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## Dynamics and Control of Tethered Formation Flight Spacecraft Using (2005)

Venue: | the SPHERES Testbed,” 2005 AIAA Guidance, Navigation and Control Conference |

Citations: | 4 - 2 self |

### Citations

1321 |
Applied Nonlinear Control
- Slotine, Li
- 2004
(Show Context)
Citation Context ...tethered spacecraft. A brief review of the results from Ref20–22 is presented in this section. A nonlinear system, possibly a time varying non-autonomous system is formulated as: ẋ = f(x,u(x, t), t) =-=(19)-=- Theorem 1 For the system in Eq. (19), if there exists a uniformly positive definite metric M(x, t) = Θ(x, t)TΘ(x, t) (20) where Θ is some smooth coordinate transformation of the virtual displacement,... |

224 | On contraction analysis for nonlinear systems
- Lohmiller, Slotine
- 1998
(Show Context)
Citation Context ...y a time varying non-autonomous system is formulated as: ẋ = f(x,u(x, t), t) (19) Theorem 1 For the system in Eq. (19), if there exists a uniformly positive definite metric M(x, t) = Θ(x, t)TΘ(x, t) =-=(20)-=- where Θ is some smooth coordinate transformation of the virtual displacement, δz = Θδx such that the associated generalized Jacobian, F = ( Θ̇ + Θ ∂f ∂x ) Θ−1 (21) is uniformly negative definite, all... |

117 | On partial contraction analysis for coupled nonlinear oscillators
- Wang, Slotine
(Show Context)
Citation Context ...ed to derive stability of the coupled dynamical systems. Theorem 2: Parallel Combination21,22 Consider two systems of the same dimension, contraction in the same metric, ẋ = fi(x,u(x, t), t) i = 1,2 =-=(22)-=- Assume further that the metric depends only on the state x and not explicitly on time. Then, any uniformly positive superposition (where ∃α > 0, ∀t ≥ 0, ∃i, αi(t) ≥ α) ẋ = α1(t)f1(x,u(x, t), t) + α2... |

28 |
The Terrestrial Planet Finder (TPF) : a NASA Origins Program to search for habitable planets. The Terrestrial Planet Finder (TPF) : a NASA Origins Program to search for habitable planets / the TPF Science Working Group ; edited by
- Beichman, Woolf, et al.
- 1999
(Show Context)
Citation Context ...The relationship between the absolute and relative acceleration is obtained by differentiating the relative velocity equation to get aB = aA + θ̈ez × rB/A + θ̇ez × (θ̇ez × rB/A) + 2θ̇ez × vB/A + aB/A =-=(2)-=- Since the SPHERES is rotating in the x-y frame that is revolving around the center of the fixed inertial X-Y frame, we are observing the coriolis term (2θ̇ez×vB/A) as well as the centrifugal force (θ... |

22 |
Autonomous docking algorithm development and experimentation using the SPHERES testbed
- NOLET, KONG, et al.
- 2004
(Show Context)
Citation Context ... θ φ θ̇ φ̇ = 0 0 1 0 0 0 0 1 0 rω2 ( IG+mr(r+l) ) lIG −2 l̇l 0 0 − rω 2 ( IG+m(r+l) 2 ) lIG 2 l̇l 0 θ φ θ̇ φ̇ + 0 0 0 0 1 ml − rIGl − 1ml r+lIGl ( F u ) =-=(18)-=- The nonlinear equations can be easily modified in the same fashion. 8 of 26 American Institute of Aeronautics and Astronautics Instead of showing an analytic solutions of the eigenvalues of Eq. (18),... |

8 |
Control of Deep-Space FormationFlying Spacecraft; Relative Sensing and Switched Information.
- Hadaegh, Smith
- 2005
(Show Context)
Citation Context ...an Institute of Aeronautics and Astronautics The equations of the tethered system is derived using Eq. (4): ∑ Fex = −Fx − T = −F sinφ− T = max∑ Fey = Fy = F cosφ = may∑ MG = −Tr sinφ+ u = IG(θ̈ + φ̈) =-=(5)-=- where MG is the moment around CM and IG denotes the moment of inertia around CM. ax and ay are the x,y acceleration components of Eq. (4) respectively. T can be eliminated and the following different... |

6 | ARGOS testbed: study of multidisciplinary challenges of future spaceborne interferometric arrays
- Chung, Miller, et al.
- 2004
(Show Context)
Citation Context ...expressed in the small x-y rotating frame and the corresponding unit directional vectors are ex and ey. The velocity of B is characterized as vB = [l̇ − r sinφ(θ̇ + φ̇)]ex + [r cosφ(θ̇ + φ̇) + lθ̇]ey =-=(1)-=- The relationship between the absolute and relative acceleration is obtained by differentiating the relative velocity equation to get aB = aA + θ̈ez × rB/A + θ̇ez × (θ̇ez × rB/A) + 2θ̇ez × vB/A + aB/A... |

6 | Control of a Rotating Variable-Length Tethered System,
- Kim, Hall
- 2003
(Show Context)
Citation Context ...eck if the system is really controllable around nominal points by calculating controllability matrix. [ B AB A2B A3B ] = 0 0 1ml − rIGl 0 0 − 1ml r+lIGl 1 ml − rIGl 0 0 − 1ml r+lIGl 0 0 =-=(15)-=- Where the calculation of A2B and A3B is omitted since the first two matrices result in the full rank (n = 4) and the rest of them are redundant (dependent). So the system is fully controllable with u... |

6 |
Retargetting Dynamics of a Linear Tethered Interfer
- Bombardelli, Lorenzini, et al.
(Show Context)
Citation Context ...3B2 ] = 0 − rIGl 0 r(r+l)ω2 ( IG+mr(r+l) ) l2IG2 0 r+lIGl 0 − r(r+l)ω2 ( IG+m(r+l) 2 ) l2IG2 − rIGl 0 r(r+l)ω2 ( IG+mr(r+l) ) l2IG2 0 r+l IGl 0 − r(r+l)ω 2 ( IG+m(r+l) 2 ) l2IG2 0 =-=(16)-=- This is a full rank (n = 4) matrix for any nonzero ω and tether length, l. Its implication to the future tethered systems is enormous: the tethered satellite systems will be able to spin up and re-si... |

1 |
et al., ”The Submillimeter Frontier: A Space Science Imperative,” Astrophysics
- 3Mather
- 1998
(Show Context)
Citation Context ...̇θ̇)ey θ̈ez × rB/A = −r sinφθ̈ex + r cosφθ̈ey θ̇ez × (θ̇ez × rB/A) = −r cosφθ̇2ex − r sinφθ̇2ey 2θ̇ez × vB/A = −2r cosφθ̇φ̇ex − 2r sinφθ̇φ̇ey aB/A = (−rφ̈ sinφ− rφ̇2 cosφ)ex + (rφ̈ cosφ− rφ̇2 sinφ)ey =-=(3)-=- Let’s assume for now the tether length is fixed. This means that aA = (−lθ̇2 + l̈)ex + (lθ̈ + 2l̇θ̇)ey is reduced to aA = (−lθ̇2)ex+ (lθ̈)ey Then, Eq.(2) becomes aB = ex[−lθ̇2 − r sinφθ̈ − r cosφθ̇2 ... |

1 |
et al., ”Decentralized Control of Satellite Clusters Under
- 4Belanger
- 2004
(Show Context)
Citation Context ...is reduced to aA = (−lθ̇2)ex+ (lθ̈)ey Then, Eq.(2) becomes aB = ex[−lθ̇2 − r sinφθ̈ − r cosφθ̇2 − 2r cosφθ̇φ̇− rφ̈ sinφ− rφ̇2 cosφ] + ey[(l + r cosφ)θ̈ − r sinφθ̇2 − 2r sinφθ̇φ̇+ rφ̈ cosφ− rφ̇2 sinφ] =-=(4)-=- 5 of 26 American Institute of Aeronautics and Astronautics The equations of the tethered system is derived using Eq. (4): ∑ Fex = −Fx − T = −F sinφ− T = max∑ Fey = Fy = F cosφ = may∑ MG = −Tr sinφ+ u... |

1 |
et al, ”SPHERES as a Formation Flight Algorithm Development and Validation Testbed
- 6Kong
(Show Context)
Citation Context ...ax and ay are the x,y acceleration components of Eq. (4) respectively. T can be eliminated and the following differential equation is obtained, [M(φ)] ( θ̈ φ̈ ) + [ C(φ, θ̇, φ̇) ] = ( F cosφ Fr + u ) =-=(6)-=- where [M(φ)] = [ ml +mr cosφ mr cosφ IG +mr2 +mrl cosφ IG +mr2 ] and [ C(φ, θ̇, φ̇) ] = [ −2mr sinφθ̇φ̇−mr sinφθ̇2 −mr sinφφ̇2 mrl sinφθ̇2 ] = [ −mr sinφ(θ̇ + φ̇)2 mrl sinφθ̇2 ] When the tether motor... |

1 |
Control of a Tethered Artificial Gravity Spacecraft
- 7Wilson
- 1990
(Show Context)
Citation Context ... mrl sinφθ̇2 ] = [ −mr sinφ(θ̇ + φ̇)2 mrl sinφθ̇2 ] When the tether motor reels in or out at a constant speed (l̇ =constant), the force term in Eq. (6) is characterized as: ( F u ) ⇒ ( F − 2ml̇θ̇ u ) =-=(7)-=- The equations of the motion can be also derived by exploiting the technology developed for multi-link robot kinematics or Lagrange’s equation. Those equations are simplified assuming the mass of the ... |

1 |
On Formation Deployment for Spinning Tethered Formation Flying and Experimental
- 9Nakaya
- 2004
(Show Context)
Citation Context ...ince 1st row times l plus 2nd row of Eq. (6) results in the first equation of Eq. (8). So the external force(or torque) terms can be matched like the following:( τ1 τ2 ) = [ r + l cosφ 1 r 1 ]( F u ) =-=(9)-=- C. Linearization and Pendulum Mode Frequency We linearize Eq. (8) about θ̇ = ω, and φ̇, φ = 0. Each term can be linearized as the following: mrl sinφθ̇2 ≈ mrlω2φ,mrl sinφφ̇2 ≈ 0,mrl sinφθ̇φ̇ ≈ 0, cos... |

1 |
Dynamic Model and Control of Mass-Distributed Tether Satellite
- 10Yu
- 2002
(Show Context)
Citation Context ...rlω2φ,mrl sinφφ̇2 ≈ 0,mrl sinφθ̇φ̇ ≈ 0, cosφ ≈ 1 The linearized equation of the motion is presented:[ IG +m(r + l)2 IG +mr(r + l) IG +mr(r + l) IG +mr2 ]( θ̈ φ̈ ) + [ 0 0 0 mrlω2 ]( θ φ ) = ( τ1 τ2 ) =-=(10)-=- 6 of 26 American Institute of Aeronautics and Astronautics Similarly, we can linearize Eq. (6) and transformation between the two system equations is easily performed using Eq. (9). It is observed th... |

1 |
et al, ”Nonlinear Control of Librational Motion of Tethered Satellites in Elliptic Orbits
- 11Kojima
- 2004
(Show Context)
Citation Context ... Eq. (10) by the inverse of M matrix and use the linearized relationship of Eq. (9):( θ̈ φ̈ ) + 0 − rω2 ( IG+mr(r+l) ) lIG 0 rω2 ( IG+m(r+l) 2 ) lIG ( θ φ ) = [ 1 ml − rIGl − 1ml r+lIGl ]( F u ) =-=(11)-=- The second-order nonlinear equation of motion of φ from the second line of (11) becomes: φ̈+ ωφ2φ = − 1 ml Fy + r + l IGl u (12) where ωφ is the frequency of the pendular libration mode: ωφ = √ r(IG ... |

1 |
et al, ”Deployment,Retrieval Control of Tethered Subsatellite Under Effect of Tether Elasticity
- 12Kokubun
- 1996
(Show Context)
Citation Context ...ω2 ( IG+m(r+l) 2 ) lIG ( θ φ ) = [ 1 ml − rIGl − 1ml r+lIGl ]( F u ) (11) The second-order nonlinear equation of motion of φ from the second line of (11) becomes: φ̈+ ωφ2φ = − 1 ml Fy + r + l IGl u =-=(12)-=- where ωφ is the frequency of the pendular libration mode: ωφ = √ r(IG +m(r + l)2) lIG ω[rad/s] (13) The actual raw gyro data from a single tethered SPHERES exhibit a high frequency oscillation (the p... |

1 |
Wave-Absorbing Control of Transverse Vibration of Tethered System
- 13Fujii
- 2004
(Show Context)
Citation Context ...ar equation of motion of φ from the second line of (11) becomes: φ̈+ ωφ2φ = − 1 ml Fy + r + l IGl u (12) where ωφ is the frequency of the pendular libration mode: ωφ = √ r(IG +m(r + l)2) lIG ω[rad/s] =-=(13)-=- The actual raw gyro data from a single tethered SPHERES exhibit a high frequency oscillation (the pendulum mode) and the DC component (a rigid body mode of a certain rotational rate). The frequency o... |

1 |
Modular Stability Tools for Distributed Computation and Control
- 21Slotine
(Show Context)
Citation Context ...te metric M(x, t) = Θ(x, t)TΘ(x, t) (20) where Θ is some smooth coordinate transformation of the virtual displacement, δz = Θδx such that the associated generalized Jacobian, F = ( Θ̇ + Θ ∂f ∂x ) Θ−1 =-=(21)-=- is uniformly negative definite, all system trajectories then converge exponentially to a single trajectory regardless of the initial conditions, with convergence rate of the largest eigenvalues of th... |

1 |
The Swing Up Control Problem For The Acrobot
- 23Spong
- 1995
(Show Context)
Citation Context ... the metric depends only on the state x and not explicitly on time. Then, any uniformly positive superposition (where ∃α > 0, ∀t ≥ 0, ∃i, αi(t) ≥ α) ẋ = α1(t)f1(x,u(x, t), t) + α2(t)f2(x,u(x, t), t) =-=(23)-=- is contracting in the same metric. By recursion, this property of parallel combination can be extended to any number of systems. Theorem 3: Synchronization22 Consider two coupled systems. If the dyna... |