Modeling Deployment of Tape Spring Antennas and Its Effects on CubeSat Dynamics

Abstract

Tape spring mechanisms have been widely used to design deployable structures for satellites. In this paper, the dynamics of the deployment of VHF tape spring antennas from a CubeSat have been discussed. A set of nonlinear equations of motion representing the coupled dynamics of the deployment and the overall motion of the CubeSat were obtained using Kane’s method. These were integrated numerically to obtain parameters of interest like the time taken for deployment, the trajectory of the antenna, and variation of the CubeSat angular velocities during the interval of deployment. These models have been used to analyze the design of Advitiy (1U CubeSat conceptualized by IIT Bombay’s Student Satellite Team). Based on simulations, comments have been made on this design of the antenna deployment system, and few performance specifications have been laid down for the CubeSat.

1 Introduction

Over the last decade, rapid growth has been seen in the CubeSat market. Due to the miniaturization of technology, it is possible to send numerous small satellites or CubeSats rather than one “mega” satellite. Several deployable members like antennas and solar panels are typically incorporated in CubeSats, allowing a smaller volume during the launch. These members are deployed after reaching the desired orbit. The design of such mechanisms is a complicated task. Investigations to develop passive and active control for primitive spacecraft, which were typically rigid, have been carried out. The same analysis cannot be directly extended to modern spacecraft, the primary reason being the inclusion of several flexible members and deployable structures [5].

This paper analyzes the antenna deployment system for CubeSats being developed by the Student Satellite Team, IIT Bombay.¹ The dynamics of the deployment are of interest as it influences several mission parameters. It is essential to know at what angular rates of the satellite can the antennas be deployed. Time taken for the deployment to complete is of interest. Based on the analysis, some remarks have been made on the design of other subsystems.

2 Design of the Antenna Deployment System for CubeSat Application

Student Satellite Team, IIT Bombay, has been developing an antenna deployment system for CubeSat applications [6]. This system comprises two tape spring antennas wrapped around a circular ring supported by a PCB. Advitiy, the second student satellite of IIT Bombay, was being designed with this antenna deployment system [4]. In Fig. 1, Advitiy, along with the antenna deployment system, is shown. Advitiy, a 1U CubeSat, was being designed for a Low Earth Orbit (LEO) mission. The amateur band 144–146 MHz was selected for communication. The monopole or dipole antennas for this band cannot be accommodated inside 1U volume constraints. To meet this, an antenna deployment system was designed. An antenna deployment system, capable of deploying dipole antennas of 51 cm length, was designed and prototyped by the team.

Fig. 1

Advitiy

3 Dynamics of the CubeSat Over the Interval of Deployment

Tape springs are straight, thin-walled strips with a curved cross-section [10]. A carpenter’s tape is a tape spring. Due to its curvatures, it exhibits unique behavior. It is stored as a coil but can be extended into a stiff tape that can be easily folded. Except for the region of the fold, the rest of the tape spring is mostly straight and undeformed. Tape springs store elastic strain energy during folding and, in principle, would deploy freely into the straight unstrained configuration once the restraining forces have been released. Tape spring can be folded in two ways, depending upon the sense of curvatures. Several deployable structures that make use of tape springs have been analyzed [2, 9, 11]. Different modeling approaches have been proposed, ranging from a theoretical model [10] to develop a FE-based model [3, 7]. In this paper, a suitable extension is proposed to the theoretical model proposed by Seffen [10].

A model to study the dynamics of uncoiling of a tape spring mounted on a spool and its experimental verification is presented by Seffen [10]. This analysis is applicable when the radius of curvature of the tape spring and tape spool is nearly the same. The tape spring is modeled as a rigid body of varying length, which uncoils from the circular base. This modeling approach assumes the conservation of energy.

The antenna deployment system consists of a disk around which the antennas are wrapped. The CubeSat can be assumed to consist of a rigid cube with two flexible tape spring antennas wrapped around a disk mounted on one of the faces. Figure 2 shows the schematic and parameters used to denote the unwrapping of the antenna from the deployment system. The x and y axes are parallel with the face containing the antenna deployment system, and the z-axis is normal to the face with the antenna deployment system, pointing towards the reader. The x-axis is parallel to the deployed antenna and points towards the right side. It was assumed that no external torques and forces are acting on the system during the deployment. It is expected that the angular momentum of the system remains conserved. The system gains kinetic energy at the expense of the potential energy stored in the coiled tape spring.

Fig. 2

Schematic of the model

Before developing the full model for the CubeSat, a more straightforward problem was analyzed - a disk with one or two antennas in a plane, where translation in two axes and rotation about one axis were permitted.² Kane’s method was used to obtain the dynamics equations. This set of nonlinear differential equations was then integrated numerically. A set of simulations were performed to understand the effect of the number of antennas, thickness, initial energy, and the sense of folding for the tape springs on the final angular velocity attained and the time taken for the deployment. This was later extended for the CubeSat by simulating a cube with one or two antennas, where translations in three axes and rotations about three axes were permitted [8]. Values assigned for various parameters were obtained from the design of Advitiy.

4 Results

4.1 Simulations on Disk

Different test cases, their parameters, and the results have been tabulated in Table 1. For all these simulations, initially, the disk is at rest, and 0.99 fraction of the antenna is wrapped around the disk. The numerical integration is performed over two intervals. Timestep of 0.00001 was used from 0 s to 0.001 s, and a timestep of 0.001 was used from 0.001 s for the rest of the period of analysis. The mass of the disk is the same as that of the satellite, and the moment of inertia for the disk is the same as that for the satellite, about the z-axis.

Table 1 Results of simulation on disk

4.2 Simulations on Cube

Different test cases, their parameters, and the results have been tabulated in Table 2. For all these simulations, initially, the cube is at rest, and 0.99 fraction of the antenna is wrapped around. The numerical integration is performed over two intervals. Timestep of 0.00002 was used from 0 to 0.001 s, and a timestep of 0.001 for FT09 and 0.002 for FT10 was used from 0.001 s for the rest of the period of analysis.

Table 2 Results of simulations on cube

For the simulations performed, the velocity of the center of mass and the angular momentum were monitored. It was observed that the velocity of the center of mass remains nearly zero, and the angular momentum is conserved. These slight variations have been attributed to numerical errors, but they are within engineering approximations and can be neglected. Since the system begins from rest, the total energy is equal to the potential energy stored due to the tape spring coiling. This potential energy depends on the strain energy per unit area and the coiled area of the tape. The opposite sense of folding is responsible for a higher value of the strain energy, while the equal sense of folding is responsible for a lower value. A thicker antenna, when coiled, stores more strain energy than a thinner antenna.

The plots for test case FT10 are shown. Figure 3 denotes the plot of the fraction of deployment for both the antennas over time. Both the antennas take nearly the same amount of time for deployment. Figure 4 shows the angular velocity of the cube. ω₁, ω₂, and ω₃ denote angular rates about the 3 axes, where ω₃ is about the z-axis, about which the antennas are wrapped around. Hence, maximum change is observed about the z-axis. The non-zero cross-terms in the moment of the inertia matrix are responsible for angular rates about the other two axes.

Fig. 3

Fraction of deployment

Fig. 4

Angular rates of CubeSat

5 Conclusions

In this paper, a model for coupled dynamics of the deployment of the tape spring antennas mounted on disk and cube were derived, and simulations were performed. The following qualitative remarks can be made from the analysis:

• The maximum change in angular velocity is observed along the axis about which the antennas are wrapped around.

• A thicker antenna with an opposite sense of bending produces the maximum angular velocity, while a thinner antenna with an equal sense of bending produces the least angular velocity.

• For the same thickness and direction of folding, the system with one antenna and two antennas have nearly the same deployment time, with the system with two antennas requiring slightly less time than the system with one antenna.

Following are some considerations which shall influence the design of other subsystems:

• The initial angular velocity of Advitiy, while in orbit, may be non-zero. Depending on the direction of the unfolding of the antennas, this can either increase or decrease the angular velocity. Hence, the onboard logic should check the angular rates and factor in the direction of the unfolding of antennas. Based on this, an appropriate instant should be decided to issue the command for deployment.

• From the model, the anticipated angular rates with which Advitiy spins after deployment can be evaluated. The attitude determination and control subsystem should be able to control Advitiy under these conditions. Thus, a set of requirements must be generated by performing a more exhaustive set of simulations on the model.

Appendix—Derivation of Dynamics of Disk with 1 Coiled Antenna

The system with a uniform disk and one tape spring antenna wrapped around its periphery is considered. The schematic is shown in Fig. 5. The disk has radius r, mass m_d, and moment of inertia about its center of mass I_d. The tape spring has length L, mass per unit length ρ, the radius of curvature R, and thickness t. The inertial frame is defined using N. The frame attached to the disk, located at its center of mass, is denoted by A. At ψ = 0, the frames N and A are parallel to each other. x and y denote the position of the center of mass of the disk in N frame. Let λ₁ denote the fraction of length of tape spring, which has been unwrapped. Let δ₁ denote the angle subtended at the center of the disk by the portion of tape spring coiled, at a given instant. The subscript has been added to identify the antenna. There are two parts of the tape spring, a coiled section, and an unwrapped section. A generic point on the coiled section is denoted by xP11Aa and on the unwrapped section by xP12Aa. ζ₁ is the parameter, which is dependent on δ₁. Kane’s method has been used to derive these equations [1, 8]. xANn denotes the position vector of the center of mass of disk with respect to frame N, expressed in unit vectors of N frame. Similar notations have been used throughout.

Fig. 5

Schematic for 1 antenna uncoiling from disk

$${\mathbf{vANa}} = \left\{ {u_1 , \, u_2 ,0} \right\}^T$$

(1)

$$\mathbf{\upomega}\mathbf{ANa} = \left\{ {0,0,u_3 } \right\}^T$$

(2)

u_1, u_2, u₃ are generalized speeds.

$$L\left( {1 - \lambda_1 } \right) = r\delta_1$$

(3)

$${\mathbf{xP11Aa}} = \left\{ {r\;\sin \left( {\zeta_1 L/r} \right),\;r\;\cos \left( {\zeta_1 L/r} \right),\;0} \right\}^T$$

(4)

$$\begin{aligned} {\mathbf{xP12Aa}} & = \left\{ {r\;\sin \left( {\delta_1 } \right) + \left( {L\;\zeta_1 - r\;\delta_1 } \right)\;\cos \left( {\delta_1 } \right),\;r\cos \left( {\delta_1 } \right)} \right. \\ & \quad \left. {-\left( {L\;\zeta_1 - r\;\delta_1 } \right)\;\sin \;\left( {\delta_1 } \right),\;0} \right\}^T \\ \end{aligned}$$

(5)

Velocities are written as follows. dδ₁/dt is set as u₄, the 4th generalized speed, in addition to the 3, defined before.

$${\mathbf{vP11Na}} = {\mathbf{vANa}} + {\text{d}}{\mathbf{xP11Aa}}/{\text{d}}t + {\mathbf{\upomega}\mathbf{ANa}} \times {\mathbf{xP11Aa}}$$

(6)

$${\mathbf{vP12Na}} = {\mathbf{vANa}} + {\text{d}}{\mathbf{xP12Aa}}/{\text{d}}t + {\mathbf{\upomega}\mathbf{ANa}} \times {\mathbf{xP12Aa}}$$

(7)

Using these, partial velocities (i =4 terms each. p indicates that these are partial velocities)-vANap_i, vP11Nap_i, vP12Nap_i, and partial angular velocities ωANap_i are evaluated by taking derivatives with generalized speeds. Accelerations are evaluated as follows.

$${\mathbf{aANa}} = {\text{d}}{\mathbf{vANA}}/{\text{d}}t + {\mathbf{\upomega}\mathbf{ANa}} \times {\mathbf{vANa}}$$

(8)

$${\mathbf{\upalpha}\mathbf{ANa}} = {\text{d}}{\mathbf{\upomega}\mathbf{ANa}}/{\text{d}}t$$

(9)

$${\mathbf{aP11Na}} = {\text{d}}{\mathbf{vP11Na}}/{\text{d}}t + {\mathbf{\upomega}\mathbf{ANa}} \times {\mathbf{vP11Na}}$$

(10)

$${\mathbf{aP12Na}} = {\text{d}}{\mathbf{vP12Na}}/{\text{d}}t + {\mathbf{\upomega}\mathbf{ANa}} \times {\mathbf{vP12Na}}$$

(11)

The potential energy for the coiled section of the tape spring is written as PE = μR²αδ₁ where μ denotes strain energy per unit area, α angle of the embrace of the cross-section. Generalized inertia forces, F^*_i, and generalized active force, F_i, are written

$$\begin{aligned} F_i^* & = -\left[ {m_d {\mathbf{aANa \cdot vANap_i}} + I_d {\mathbf{\upalpha}\mathbf{ANa} \cdot \mathbf{\upomega}\mathbf{ANap_{i}}} } \right. \\ & \quad + \int_0^{r\;\delta 1\;/L} {\rho L\left( {{\mathbf{aP11Na \cdot vP11Nap_i}} } \right){\text{d}}\;\zeta_1 } \\ & \quad \left. { + \int_{r \, \delta 1 \, /L}^1 {\rho L\left( {{\mathbf{aP12Na \cdot vP12Nap_i}} } \right)d\;\zeta_1 } } \right] \\ \end{aligned}$$

(12)

$$F_4 = -{\text{dPE}}/{\text{d}}\delta_1$$

(13)

Other generalized active force terms are zero. Using these, equations of motion are obtained.

$$\begin{aligned} & -\left( {L\rho + m_d } \right){\text{d}}u_1 /{\text{d}}t \\ & -0.5\rho \left[ {-2r^2 \sin \left( {\delta_1 } \right)-2r\left( {L-r\;\delta_1 } \right)\cos \left( {\delta_1 } \right) + \left( {L-r\;\delta_1 } \right)^2 \sin \left( {\delta_1 } \right)} \right]{\text{d}}u_3 /{\text{d}}t \\ & + 0.5\rho \left[ {\left( {L-r\;\delta_1 } \right)^2 \sin \left( {\delta_1 } \right)} \right] \, {\text{d}}u_4 /dt = -\left( {L\rho + m_d } \right)u_2 u_3 \\ & + 2r^2 u_3^2 \left( {1-\cos \left( {\delta_1 } \right)} \right) + 2r\left( {L-r\;\delta_1 } \right)\left( {u_4^2 -u_3^2 } \right)\sin \left( {\delta_1 } \right) \\ & + \left( {L-r\;\delta_1 } \right)^2 \left( {u_4 -u_3 } \right)^2 \cos \left( {\delta_1 } \right) \\ \end{aligned}$$

(14)

$$\begin{aligned} & -\left( {L\rho + m_d } \right){\text{d}}u_2 /{\text{d}}t \\ & -0.5\rho \left[ {2r^2 \left( {1-\cos \left( {\delta_1 } \right)} \right) + 2r\left( {L-r\;\delta_1 } \right)\sin \left( {\delta_1 } \right) + \left( {L-r\;\delta_1 } \right)^2 \cos \left( {\delta_1 } \right)} \right]{\text{d}}u_3 /{\text{d}}t \\ & + 0.5\rho \left[ {\left( {L-r\;\delta_1 } \right)^2 \cos \left( {\delta_1 } \right)} \right]{\text{ d}}u_4 /{\text{d}}t = \left( {L\rho + m_d } \right)u_1 u_{3 } - \, 2r^2 u_3^2 \sin \left( {\delta_1 } \right) \\ & + 2r\left( {L-r\;\delta_1 } \right)\left( {u_4^2 - \, u_3^2 } \right)\cos \left( {\delta_1 } \right) + \left( {L-r\;\delta_1 } \right)^2 \left( {u_4 - \, u_3 } \right)^2 \sin \left( {\delta_1 } \right) \\ \end{aligned}$$

(15)

$$\begin{aligned} & 0.5\rho \left[ {-2r^2 \sin \left( {\delta_1 } \right)-2r\left( {L-r\;\delta_1 } \right)\cos \left( {\delta_1 } \right) + \left( {L-r\;\delta_1 } \right)^2 \sin \left( {\delta_1 } \right)} \right]{\text{d}}u_1 /{\text{d}}t \\ & + 0.5\rho \left[ {2r^2 \left( {1-\cos \left( {\delta_1 } \right)} \right) + 2r\left( {L-r\;\delta_1 } \right)\sin \left( {\delta_1 } \right) + \left( {L-r\;\delta_1 } \right)^2 \cos \left( {\delta_1 } \right)} \right]{\text{d}}u_2 /{\text{d}}t \\ & + \left[ {I_d + \rho Lr^2 + \left( {\rho /3} \right)\left( {L-r\;\delta_1 } \right)^3 } \right]{\text{d}}u_3 /{\text{d}}t \\ & - \left( {\rho /3} \right)\left( {L-r\;\delta_1 } \right)^3 {\text{d}}u_4 /{\text{d}}t \\ & \quad = - \, r^2 u_1 u_3 \rho \left( {1-\cos \left( {\delta_1 } \right)} \right)-r\rho \left( {L-r\;\delta_1 } \right)^2 \left( {u_4 - \, u_3 } \right)u_4 \\ & -r\rho u_3 \left( {L-r\;\delta_1 } \right)\left[ { \, u_2 \cos \left( {\delta_1 } \right) + u_1 \sin \left( {\delta_1 } \right)} \right] \\ & -0.5\rho u_3 \left( {L-r\;\delta_1 } \right)^2 \left[ {u_1 \cos \left( {\delta_1 } \right)-u_2 \sin \left( {\delta_1 } \right)} \right] \\ \end{aligned}$$

(16)

$$\begin{aligned} & 0.5\rho \left( {L-r\;\delta_1 } \right)^2 \left[ {{\text{d}}u_2 /{\text{d}}t\;\cos \left( {\delta_1 } \right) + {\text{d}}u_1 /{\text{d}}t\;\sin \left( {\delta_1 } \right)} \right] \\ & -\left( {\rho /3} \right)\left( {L-r\;\delta_1 } \right)^3 \left[ {{\text{d}}u_4 /{\text{d}}t-{\text{d}}u_3 /{\text{d}}t} \right] \\ & = -0.5\rho u_3 \left( {L-r\;\delta_1 } \right)^2 \left[ {u_1 \cos \left( {\delta_1 } \right)-u_2 \sin \left( {\delta_1 } \right)} \right] \\ & -0.5r\rho \left( {L-r\;\delta_1 } \right)^2 \left( {u_4^2 -u_3^2 } \right) + \mu R^2 \alpha \\ \end{aligned}$$

(17)

Note that these equations of motion (14–17) are functions of the generalized speeds, their first derivatives, and only one generalized coordinate δ₁. These are solved simultaneously with the following four Eqs. (18–21) to obtain the trajectory of the system in terms of the generalized coordinates of the system.

$${\text{d}}x/{\text{d}}t = u_1 \cos \left( \psi \right)-u_2 \sin \left( \psi \right)$$

(18)

$${\text{d}}y/{\text{d}}t = u_2 \cos \left( \psi \right) + u_1 \sin \left( \psi \right)$$

(19)

$${\text{d}}\psi /{\text{d}}t = u_3$$

(20)

$${{\text{d}}\delta_1}/{\text{d}}t=u_{4}$$

(21)

The example considered by Seffen [10] restricts the translation of the disk. The set of equations can be reduced by setting u₁ = u₂ = 0 and du₁/dt = du₂/dt = 0 to verify these against the equations presented by Seffen [10].

This approach of deriving dynamics has been extended for a disk with two antennas and a cube with one or two antennas [8]. Use of symbolic manipulation software is highly recommended to obtain and solve these equations.