# Tracking Control of Nanosatellites with Uncertain Time Varying Parameters

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Published on March 21, 2014

Author: BelindaMarchand

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Talk from AAS/AIAA Astrodynamics Specialist Conference, held July 31 - August 4, 2011 in Girdwood, Alaska.

Problem Statement Control Formulation Conclusions Tracking Control of Nanosatellites with Uncertain Time Varying Parameters Divya Thakur 1 and Belinda G. Marchand 2 Department of Aerospace Engineering and Engineering Mechanics The University of Texas at Austin AAS/AIAA Astrodynamics Specialist Conference July 31 - August 4, 2011 Girdwood, Alaska 1 Graduate Student, Department of Aerospace Engineering 2 Assistant Professor, Department of Aerospace Engineering, AIAA Associate Fellow Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 1/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Motivation ◮ Spacecraft tracking problem is widely studied. ◮ Many adaptive control solutions for systems with constant uncertain inertia parameters. ◮ Limited research in adaptive control of time-varying inertia matrix. ◮ Focus of study: Adaptation mechanism that maintains consistent tracking performance in the face of uncertain time-varying inertia matrix. Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 2/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Attitude-Tracking Error Dynamics ◮ Attitude-error dynamics: ˙qe0 = − 1 2 qT ev ωe ˙qev = 1 2 qe0 I + qev × ωe Angular-velocity tracking error dynamics: ˙ωe = J−1 −˙Jω − [ω×]Jω + u + [ωe×]B CR (qe)ωr − B CR (qe) ˙ωr ◮ Control objective: Find u(t) s.t. limt→∞ qe, ωe = 0 for any [qr(t), ωr(t)] for all [q(0), ω(0)], assuming full feedback [q(t), ω(t)] and uncertainty in J(t). Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Attitude-Tracking Error Dynamics ◮ Attitude-error dynamics: ˙qe0 = − 1 2 qT ev ωe ˙qev = 1 2 qe0 I + qev × ωe Angular-velocity tracking error dynamics: ˙ωe = J−1 −˙Jω − [ω×]Jω + u + [ωe×]B CR (qe)ωr − B CR (qe) ˙ωr ◮ Control objective: Find u(t) s.t. limt→∞ qe, ωe = 0 for any [qr(t), ωr(t)] for all [q(0), ω(0)], assuming full feedback [q(t), ω(t)] and uncertainty in J(t). Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Attitude-Tracking Error Dynamics ◮ Attitude-error dynamics: ˙qe0 = − 1 2 qT ev ωe ˙qev = 1 2 qe0 I + qev × ωe Angular-velocity tracking error dynamics: ˙ωe = J−1 −˙Jω − [ω×]Jω + u + [ωe×]B CR (qe)ωr − B CR (qe) ˙ωr ◮ Control objective: Find u(t) s.t. limt→∞ qe, ωe = 0 for any [qr(t), ωr(t)] for all [q(0), ω(0)], assuming full feedback [q(t), ω(t)] and uncertainty in J(t). Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Attitude-Tracking Error Dynamics ◮ Attitude-error dynamics: ˙qe0 = − 1 2 qT ev ωe ˙qev = 1 2 qe0 I + qev × ωe Angular-velocity tracking error dynamics: ˙ωe = J−1 −˙Jω − [ω×]Jω + u + [ωe×]B CR (qe)ωr − B CR (qe) ˙ωr ◮ Control objective: Find u(t) s.t. limt→∞ qe, ωe = 0 for any [qr(t), ωr(t)] for all [q(0), ω(0)], assuming full feedback [q(t), ω(t)] and uncertainty in J(t). Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 3/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Type of Inertia Matrix Considered ◮ Time-varying inertia matrix of the form J(t) = JoΨ(t) ◮ Jo: Jo > 0, JT o = Jo constant, unknown or uncertain ◮ Ψ(t): Ψ > 0, ΨT = Ψ time-varying, known ◮ Uncertainty itself is constant, multiplicative ◮ May be used to model spacecraft undergoing 1. Thermal Variations 2. Fuel slosh 3. Appendage deployment (sensor booms, solar sails, antennas, etc.) Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 4/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Example of a Time-Varying Inertia Matrix (1/2) ◮ Consider a spacecraft undergoing boom deployment (e.g., GOES-R spacecraft): ◮ Boom extension rate controlled by miniature DC-torque motors ◮ Initial mass of prism: m0 ◮ Mass of fully extended boom: αm0, 0 < α < 1 DEPLOYED BOOM 12l SENSOR SATELLITE MAIN BODY 1l 13l 1l STOWED COLLABPSIBLE BOOM SATELLITE MAIN BODY STOWED BOOM Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 5/ 18

Problem Statement Control Formulation Conclusions Motivation Error Dynamics Modeling a Time-Varying Inertia Matrix Example of a Time-Varying Inertia Matrix (2/2) ◮ Rod length, rod mass, and prism mass are (respectively) r(t) = 2l1 τ , mp(t) = αm0 τ t, mc(t) = m0 − 2mp(t) ◮ Inertia matrix given by Jo =   5 6 m0l2 1 0 0 5 6 m0l2 1 0 0 0 1 6 m0l2 1   For 0 ≤ t ≤ τ, Ψ(t) =   1 − 2α τ t 0 0 0 1 − 7 5 α τ t + 12 5 α τ2 t2 + 16 5 α τ3 t3 0 0 0 1 + α τ t + 12 α τ2 t2 + 16 α τ3 t3   , for t > τ, Ψ(t) =   1 − 2α 0 0 0 1 − 7 5 α + 12 5 α + 16 5 α 0 0 0 1 + α + 12α + 16α   . Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 6/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Control Formulation ◮ Control method based on the non-certainty equivalence (non-CE) adaptive control results of Seo and Akella (2008)3 . ◮ Provides superior performance over traditional CE based methods when reference trajectory does not satisfy certain persistence of excitation (PE) conditions. ◮ Original result treats constant inertia matrix. ◮ Present investigation modiﬁes original result to handle time-varying inertia matrix of the speciﬁc form J(t) = JoΨ(t). 3 Seo, D. and Akella, M. R., High-Performance Spacecraft Adaptive Attitude-Tracking Control Through Attracting-Manifold Design, Journal of Guidance, Control, and Dynamics, Vol. 31, No. 4, 2008, pp. 884–891 Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 7/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-CE Adaptive Controller ◮ For problem described by tracking-error equations and inertia matrix J = JoΨ(t), control input is u = Ψ −W ˆθ + δ + Wf ΓWT f kp(qev − ωef ) + ωe ˙ˆθ = ΓWT f (β + kv)ωef + kpqev − ΓWT ωef δ = ΓWT f ωef , ◮ Regressor matrix Wθ∗ = −Ψ−1 Jo ˙Ψω − Ψ−1 [ω×]JoΨω + Jo [ω×]B CR (qe)ωr − B CR (qe) ˙ωr + Jo kpβqev + kp ˙qev + kvωe , ◮ Parameters: θ∗ = [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T . ◮ Filter variables ˙ωef = −βωef + ωe ˙Wf = −βWf + W, Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-CE Adaptive Controller ◮ For problem described by tracking-error equations and inertia matrix J = JoΨ(t), control input is u = Ψ −W ˆθ + δ + Wf ΓWT f kp(qev − ωef ) + ωe ˙ˆθ = ΓWT f (β + kv)ωef + kpqev − ΓWT ωef δ = ΓWT f ωef , ◮ Regressor matrix Wθ∗ = −Ψ−1 Jo ˙Ψω − Ψ−1 [ω×]JoΨω + Jo [ω×]B CR (qe)ωr − B CR (qe) ˙ωr + Jo kpβqev + kp ˙qev + kvωe , ◮ Parameters: θ∗ = [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T . ◮ Filter variables ˙ωef = −βωef + ωe ˙Wf = −βWf + W, Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-CE Adaptive Controller ◮ For problem described by tracking-error equations and inertia matrix J = JoΨ(t), control input is u = Ψ −W ˆθ + δ + Wf ΓWT f kp(qev − ωef ) + ωe ˙ˆθ = ΓWT f (β + kv)ωef + kpqev − ΓWT ωef δ = ΓWT f ωef , ◮ Regressor matrix Wθ∗ = −Ψ−1 Jo ˙Ψω − Ψ−1 [ω×]JoΨω + Jo [ω×]B CR (qe)ωr − B CR (qe) ˙ωr + Jo kpβqev + kp ˙qev + kvωe , ◮ Parameters: θ∗ = [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T . ◮ Filter variables ˙ωef = −βωef + ωe ˙Wf = −βWf + W, Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-CE Adaptive Controller ◮ For problem described by tracking-error equations and inertia matrix J = JoΨ(t), control input is u = Ψ −W ˆθ + δ + Wf ΓWT f kp(qev − ωef ) + ωe ˙ˆθ = ΓWT f (β + kv)ωef + kpqev − ΓWT ωef δ = ΓWT f ωef , ◮ Regressor matrix Wθ∗ = −Ψ−1 Jo ˙Ψω − Ψ−1 [ω×]JoΨω + Jo [ω×]B CR (qe)ωr − B CR (qe) ˙ωr + Jo kpβqev + kp ˙qev + kvωe , ◮ Parameters: θ∗ = [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T . ◮ Filter variables ˙ωef = −βωef + ωe ˙Wf = −βWf + W, Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-CE Adaptive Controller ◮ For problem described by tracking-error equations and inertia matrix J = JoΨ(t), control input is u = Ψ −W ˆθ + δ + Wf ΓWT f kp(qev − ωef ) + ωe ˙ˆθ = ΓWT f (β + kv)ωef + kpqev − ΓWT ωef δ = ΓWT f ωef , ◮ Regressor matrix Wθ∗ = −Ψ−1 Jo ˙Ψω − Ψ−1 [ω×]JoΨω + Jo [ω×]B CR (qe)ωr − B CR (qe) ˙ωr + Jo kpβqev + kp ˙qev + kvωe , ◮ Parameters: θ∗ = [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T . ◮ Filter variables ˙ωef = −βωef + ωe ˙Wf = −βWf + W, Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-CE Adaptive Controller ◮ For problem described by tracking-error equations and inertia matrix J = JoΨ(t), control input is u = Ψ −W ˆθ + δ + Wf ΓWT f kp(qev − ωef ) + ωe ˙ˆθ = ΓWT f (β + kv)ωef + kpqev − ΓWT ωef δ = ΓWT f ωef , ◮ Regressor matrix Wθ∗ = −Ψ−1 Jo ˙Ψω − Ψ−1 [ω×]JoΨω + Jo [ω×]B CR (qe)ωr − B CR (qe) ˙ωr + Jo kpβqev + kp ˙qev + kvωe , ◮ Parameters: θ∗ = [Jo11 , Jo12 , Jo13 , Jo22 , Jo23 , Jo33 ]T . ◮ Filter variables ˙ωef = −βωef + ωe ˙Wf = −βWf + W, Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 8/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Numerical Simulations ◮ Two sets of simulations: 1. Non-PE reference trajectory 2. PE reference trajectory ◮ Simulation features ◮ Quantities used to calculate Jo and Ψ m0 = 30 kg, l = 0.2 m, α = 0.1, τ = 200 s ◮ Uncertain parameter Jo =   0.2 0 0 0.2 0 0 0 1.0   −→ θ∗ = [0.2, 0, 0, 0.2, 0, 1.0]T ◮ Initial parameter estimate: ˆθ(0) + δ(0) = 1.3 θ∗ ◮ Simulation period is τ = 200 seconds. Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 9/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-PE Reference Trajectory (1/3) ◮ ωr = 0.1 cos(t)(1 − e0.01t2 ) + (0.08π + 0.006 sin(t))te−0.01t2 · [1, 1, 1]T ◮ Gain values kp = 0.08, kv = 0.07, Γ = diag {100, 0.01, 0.01, 200, 0.01, 100}. 0 50 100 150 200 1e−007 1e−006 1e−005 0.0001 0.001 0.01 0.1 time (s) AngularVel.ErrorVectorNorm Norm of angular velocity error vector ωe 0 50 100 150 200 1e−007 1e−006 1e−005 0.0001 0.001 0.01 0.1 1 time (s) QuaternionErrorVectorNorm Norm of quaternion error vector qev Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 10/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-PE Reference Trajectory (2/3) 10 −2 10 −1 10 0 10 1 10 2 10 3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 time(s) ControlTorqueNorm(N−m) Norm of control vector u Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 11/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations Non-PE Reference Trajectory (3/3) 10 −1 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 time(s) J 0 Parameters J o (3,3) J o (1,1) = J o (2,2) Estimated True Parameter estimates converge to true values due to additional persistence of excitation introduced by Ψ(t) Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 12/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations PE Reference Trajectory (1/3) ◮ ωr = cos(t) + 2 5 cos(t) sin(t) + 2 T ◮ Gain values: kp = 0.8, kv = 0.8, Γ = diag {1, 0.001, 0.001, 1, 0.001, 1}. 0 50 100 150 200 0.0001 0.001 0.01 0.1 1 10 time (s) AngularVel.ErrorVectorNorm Norm of angular velocity error vector ωe 0 50 100 150 200 0.0001 0.001 0.01 0.1 1 time (s) QuaternionErrorVectorNorm Norm of quaternion error vector qev Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 13/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations PE Reference Trajectory (2/3) 10 −2 10 −1 10 0 10 1 10 2 10 3 0 2 4 6 8 10 12 time(s) ControlTorqueNorm(N−m) Norm of control vector u Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 14/ 18

Problem Statement Control Formulation Conclusions Adaptive Control Numerical Simulations PE Reference Trajectory (3/3) 10 −1 10 0 10 1 10 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 time(s) J 0 Parameters J o (3,3) J o (1,1) = J o (2,2) Estimated True Parameter estimates converge to true values Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 15/ 18

Problem Statement Control Formulation Conclusions Conclusions ◮ A non-CE adaptive control law employed for spacecraft attitude tracking in the presence of uncertain time-varying inertia matrix. ◮ Uncertainty has special multiplicative structure. ◮ Numerical simulations performed for PE and non-PE reference signals. ◮ Attitude and angular-velocity tracking errors converge to zero. ◮ Parameter estimates converge to true values even when reference signal is non-PE. Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 16/ 18

Problem Statement Control Formulation Conclusions Extra 1: Some Necessary Manipulations ◮ The following algebraic manipulations are necessary to enable the adaptive control derivation ˙ωe = −kpβqev − kp ˙qev − kvωe subtracted term + J−1 o   Ψ−1 u − Jo ˙Ψω − [ω×]JoΨω − Joφ + Jo kpβqev + kp ˙qev + kvωe added term    , where φ = [ωe×]B CR (qe)ωr − B CR (qe) ˙ωr ◮ kp, kv > 0 and β = kp + kv ◮ Note: Dynamics are unchanged Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 17/ 18

Problem Statement Control Formulation Conclusions Extra 2: Initial Conditions for Simulations ◮ Initial conditions q(0) = 0.9487, 0.1826, 0.1826, 0.18268 T ω(0) = 0, 0, 0 T rad/s qr(0) = [1, 0, 0, 0]T Wf (0) = 0 , ωf (0) = ωe(0) + kpqve (0) kp ◮ Initial ﬁlter-states1 are Wf (0) = 0 and ωf (0) = ωe(0)+kpqve (0) kp . Thakur, Marchand Tracking Control for Uncertain Time-Varying Matrix 18/ 18

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