Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field

Transcription

1 Dynamics and Control of an Elastic Dumbbell Spacecraft in a Central Gravitational Field Amit K. Sanyal Jinglai Shen N. Harris McClamroch Department of Aerospace Engineering University of Michigan Conference on Decision and Control December 10, 2003

2 Related Prior Literature Relatively few publications treat the coupling between orbit, attitude, and shape degrees of freedom for multi-body spacecraft This is important for spatially large spacecraft Multi-body spacecraft examples studied in the literature Space robots Tethered spacecraft Dumbbell spacecraft Connections with full-body problems Spatially distributed, elastic spacecraft orbiting a massive central body Spherical spacecraft orbiting a massive spatially distributed, elastic central body N-body problems for spatially distributed, elastic bodies Applications to asteroids and asteroid pairs

3 Remainder of this Presentation Models of dumbbell spacecraft Determine equations of motion Determine reduced equations of motion Determine relative equilibria Controllability results for dumbbell spacecraft Linear controllability of full equations Linear controllability of reduced equations Comments on spacecraft control problems

4 Description of Elastic Dumbbell Spacecraft Dumbbell spacecraft consists of two identical mass particles connected by an elastic link Central body gravity forces act on each of the mass particles Control forces act on each of the mass particles Simplifying assumption: all motion occurs in a fixed orbital plane Generalized coordinates Orbit DOFs: orbit variables r,f Attitude DOF: attitude variable q Shape DOF: shape variable q

5 Dumbbell Spacecraft Figure: Full Actuation

6 Full Equations of Motion of Dumbbell Spacecraft Lagrangian formulation Gravity potential (approximation) V g = - mm r Ê 2 - q2 r 2 1- ˆ ( 3cos2 q) Ë Elastic potential V s = k( q - l) 2 Lagrangian function ( q 2 ) L = m r 2 + r 2 f 2 + q 2 q 2 + 2q 2 q f + q 2 f 2 + -V g -V s

7 Equations of motion Ê 2m r - m 2rf 2-2m r 2 + 3mq2 Ë r 4 1-3cos 2 ˆ ( q) = F r ( ) = F f 2m r 2 f + 2r r f + q 2 f + q 2 q + 2q q f + 2q q q ( ) + 6mmq2 2m q 2 q + q 2 f + 2q q f + 2q q q r 3 ( f 2 ) + 2mmq r 3 2m q - m 2qq 2 + 4q q f + 2q cosq sinq = 2qN ( 1-3cos 2 q) + 2k(q - l) = T

8 Dumbbell Spacecraft Figure: No Orbital Actuation

9 Reduced Equations of Motion of Dumbbell Spacecraft Assume attitude and shape control inputs only, i.e. no orbital inputs Since f is cyclic, angular momentum is a conserved quantity: is constant ( ) p = L f = 2m ( q2 q + r 2 + q 2 f ) Form the classical Routhian R = L - pf = m r 2 + q 2 q 2 + -V g -V s ( q ) 2 ( q 2 ) - p - 2mq2 ( ) 4m r 2 + q 2

10 Reduced equations of motion r = rq 4 ( r 2 + q 2 ) 2 q 2 p 2 r + 4m 2 r 2 + q 2 m( r - qcosq) - 2 r 2 + q 2-2qr cosq ( ) 2 - rq 2 p ( ) 2 m r 2 + q 2 m r + qcosq ( ) 3/2-2( r 2 + q 2 + 2qr cosq) 3/2 ( r + r 3 q ) rq( r 2 + q 2 ) m( r 2 + q 2 ) 2 q - sinq 2qr( r 2 + q 2-2qr cosq) 3/2 + m( r 2 + q 2 ) sinq 2qr( r 2 + q 2 + 2qr cosq) 3/2 ( q - q r ) ( ( ) + r 2 + q 2 ) mr 2 q 2 Nq q = - 2 q3 - p r mqr r 2 + q 2 q ( )

11 q = r 4 q ( r 2 + q 2 ) 2 q 2 p 2 q + 4m 2 r 2 + q 2 m( q - r cosq) - 2 r 2 + q 2-2qr cosq ( ) 2 + ( ) 3/ k m (q - l)+ T 2m r 2 pq ( ) 2 m r 2 + q 2 m q + r cosq q ( ) ( ) 3/2 2 r 2 + q 2 + 2qr cosq f = p - 2mq2 q 2m r 2 + q 2 ( )

12 Relative Equilibria for the Dumbbell Spacecraft Assume there are no control force inputs The dumbbell spacecraft has two classes of relative equilibrium solutions (equilibrium solutions for the reduced equations) The longitudinal axis of the dumbbell spacecraft is aligned with the local vertical q e = 0 (or p ), f 2 e = m 3 r + 3mq e 2, 3mq e 5 3 e r e r e = k ( m q e - l) The longitudinal axis of the dumbbell spacecraft is aligned with the local horizontal q e = p 2 (or 3p 2 ), f 2 e = m 3 r - 3mq e 2 5 e 2r, q e = l e

13 Linearized Full Orbit, Attitude, and Shape Equations Consider a neighborhood of a relative equilibrium for which the longitudinal axis of the dumbbell spacecraft is aligned with the local vertical Define the full perturbations from this relative equilibrium as [ ] T x = dr df dq dq Define the orbit, attitude and shape control force inputs as [ ] T u = F r F f N T The linearized full equations of motion have the form M x +C x + Kx = Bu Define the parameter w 2 = m 3 r + 3mq e 2 e r e 5

14 È 2m m(r 2 M = e + q 2 2 e ) 2mq e mq e 2mq e 0 Î m È 0-4mr e w 0 0 4mr C = e w 0 0 4mq e w mq e w Î 0-4mq e w -4mq e w 0 È Ê -2m w 2 + 2m 2 3 r +12mq ˆ e 12mmq e 4 Ë e r e r e È K = 6mmq 0 0 e 0, B = q 0 r e 12mmq Ê e 0 0 2k - 2m w 2 + 2m ˆ Î Î r e Ë r e

15 Linearized Reduced Orbit, Attitude, and Shape Equations Consider a neighborhood of a relative equilibrium for which the longitudinal axis of the dumbbell spacecraft is aligned with the local vertical Define the reduced perturbations from this relative equilibrium as [ ] T x = dr dq dq Define the attitude and shape control force inputs as u = [ N T ] T The linearized full equations of motion are M x +C x + Kx = Bu

16 C = 0 2 pq e 2 r e r e 2 + q e 2 ( ) 2 Ê Ë ˆ 0-2 pq e 2 r e r e 2 + q e 2 ( ) 2 Ê Ë ˆ 0 2 pq e r e 2 r e 2 + q e 2 ( ) 2 Ê Ë ˆ 0-2 pq e r e 2 r e 2 + q e 2 ( ) 2 Ê Ë ˆ 0 È Î K = k +... È Î M = 2m mr e 2 q e 2 r e 2 + q e 2 Ê Ë ˆ m È Î B = 0 0 2q e È Î

17 Linearized Equations of Motion The linearized orbit, attitude and shape equations of motion are coupled, with a special mathematical structure that reflects the Hamiltonian features of the dumbbell spacecraft The linearized orbit, attitude and shape equations of motion generalize the Clohessy Wiltshire equations for orbit perturbations only Linear gravity gradient equations for attitude perturbations only Linear equations for elastic shape perturbations only Linearized orbit, attitude and shape equations can be obtained in a neighborhood of the horizontally aligned relative equilibrium

18 System theoretic properties of these linearized models can be studied using the linear models Stability Multivariable transfer function properties Controllability Observability Control problems for perturbations from a relative equilibrium can be formulated in terms of the linearized orbit, attitude and shape equations of motion Open loop maneuvers High thrust impulsive maneuvers Low thrust maneuvers Stabilization problems

19 Controllability Properties of the Dumbbell Spacecraft Linear full equations of motion Complete controllability using orbit, attitude and shape actuation Full actuation assumption Can cancel all orbit, attitude, and shape coupling Complete controllability using orbit actuation only Orbit actuation can be used to control attitude and shape through coupling Attitude and shape can be controlled without direct actuation Not controllable using attitude and shape actuation only Angular momentum is conserved

20 Linear reduced equations of motion Complete controllability using attitude and shape actuation only Can perform orbit maneuvers on a constant angular momentum surface Complete controllability using attitude actuation only Can perform orbit maneuvers on a constant angular momentum surface Complete controllability using shape actuation only Can perform orbit maneuvers on a constant angular momentum surface

21 Conclusions Major themes of this presentation Multi-body spacecraft control problems that take an integrated approach to control of orbit, attitude and shape dynamics have been neglected in the published literature These integrated multi-body spacecraft control problems are conceptually and computationally challenging Controllability results have been obtained for the dumbbell spacecraft example There are many open theoretical and applied research problems on integrated orbit, attitude and shape control of underactuated multi-body spacecraft Can this integrated spacecraft control approach be made into an effective spacecraft technology for spatially large spacecraft?