 a 2r sinh − l + ðÞyb − ya =0: ð8Þ 2r This is a nonlinear equation, only depending on r, which y cannot be solved analytically.