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.
Recent and future missions involve a precise descent and landing in addition to the ascent phase to reach the target orbit. This can be on the one hand the powered descent and landing of a reusable first stage of a launch vehicle as it was demonstrated several times by SpaceX with its Falcon 9 launch system. On the other hand, precise descent and landing has been applied to planetary missions and is foreseen for many more future missions to Mars and to the Moon. The development of guidance, navigation, and control (GNC) techniques for these applications remains a challenging task although several missions have been already successfully completed. Several ideas to support and accelerate the GNC development using demonstrators have been conceived in the past, for example NASA’s Morpheus lander [
For vertical-take off, vertical landing (VTVL) vehicles, the use of tethered solutions has pros and cons. One of the main drawbacks is the impact of the tethers on the flight characteristics of the vehicle. They introduce effects which do not exist in the final free-flight scenario. A further downside is that the allowed flight envelope is usually quite small due to limited tether length. Nevertheless, it is sufficient to test the hovering capability, one of the first milestones to be achieved towards the development of the full free-flight capability. On the other side, a tethered configuration provides a safe environment for the vehicle. It is mitigating the effect of failures until the technology under development is mature enough to allow a reliable free flight. Moreover, the use of tethered configurations allows for making the test facility a protective area for the team of engineers and researchers. Thus, they can safely and closely track the progress in the development of the vehicle.
This paper addresses the problem of modeling the tethered testbed during the hovering experiments of the VTVL vehicle EAGLE (Environment for Autonomous GNC Landing Experiments) developed by the German Aerospace Center (DLR) [
The modeling starts with the use of the catenary, a well-known concept in the field of structures and mechanics [
In [
This paper focuses on the modeling and the analysis of complex tethered configurations including hanging ropes attached to the vehicle while hovering. In addition, elasticity is included in the model. The purpose is to know the forces which act on the VTVL vehicle and can be included as feed-forward contribution in the design of the EAGLE controller. We propose an algorithm implementing the aforementioned theories of catenary and elasticity that computes the forces acting on EAGLE in an iterative way.
The paper is structured as follows. In Section
In Section
In this section, a brief introduction about the modeling of a static rope is given, and the computation procedure of the forces at the suspension points is explained.
The function describing the shape of a hanging rope subject to a constant gravity is called
A two-dimensional
Example of catenary curve. Point
We can derive three equations for three unknown parameters
With this equations, we get the following system:
For a simplification of the system of equations, the catenary curve is shifted. The suspension points are moved, such that the first suspension point is on the origin. Thus, a local coordinate system positioned at the first suspension point is used.
Now, Equations (
Here,
We can rewrite the condition on the length of the rope represented by Equation (
This is a nonlinear equation, only depending on
Different methods can be employed. Classical Newton methods for this specific problem can experience numerical issues. This happens because the slope of the function rapidly tends to infinity. A better alternative is the Halley-method [
The Halley method works with the following iterative scheme:
Therefore, the derivatives of the functions
A reasonable initial guess for starting the algorithm is required. In this case, we have to look at the double dependency of the radius
To get the forces acting on the suspension points, the force equilibrium and the knowledge about the orientation of the forces are used. An example of configuration where we can benefit from this knowledge is depicted in Figure
Balance of forces acting on the rope.
So far, we considered two-dimensional representations of the problem. The general three-dimensional problem can always be reduced to the two-dimensional version as pictured in Figure
Three-dimensional catenary. The suspension points are
Therefore, the whole catenary is shifted by
After this transformation, all the points of the catenary lie in the
Once the forces are computed according to Equations (
After the rotation, we have to back transform in terms of translation. This means that also the vertex of the catenary needs to be shifted.
The catenary is now described by
The back-transformation for the forces is the same as with the displacements of the vertex.
In the following, some simple implementation results of forces at the suspension points of a simple hanging rope without elasticity are illustrated. For a 2.90 m rope with a mass of
Different positions of vertex. (a) The vertex can be directly seen at a hanging rope with suspension points (0, 4) and (2, 5) and a length of 2.90 m. (b) The vertex can be positioned outside of the suspension points (b) with suspension points (0, 3) and (2, 5) and a length of 2.90 m.
It can be seen that the
If the left suspension point is changed to
In this case, we observe that the sign of the
In both cases, the length of the rope is longer than the distance between the suspension points. But if the length is shorter, the rope has to be stretched. Thus, it is necessary to consider the elasticity. Under the assumption that the rope is linear-elastic, the behavior of the rope can be modeled by using Hooke’s law:
There are two cases which has to be described separately:
The length of the rope is at least the distance between the two suspension points The length of the rope is shorter than the distance between the two suspension points
In the first case, the rope has to be stretched by its own weight. To solve this case, we implement the following procedure: first the forces at the suspension points are computed for the non-stretched rope. These forces are used to calculate the corresponding stretch with Hooke’s law through Equation (
For the second case, the superposition principle is used. A massless rope, stretched exactly to the length of the distance between the suspension points, is overlaid with a rope which is stretched by its own weight. The force, which is needed to stretch the rope until it is as long as the difference between the suspension points, is calculated with Hooke’s law through Equation (
As example consider the case represented by Equation (
The decrease in the horizontal components of the forces shows that the rope sags more than in the previous case. The vertical component of the force at the lower suspension point is larger, while the same component on the higher suspension point becomes smaller. In other words, the elasticity of the rope tends to slightly reduce the force gap at the suspension nodes. In the above second case as represented in Equation (
The more important case is the one where the distance between the suspension points is larger than the length of the rope. For instance, if the length is changed to 2 m, the weight is
As expected, these forces are large compared to the forces in the first two cases. This is because the stretch generates a significant additional force, which is significant larger than the forces discussed in (
In this section, we want to check our results to see whether they fulfill the forces equilibrium, described by Equations (
In the above first case, Equation (
Thus, the catenary function with the given parameters connects the suspension points and the rope is as long as expected.
The same can be done for the above second case (
Thus, the catenary function with the given parameters connects the suspension points and the rope is as long as expected.
In the above third case (
Thus, the catenary function with the given parameters connects the suspension points. As expected, the rope is longer than the initial length caused by stretch.
Finally for the above fourth case (
Thus, the catenary function with the given parameters connects the suspension points. As expected, the rope is significantly longer than the initial length caused by stretch. It is even a bit longer than the direct connection between the suspension points, which is
First, the given experimental set-up is introduced followed by the explanation of the modeling procedure for this set-up.
Tethered configurations have been used for testing several vehicles and spacecraft in a secured flight environment. In this section, DLR’s experimental facility NEST (Nest Environment for Suspended flight Tests) for the vehicle EAGLE, depicted in Figure
DLR’s experimental facility NEST (NEST Environment for Suspended flight Tests) for the vehicle EAGLE as an application for the modeling approach. The vehicle EAGLE is connected to the three ropes (visible in the red circles), which are attached to the top of the support structure. Three more ropes are used to limit the altitude. They are attached to the lower part of the support structure.
The legs of the VTVL vehicle EAGLE are connected to three ropes, one for each of its three legs. These ropes are led over a turning wheel at the top of a traverse pole with height of
Top view on NEST experimental setup. The traverse poles are arranged as a triangle with a distance of 6 m. The lander is fixed with three tethers which are led over turning wheels at the top part of a traverse pole to the bottom inside the pole. Here, the lander rests at the middle of the triangle.
The coordinate system has its origin at the base plate in the middle of the traverse pole. The
Coordinate system in NEST. The coordinate system’s origin is the base plate in the middle of the traverse poles.
We will limit the analysis to only one rope which is fixed at the lander and led over a turning wheel through a traverse pole. The procedure can be repeated for the other two ropes. Figure
NEST configuration for each of the ropes. The main part of the rope is a very stiff rope. At the bottom part, an expander rope (less stiff rope) with length of 1.50m is added, therefore pictured as a spring. At the top part of the traverse pole, the turning wheel can be seen. For a better handling in the solving procedure, the rope can be split into a lander and a traverse part.
Now that the scenario has been defined, we can see how to solve the problem. The proposed iterative method is the subject of the next section.
In this part, the solving procedure is explained. First, the cutting clear technique is described and applied to the NEST facility. Two resulting systems are calculated individually considering the elasticity of the ropes.
It is assumed that the elastic behavior is linear, and Hooke’s law can be applied. In addition, it is assumed that the expander rope does not get stretched above the turning wheel, which is geometrically represented only by a point (In the hypothetical case of the lander position outside of the triangular flight area the ropes are led over the turning wheels anyway, but this case is clearly excluded from this analysis, as we assume that the controller is able to keep EAGLE within the prescribed area).
A further assumption is that the rope is treated as elastostatic. This means that the catenary curve is not depending on time, and only the hanging of the rope with its forces at the suspension points is modeled. This assumption is justified by the small motion characterizing EAGLE during its hovering. Finally, as said, the rope is frictionless at the turning wheel.
To model the forces at the rope as it is seen at Figure
Free-body principle for a rope in NEST experimental setup. Intersect the rope at the turning wheel leads to virtual forces
Whole System
Free-body diagram traverse system
Free-body diagram lander system
If the lengths of the ropes in the single systems are known, it will be possible to calculate the forces in the single systems. The problem is to choose the rope lengths in the single systems such that the forces
Choose a specific starting shift; Calculate the forces Fb and Ft in the single systems; while if Change the split such that lander system gets a longer section of rope; else Change the split such that traverse system gets a longer section of rope; end end
At the beginning, the stiff rope is divided into a lander-side part and a traverse-side part. A specific offset is chosen as a starting shift.
Then, the forces in the single systems
To calculate the single systems with given rope lengths, it is necessary to consider the elasticity as described in Section
After calculating the stretch, we can distinguish three different cases for the length of the rope in the traverse system:
The length is shorter than the height of the traverse The length is equal to the height of the traverse The length is longer than the height of the traverse
In the first case, the rope has to be further stretched. The required force can be calculated with Hooke’s law described in Equation (
In this section, more complex examples than in Section
Operating system: Windows 7 Enterprise 64 bits
Processor: AMD FX(tm)-6300 Six-Core Processor; 3.50 GHz
Working storage: 32 GB
Software: MATLAB 2017a
To simulate the proposed approach, the VTVL vehicle is assumed to be a point mass and is attached to three ropes. In the first example, we use a vertical trajectory resembling a lift-off followed by a landing maneuver. EAGLE is placed in the barycenter of the equilateral triangle formed by the supports. In this case, we model a 7 m-long rope, with the expander rope that has a length of 1.5 m when not stretched. The scenario is depicted in Figure
EAGLE’s trajectory 2 (in yellow) and the three ropes (fixed at the lander and led to the supporters). (a) 3D view of the motion. (b)
In Figure
Forces obtained for the trajectory 1. (a) Sum of forces at the lander (in red) and the corresponding individual forces (in blue, green, and yellow), over the time steps of the lander’s movement. (b) Virtual forces and their difference (in blue, green, and yellow) for each rope.
The second plot of Figure
The second trajectory is a helix-shaped profile. EAGLE changes its altitude slowly, while constantly staying within a given distance from the center of the NEST. This scenario is represented in Figure
EAGLE’s trajectory 1 (in yellow) and the three ropes (fixed at the lander and led to the supporters). (a) 3D view of the motion. (b)
In Figure
Forces obtained for the trajectory 1. (a) Sum of forces at the lander (in red) and the corresponding individual forces (in blue, green, and yellow), over the time steps of the lander’s movement. (b) Virtual forces and their difference (in blue, green, and yellow) for each rope.
Furthermore, there is an error plot in Figure
In this paper, the modeling of a tethered configuration for testing a VTVL vehicle is described. After a short introduction of the catenary curve and the calculation of the forces at the suspension points of a single hanging rope, a specific geometry (NEST) including different values of length and rope stiffness is considered.
The model computes the forces of each rope at the lander and at the bottom of the traverse, where the ropes are fixed. The use of equilibrium of forces, together with Hooke’s law provide a viable way to compute the forces acting on the vehicle in conditions of hovering flight.
The examples show that in the relevant area of the testbed the algorithm provides a high accuracy solution. So, it can be included in the design of the controller as feed-forward contribution to compensate for the effects of the tethers. Based on an implementation in MATLAB and Simulink, this contribution can be run
The model has been developed and can be applied also for other configurations of the tethering. For example, after further maturing of the control system of EAGLE, all tethers on NEST but one could be removed (see
The data used to support the findings of this study are included within the supplementary information files.
The authors declare that they have no conflicts of interest.
Video_Supports_Vertical_Motion: animation showing the entire behavior of the ropes for simulation of example 1. Video_Supports_Helix: animation showing the entire behavior of the ropes for simulation of example 2. Video_Forces_Vertical_Motion: animation showing the forces evolution for simulation of example 1. Video_Forces_Helix: animation showing the forces evolution for simulation of example 2.