# Geometric Adaptive Control of Quadrotor UAVs Transporting a Cable-Suspended Rigid Body

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8 D Ωi L = J i Ω i (47) D x L = m T ge 3 (48) D qi L = m i l i ge 3 (49) where J = J n m i ˆρ i. The variation of a rotation matrix is represented by δr j = R j ˆη j for η j R 3. Using this the derivative of the Lagrangian with respect to R j can be written as D R L δr = m i R ˆη ˆΩ ρ i (ẋ i l i q i ) + m i ge 3 R ˆη ρ i = m i { ˆΩ ρ i R T (ẋ l i q i ) + g ˆρ i R T e 3 } η d R L η (5) where d R L (R 3 ) R 3 is referred to as left-trivialized derivatives. Substituting δr j = R j ˆη j into the attitude kinematic equations () and rearranging the variation of the angular velocity can be written as δω j = η j + Ω j η j. For the variation model of q i given at (9) we have δq i = ξ i q i and ξ i = ξ i q i + ξ i q i. b) Lagrange-d Alembert Principle: Let G = t f t L dt be the action integral. Using the above equations the infinitesimal variation of the action integral can be written as tf δg = Dẋ L δẋ + D x L δx t + D Ω L ( η + Ω η ) + d R L η + D qi L ( ξ i q i + ξ i q i ) + D qi L (ξ i q i ) + D Ωi L ( η i + Ω i η i ) dt. The total control thrust at the i-th quadrotor with respect to the inertial frame is denoted by u i = f i R i e 3 R 3 and the total control moment at the i-th quadrotor is defined as M i R 3. There exist the disturbances x R for the payload and the disturbances xi Ri for the i-th quadrotor. The corresponding virtual work can be written as tf δw = (u i + xi ) {δx + R ˆη ρ i l i ξ i q i } + t (M i + Ri ) η i + x δx + R η dt. According to Lagrange-d Alembert principle we have δg = δw for any variation of trajectories with fixed end points. By using the integration by parts and rearranging we obtain the following Euler-Lagrange equations: d dt D ẋ L D x L = x + (u i + xi ) d dt D Ω L + Ω D Ω L d R L = R + We have µ id = q id qi T d µ id from (3). Using this the error term ˆρ i R T (u i + xi ) can be written in terms of e qi as Y x = n (qi T m d µ id ){(qi T q id )q i q id } ˆq i d dt D q i L ˆq i D qi L = l iˆq i (u i + xi ) d dt D Ω i L + Ω i D Ωi L = M i + Ri. Substituting (44)-(5) into these and rearranging by the fact that q i = ˆq i ω i ω i q i and ˆq i q i = ˆq i ω i = ω i the equations of motion can be rewritten as m T ẍ + m i ( R ˆρ i Ω + l iˆq i ω i ) + +m i l i ω i q i = m T ge 3 + x + J Ω + m i R ˆΩ ρ i (u i + xi ) (5) m i ˆρ i R T (ẍ + l iˆq i ω i + l i ω i q i ) + ˆΩ J Ω = R + ˆρ i R T (u i + xi + m i ge 3 ) (5) m i l i ω i m iˆq i ẍ + m iˆq i R ˆρ i Ω m iˆq i R ˆΩ ρ i = ˆq i (u i + xi + m i ge 3 ) (53) J i Ωi + Ω i J i Ω i = M i + Ri (54) where m T = m + n m i R 3 and J = J n m i ˆρ i R3 3. Next we substitute (53) into (5) and (5) to eliminate the dependency of ω i in the expressions for ẍ and Ω. Using the fact that I + ˆq i = q i q T i for any q i S and ˆΩ ˆρ i Ω = ˆρ i ˆΩ ρ i for any Ω ρ i R 3 we obtain () and () after rearrangements and simplifications. It is straightforward to see that (53) is equivalent to (3). B. Proof of Proposition c) Error Dynamics: From () and (3) the dynamics of the position tracking error is given by m ë x = m (ge 3 ẍ d ) + x + q i qi T µ id + x i. From (8) and (6) this can be rearranged as ë x = ge 3 ẍ d + m F d + Y x + m ( x + = k x e x kẋ ė x + m ( x + x i ) x i ) + Y x (55) where x i = x i x i R 3 corresponds to the parallel component of the estimation error. The last term Y x R 3 represents the error caused by the difference between q i and q id and it is given by Y x = n (q i qi T I)µ id. m

9 = n (qi T m d µ id )ˆq i e qi. Using (9) an upper bound of Y x can be obtained as Y x n µ id e qi m γ( F d + M d ) e qi where γ =. From (6) and (7) this can be m λmpp T further bounded by Y x {β(k x e x + kẋ ė x ) + γ(k R e R + k Ω e Ω ) + B} e qi (56) where β = m γ and the constant B is determined by the given desired trajectories of the payload and (6) which defines the domain D of the error variables that the presented stability proof is considered. Throughout the remaining parts of the proof any bound that can be obtained from x d R d or (6) is denoted by B for simplicity. In short the position tracking error dynamics of the payload can be written as (55) where the error term is bounded by (56). Similarly we find the attitude tracking error dynamics for the payload as follows. Using () (7) and (3) the timederivative of J e Ω can be written as J ė Ω = (J e Ω + d) e Ω k R e R k Ω e Ω + R + ˆρ i R T x i + Y R (57) where d = (J trj I)R T R d Ω d R 3 3 that is bounded and R R 3 denotes the estimation error given by R = R ˆ R. The error term in the attitude dynamics of the payload namely Y R R 3 is given by Y R = ˆρ i R T (q i qi T I)µ id = ˆρ i R T (qi T d µ id )ˆq i e qi. Similar with (56) an upper bound of Y R can be obtained as Y R {δ i (k x e x + kẋ ė x ) + σ i (k R e R + k Ω e Ω ) + B} e qi (58) ˆρ i where δ i = m σ λmpp T i = δi m R. Next from (34) the time-derivative of the angular velocity error projected on to the plane normal to q i is given as ˆq i ė ωi = ω + (q ω d ) q + ˆq ω d = k q e qi k ω e ωi m i l i ˆq i xi. (59) In summary the error dynamics of the simplified dynamic model are given by (55) (57) and (59). d) Stability Proof: Define an attitude configuration error function Ψ R for the payload as Ψ R = tr I R T d R which is positive-definite about R = R d and Ψ R = e R e Ω 9 3. We also introduce a configuration error function Ψ qi for each link that is positive-definite about q i = q id as Ψ qi = q i q id. For positive constants e xmax ψ R ψ qi B δ R consider the following open domain containing the zero equilibrium of tracking error variables: D = {(e x ė x e R e Ω e qi e ωi x R xi ) (R 3 ) 4 (R 3 R 3 ) n (R 3 ) R 3n e x < e xmax Ψ R < ψ R < Ψ qi < ψ qi < x < B δ R < B δ xi < B δ }. (6) In this domain we have e R = Ψ R ( Ψ R ) ψr ( ψ R ) α < and e qi = Ψ qi ( Ψ qi ) ψqi ( ψ qi ) α i <. It is assumed that ψ qi is sufficiently small such that nα i β <. We can show that the configuration error functions are quadratic with respect to the error vectors in the sense that e R Ψ R e R ψ R e q i Ψ qi e qi ψ qi where the upper bounds are satisfied only in the domain D. Define V = ė x + k x e x + c x e x ė x + e Ω J Ω + k R Ψ R + c R e R J e Ω + e ω i + k q Ψ qi + c q e qi e ωi where c x c R c q are positive constants. This is composed of tracking error variables only and we define another function for the estimation errors of the adaptive laws as V a = h x x + h R R + h xi xi. The Lyapunov function for the complete simplified dynamic model is chosen as V = V + V a. Let z x = e x ė x T z R = e R e Ω T z qi = e qi e ωi T R. The first part of the Lyapunov function V satisfies zx T P x z x + zr T P R z R + zq T i P qi z qi V z T x P x z x + z T R P R z R + zq T i P qi z qi

10 where the matrices P x P R P qi P x P R P qi R are given by P x = kx c x c x P R = kr c R λ c R λ λ kq c q c q P qi = P x = P R = P qi = kx c x c x kr ψ R c R λ kq ψ qi c R λ λ c q c q where λ = λ m J and λ = λ M J. If the constants c x c R c q are sufficiently small all of the above matrices are positive-definite. As the second part of the Lyapunov function V a is already given as a quadratic form it is straightforward to see that the complete Lyapunov function V is positive-definite and decrescent. The time-derivative of the Lyapunov function along the error dynamics (55) (57) and (59) is given by V = (kẋ c x ) ė x c x k x e x c x kẋ e x ė x + (c x e x + ė x ) Y x k Ω e Ω + c R ė R J e Ω c R k R e R + c R e R ((J e Ω + d) e Ω k Ω e Ω ) + (e Ω + c R e R ) Y R + (k ω c q ) e ωi c q k q e qi c q k ω e qi e ωi + m (ė x + c x e x ) ( x + + (e Ω + c R e R ) ( R + x i ) ˆρ i R T x i ) ( n ) ˆq i (e ωi + c q e qi ) m i l xi i R R h R x x h x h xi ( x i xi + x i xi ) (6) where the last term has been obtained using the facts that x i xi = x i xi =. Substituting the adaptive laws given by (36) (37) and (38) into (6) the expressions at the last four lines of (6) that are dependent of estimation errors vanish. An upper bound of the remaining expressions of V can be obtained as follows. Since e R ė R e Ω and d B V (kẋ c x ) ė x c x k x e x c x kẋ e x ė x + (c x e x + ė x ) Y x (k Ω c R λ) e Ω c R k R e R + c R (k Ω + B) e R e Ω + (e Ω + c R e R ) Y R + (k ω c q ) e ωi c q k q e qi c q k ω e qi e ωi. (6) From (56) an upper bound of the fourth term of the right-hand side is given by (c x e x + ė x ) Y x α i β(c x k x e x + c x kẋ e x ė x + kẋ ė x ) + {c x B e x + (βk x e xmax + B) ė x } e qi + α i γ(c x e x + ė x )(k R e R + k Ω e Ω ). (63) Similarly using (58) (c R e R + e Ω ) Y R α i σ i (c R k R e R + c R k Ω e R e Ω + k Ω e Ω ) + {c R B e R + (α σ i k R + B) e Ω } e qi + α i δ i (c R e R + e Ω )(k x e x + kẋ ė x ). (64) Substituting these into (6) and rearranging V zi T W i z i (65) where z i = z x z R z qi T R 3 and the matrix W i R 3 3 is defined as λ m W xi W xr i W xq i W i = W xr i λ m W Ri W Rq i (66) W xq i W Rq i λ m W qi where the sub-matrices are given by W xi = c x k x ( nα i β) cxkẋ ( + nα i β) n cxkẋ ( + nα i β) kẋ ( nα i β) c x W Ri = c R k R ( nα i σ i ) c R (k Ω + B + nα i σ i ) n c R (k Ω + B + nα i σ i ) k Ω ( nα i σ i ) c R λ c W qi = q k q cqkω cqkω k ω c q γcx k W xri = α R + δ i c R k x γc x k Ω + δ i k x i γk R + δ i c R kẋ γk Ω + δ i kẋ W xqi = c x B βk x e xmax + B W xri = c R B α σ i k R + B If the constants c x c R c q that are independent of the control input are sufficiently small the matrices W xi W Ri W qi are positive-definite. Also if the error in the direction of the link is sufficiently small relative to the desired trajectory we can choose the controller gains such that the matrix W i is positive-definite which follows that the zero equilibrium of tracking errors is stable in the sense of Lyapunov and all of the tracking error variables z i and the estimation error variables are uniformly bounded i.e. e x ė x e R e Ω e qi e ωi x R xi L. These also imply that e x ė x e R e Ω e qi e ωi L from (65) and that ė x ë x ė R ė Ω ė qi ė ωi L. According to Barbalat s lemma all of the tracking error variables e x ė x e R e Ω e qi e ωi and their time-derivatives asymptotically converge to zero..

12 M. Shuster Survey of attitude representations Journal of the Astronautical Sciences vol. 4 pp T. Lee Computational geometric mechanics and control of rigid bodies Ph.D. dissertation University of Michigan 8. T. Lee M. Leok and N. McClamroch Lagrangian mechanics and variational integrators on two-spheres International Journal for Numerical Methods in Engineering vol. 79 no. 9 pp F. Goodarzi D. Lee and T. Lee Geometric nonlinear PID control of a quadrotor UAV on SE(3) in Proceedings of the European Control Conference Zurich July 3 pp F. Bullo and A. Lewis Geometric control of mechanical systems ser. Texts in Applied Mathematics. New York: Springer-Verlag 5 vol. 49 modeling analysis and design for simple mechanical control systems. 5 T. Wu Spacecraft relative attitude formation tracking on SO(3) based on line-of-sight measurements Master s thesis The George Washington University. 6 H. Khalil Nonlinear Systems nd Edition Ed. Prentice Hall S. Bhat and D. Bernstein Finite-time stability of continuous autonomous systems SIAM Journal of Control and Optimization vol. 38 no. 3 pp S. Yu X. Yu B. Shirinzadeh and Z. Man Continuous finite-time control for robotic manipulators with terminal slideing mode Automatica vol. 4 pp S. Wu G. Radice Y. Gao and Z. Sun Quaternion-based finite time control for spacecraft attitude tracking Acta Astronautica vol. 69 pp P. Ioannou and J. Sung Robust Adaptive Control. Prentice Hall 995.

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