Junjie KANG,Zheng H.ZHU,Wei WANG,Changqing WANG,Aijun LI

aDepartment of Earth and Space Science and Engineering,York University,Toronto,Ontario M3J 1P3,Canada

bDepartment of Mechanical Engineering,York University,Toronto,Ontario M3J 1P3,Canada

cSchool of Automation,Northwestern Polytechnical University,Xi'an 710072,China

KEYWORDS De-spin;Dynamics;PD control;Space tethers;Tethered tug

Abstract This paper investigates the dynamics and de-spin control of a massive target by a single tethered space tug in the post-capture phase.The dynamic model of the tethered system is derived and simplified to a dimensionless form.Further,a decoupled PD controller is proposed,and the local stability of the controller is analyzed by linearization technique.Parametric studies of the dynamics and de-spin control of a massive target are conducted to characterize the dynamic process of de-spin with the proposed control law.It is shown that the massive target can be de-span by a single and small space tug with limited thrust within finite time.The thrust tangent with the tether de-spins the target while the thrust normal to the tether prevents the tether from winding up the target.The tether length has a positive contribution to the de-spin of a target.The longer tether leads to a faster de-spin process.

1.Introduction

Asteroid redirection and debris removal have received increasing attentions for their potential applications in planetary exploration and space utilization.1-5The challenge of such task is to de-spin the rotating asteroid and debris after capture for the safety of space operations.6Many technologies have been proposed for such tasks,such as space robots,space net and gripper mechanism.6-16Among them,space tether technology is appealing due to its light weight and high flexibility.6,15For instance,Robert and Karsten proposed a concept of deploying a nano-satellite to extract the angular momentum from the rotating target based on momentum exchange to passively de-spin the target by a single tethered space tug.6Similarly,Yudintsev and Aslanov proposed a modified Yo-Yo mechanism to de-spin the space debris based on the concept of momentum exchange.17In addition to the single tether configuration,multiple tethered systems are also developed,e.g.,a configuration of four sub-tethers attached to debris is proposed to stabilize the spinning debris through the viscouselastic properties of tether.18O'Connor proposed a de-spin approach of using a tethered open-net with thrusters.19Zhang designed an offset control for the attitude stabilization of the single tethered space tug together.20

The previous researches focused on de-spinning small targets,and the de-spin of massive targets with small space tugs is rarely studied.The direction and magnitude of rotation velocity of tumbling asteroid and debris are threedimensional in general.The dynamics and control of detumbling such targets by a much smaller space tug are complicated.As a first approach to prove the concept,we simplify the problem into a two-dimensional problem where the target is assumed to rotate about a single and fixed axis.Furthermore,we ignore the translational motion of the target by assuming that the rotation axis is stationary in the space to focus on the de-spin dynamics of a massive target with a small tug and associated the control problem.This paper investigates the de-spin of a massive target by an actively controlled small space tug in finite time.Two thrusters are utilized on the space tug to efficiently de-spin the target via a single tether with finite length.The purpose is to de-spin the target to a desired state so that a redirect mission can be conducted.The control force is exerted by the thrusts at the end of tether until the spinning of the target diminishes.

2.Dynamics formulation and de-spin control law

A massive spinning target is connected to a space tug via a single massless and inextensible tether moving in a plane as shown in Fig.1,where the tether transfers the tug force to the target.The tethered system is assumed in a free space where the gravitational field is neglected.The target is treated as a rigid body with an arbitrary shape,a mass M,and a moment of inertia J.The space tug is simplified as a lumped mass with a mass m(m＜＜M)and connected to the target by a tether with a constant length of L.The rotation center and the center of mass of the target are assumed coincided and stationary in the de-spin process.Furthermore,the Center of Mass(CM)of the tethered system is assumed approximately as same as the massive target due to the large mass ratio of the target and the tug.

We de fine a planar coordinate system OXY with its origin at the CM of the massive target.The tether is anchored at the edge of the target with distance r to the CM.The target is spinning around its CM at an angular velocity ω as shown in Fig.1.

Based on the above assumptions,the position of the tug R2with respect to the CM of spinning target is de fined as

Fig.1 Sketch of spinning target and tethered space tug system.

where R1is the vector of tether anchor point on the target,α is the angle between the vector R1and the x-axis,are the unit vectors of x and y axes,and θ is the angle between the tether and the vector R1.

Correspondingly,the velocity and acceleration of the tug are derived as

Then,the equations of motion of the target and the tug can be written as

where T is the tether tension andare the components ofthrustforcealong and perpendicularto thetether respectively.

Substituting Eqs.(1)-(5)into Eqs.(6)and(7)and rearranging variables yield the differential equations of the system in terms of angular velocity ω=˙α and angle θ,such that

subject to the constraints of| θ|≤ π/2 and T ≥ 0 to avoid tether winding up the target and slack.

De fine the dimensionless variablesand then Eq.(8)can be rewritten in the dimensionless forms,such that where（）′=d（）/dτ,ω0is theinitial angular velocity of the spinning target,andis the dimensionless tether tension.

The admissible equilibrium state of the system in Eq.(10)can be determined by analyzing its phase plane and the operational constraint.Assume that the target could be stabilized to rotate with a small and constant angular velocity,and then the differential equation of θ in Eq.(10)could be reduced to

Based on Eq.(12),the phase plane of θ is shown in Fig.2 for two different values of l and η.It can be seen that the trajectories of different l and η are not the same but their trends are similar.The system has a stable equilibrium at the center point of phase planeand unstable equilibriums at the saddlesConsidering the operational constraint on the libration angle| θ|≤ π/2,the only admissible equilibrium state of the system should be（ η ，θ，θ′）=（ηd，0，0）,where ηdis the dimensionless desired spin rate of the massive target.

Once the desired equilibrium state is determined,a de-spin control law is developed as follows.De fine a new error state as eη=η-ηd.Then,the equilibrium stateis transferred into a zero-equilibrium stateand Eq.(10)is transferred to

Linearizing the nonlinear coupled Eq.(13)about the zero equilibrium state yields

Note from Eq.(14)that the state eηdepends only on utwhile the state θ depends on both utand un.To simplify the design of control law,it is intuitive to decouple these two control inputs by assuming ut=c≥0 with c being a positive constant.Obviously,this assumption satisfies the constraint of non-negative tension from Eq.(11).Then,unis the only control input that needs to be determined.

De fine a simple Proportional-Derivative(PD)control law as follows:

where ki(i=1,2,3)are control gains.

Substituting Eq.(15)into Eq.(14)yields

Fig.2 Phase planes of tethered space tug at different values of λ and η.

The eigenvalue equation of Eq.(16)can be easily deduced as

If k2＞0 and k3（1+l） ＞k1l hold,inequality Eq.(18)is always satis fi ed even when ηd=0.Therefore,under the control input in Eq.(15),Eq.(16)will be locally stable at the zero equilibriumi.e.,the spinning velocity of the target will converge to ηd.Moreover,zeroing the desired angular velocity ηd=0 leads to the zero spinning.

3.Simulations and discussion

The effectiveness of the proposed control strategy is demonstrated by parametric analysis.The initial conditions of the system are assumed the same in all cases as η，θ，θ′（ ）=（1，0，0）.A spinning rate of meters of asteroid,ω0=0.02 rad/s from Ref.6is used as the initial angular velocity of the target.The time scale in the analysis results is normalized by the nominal rotation periodof the spinning target,T0=2π/ω0.Finally,the controller gains are assumed as k1=k2=k3=1 for all cases.

3.1.Effect of tangent thrust utof space tug

Inthiscase,theparametersofthesystemareassumedasfollows:inertial ratio λ=2000 and length ratio l=10.Four different valuesoftangentthrustut=0.1，1，10，100areusedinthesimulations.The results are shown in Fig.3(a)-(e).All states of the system converge to zero as required.As the thrust utincreases from 0.1 to 100,the de-spin times decrease significantly from 10,000 to 20 cycles(see Fig.3(a)),which makes sense.Similarly,the tether libration angle is reduced to zero from 10,000 to 20 cycles(see Fig.3(b)).However,the tether libration angular velocity is reduced much faster(＜15 cycles)and the decrease rates are almost the same for all values of thrust ut(see Fig.3(c)).Accordingly,the tether tension approaches to the thrust utas the spinning rate and libration angle decrease(see Fig.3(d))and the normal thrust approaches to zero.The magnitude of the normal thrust is never greater than unit(|un|≤1)for the given control gains.Its purpose is de fined to prevent the tether from winding up the target.The analysis shows that the tangentthrustishighlyeffectiveinde-spinningthetargetifthere issufficientthrustingcapacityatthetug.Incasethethrustislimited,suchasut≤1,thetargetcanstillbede-spunsofarthereisa normal thrust unto prevent the tether winding up the target.Based on the above analysis,a tangent thrust ut=1 is used for the following parametric analysis.

3.2.Effect of inertia ratio λ

Fig.4 Effects of inertia ratio λ on de-spin process.

In this case,the effect of inertia ratio on the de-spin is studied with the following parameters of the system:Length ratio l=10 and tangent thrust ut=1.The inertia ratio is assumed as λ=500，1000，2000，10000，20000.As shown in Fig.4,the spinning velocity of the target,tether libration angle and its rate are reduced to zero for all inertia ratios.The de-spin time increases as the inertial ratio increases.The tether tension peaks at the beginning and its value never exceed 15.It converges to the given tangent thrust ut=1.Similar to the previous case,the normal thrust peaks initially and its peak value is less than unit for all inertia ratios.The normal thrust converges to zero once the system reaches its zero-equilibrium state.Therefore,de-spinning a large target by a small space tug with limited thrust is feasible.

3.3.Effect of length ratio l

In this case,different length ratios are chosen to study the effect for de-spin system caused by tether length and target radius.The parameters of the system are chosen as follows:inertia ratio λ=2000 and tangent thrust ut=1.The length ratio is assumed as l=2，5，10，100 in the simulation.The results are shown in Fig.5(a)-(e).Figs.5(a)shows that the de-spin rates of the angular velocity of the large target are enhanced as the length ratio increases.In Figs.5(b)and(c),the effect is similar in tether libration angle and its rate.With the increasing length ratio,the tether libration motion decays faster.It is noted that there is high-speed oscillation when the length ratio increases to 100.This is because the effect of libration rate(k3/l)feedback,which is equivalent to a damping term for the tether libration motion θ,decreases as the length ratio increases(see Eq.(15)).One can eliminate this phenomenon by increasing the control gain k3.As shown in Fig.6,the libration angle's oscillation was obviously controlled when k3=10 for length ratios considered.The angular rate θ′and normal thrust unare not plotted here because they are directly related to libration angle θ.The max magnitude of the tether libration angle decreases from 1 to 0.35 as the length ratio increases from 2 to 100.From Fig.5(d),we can see that the magnitude of peak tether tension increases rapidly as the length ratio increases.The max tether tension is only about 15 when length ratio is low at l=2,but increases to 120 when l=100.This is caused by the term l（ η +θ′）2in Eq.(11).Actually,this term is a centrifugal force in the de-spin process.Similarly,the normal thrust unchanges in similar manners as shown in the above case.

Fig.5 Effects of length ratio l on de-spin process.

Fig.6 Libration angle versus time for different length ratios when k3=10.

4.Conclusions

This paper characterizes the dynamics and de-spin control of a massive target by a tethered space tug for the safety and feasibility of spinning large target removal.The conclusions from the study are as follows:

(1)A decoupled PD feedback controller is proposed to regulate the thrusters at the space tug with guarantee of the local stability of the closed-loop system.

(2)Parametric analysis shows that the massive target could be effectively de-spun by a relative small thrust at the tug.For the given inertia ratios of the target over tug and thrust,a longer tether could increase the de-spin rate.

(3)De-spin is achieved by the tangent thrust while the normal thrust is necessary to prevent the tether from winding up the target.

Acknowledgements

This work is supported by the Discovery Grant(No.RGPIN-2018-05991)of the Natural Sciences and Engineering Research Council of Canada.