Nested Saturation Control of an UAV Carrying a ... - Semantic Scholar
2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA
Nested Saturation Control of an UAV Carrying a Suspended Load Marco M. Nicotra1 , Emanuele Garone1 , Roberto Naldi2 , Lorenzo Marconi2
Abstract— This paper addresses the problem of using unmanned aerial vehicles for the transportation of suspended loads. The proposed solution introduces a novel control law capable of steering the aerial robot to a desired reference while simultaneously limiting the sway of the payload. The stability of the equilibrium is proven rigorously through the application of the nested saturation formalism. Numerical simulations demonstrating the effectiveness of the controller are provided.
I. INTRODUCTION Recent advancements in the field of Unmanned Aerial Vehicles (UAVs) have lead to the availability of inexpensive aerial robots which can be deployed in a wide range of applications. The ability to fly over obstacles and maneuver in confined spaces makes quadrotors ideal candidates for the rapid transportation of light payloads. Early works in this field propose UAVs equipped with graspers [1]. However, the main drawback of this solution is that it lows down the attitude dynamics due to the increased inertia of the vehicle. An alternative solution is to suspend the load with the aid of a cable. This solution preserves the attitude response of the vehicle, but has the disadvantage of introducing an additional degree of freedom represented by the swinging motion of the cargo. This paper focuses on this second configuration and presents a novel control law capable of controlling the UAV while reducing the sway of the suspended load. This control problem is somewhat similar to the control of an overhead crane, which has been widely investigated in literature. Proposed control strategies for the overhead crane can be split in two categories. The first focuses on the generation of swing-free trajectories [2], [3]. The second is centered on the stabilization of the swinging dynamics. Possible schemes for this second category include cascade controllers [4], feedback linearization [5], Lyapunov-based controllers [6], and nested saturation controllers [7]. As for the specific case of controlling an UAV carrying a suspended mass, some solutions have been proposed. In [8], the load oscillations are dampened by pre-filtering the reference of the UAV. In [9], the oscillating load is decoupled from the UAV dynamics with the aid of a force sensor and then dampened using a linearized state-feedback controller. In [10] the authors present two different approaches. The first stabilizes the UAV through feedback linearization and adaptive control: the feedback linearization is designed using a model without swinging mass, the effects of which are *This work is supported by a FRIA scholarship grant and the FP7 European project SHERPA. 1 Universit´ e Libre de Bruxelles (ULB), Brussels, Belgium
(mnicotra,egarone)@ulb.ac.be 2 Univerisity
of
Bologna,
Bologne,
(roberto.naldi,lorenzo.marconi)@ulb.ac.be 978-1-4799-3271-9/$31.00 ©2014 AACC
Italy
Fig. 1.
Two-dimensional model of an UAV carrying a suspended mass
compensated by the adaptive part of the controller. The second approach is based on the idea of generating swingfree trajectories by means of dynamic programming. Finally, in [11], the authors take advantage of the differential flatness of the system to impose arbitrary load trajectories. In this paper, a new control law for a quadrotor carrying a suspended mass is proposed based on the idea of nested saturation approach. The stability of the approach is proven rigorously by means of ISS arguments. The paper is organized as follows. First the nonlinear model of the system will be derived and its properties emphasized. Then the control strategy is described. The main idea is that the system can be decoupled into independent dynamic systems which are stabilized using a nested saturation controller. The stability properties of the controlled system will be detailed. Numerical simulations end the paper and show the practical effectiveness of the proposed control law. II. PROBLEM STATEMENT Consider an UAV carrying a suspended mass, as shown T in Figure 1. Let pM = [x, z] ∈ R2 describe the position of the center of mass of the UAV in the inertial reference frame and θ ∈ R be the angle between the horizon and the UAV. Moreover, let α ∈ R be the rotation angle between the gravity vector and the cable of length L connecting pM to the suspended mass. The vehicle is actuated by two propellers generating the forces f1 ∈ R≥ and f2 ∈ R≥ where R≥ = {R ≥ 0}. To simplify the notation, the system inputs are
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defined as u1 = f1 + f2 and u2 = (f1 − f2 ) b where b is the distance between the motors and the center of mass. Assumption 1: The cable is rigid, inextensible and massless. The effects of air viscosity are negligible. Under this assumption, the position and velocity of the suspended mass are sin α cos α pm = pM − L p˙m = p˙M − Lα˙ cos α − sin α
For the reader’s convenience, existing results that are necessary to this paper will be recalled.
and the total kinetic and potential energy are
A. Input to State Stability with Restrictions
K = 21 M p˙2M + 21 mp˙2m + 12 J θ˙2 P = (M + m) gz − mgL cos α. Due to the absence of dissipative forces, it is possible to define the Lagrangian function L = K − P. At this point, the dynamic model of the system can be obtained via the Euler-Lagrange theorem ∂L d ∂L − = Fi dt ∂ q˙i ∂qi
i = x, z, α, θ
1) lim [x (t) , z (t)] = [xR , zR ] t→∞ 2) lim α (t) = 0 t→∞
for all [z (t0 ) , z˙ (t0 ) , x (t0 ) , x˙ (t0 )] and for [α (t0 ) , α˙ (t0 )] belonging to a (possibly large) compact set Ωα . III. PRELIMINARIES
The following definition can be found in [14]. Definition 1: Consider a system x˙ = f (x, v) where x ∈ Rn denotes the state and v ∈ Rm is an exogenous input. Let the origin be an equilibrium point for v = 0 (i.e. f (0, 0) = 0). The system is Input to State Stable (ISS) with restrictions S ⊆ Rn on the initial state and restrictions vmax on the input if the exists a class-K function γ : R≥ ⇒ R≥ and a classKL function β : R≥ × R≥ ⇒ R≥ such that for all inputs kv (t)k < vmax and for all initial states x (0) ∈ S kx (t)k ≤ β (kx (0)k , t) + γ (kv (t)k)
where Fz = u1 cos θ
Fx = u1 sin θ
∀t ≥ 0.
B. Good Saturated Linear Controller
Fα = 0 Fθ = u2 .
Let σλ (t) be the saturation function λ if t > λ t if |t| ≤ λ . σλ (t) = −λ if t < −λ
Finally, the following model is obtained (M +m)(¨ z +g)+mL cos αα˙ 2 +sin αα ¨ = u1 cos θ (M +m)¨ x +mL sin αα˙ 2 −cos αα ¨ = u1 sin θ . 2 mL α ¨ +mL sin α¨ z −mL cos α¨ x +mgL sin α = 0 ¨ J θ = u2 (1) Remark 1: Note that, compared to the standard UAV model [12], the suspended mass introduces an additional pair of state variables (α, α). ˙ These variables are governed by the third equation of system (1), which can be re-written as g + z¨ 1 α ¨=− sin α + x ¨ cos α. L L Therefore, whenever the UAV is moving at constant speed, the dynamics of α(t) become g α ¨ = − sin α L meaning that the suspended mass behaves like an undampened pendulum. The equilibrium point α = π, addressed in [13], is not of interest in this paper. As for the equilibrium α = 0, note that 1 ¨. lim α ¨= x α→0 L Therefore, in proximity of the equilibrium, the pendulum dynamics are not affected by the vertical dynamics, but remain coupled with the horizontal ones. The objective of this paper is to develop a control law able to steer the UAV to a desired reference point while simultaneously limiting the oscillations of the suspended mass. More formally, this may be expressed as: Control Objectives: Given the system (1) and given a constant reference [xR , zR ], design a control law such that
The following theorem follows from [15] and [16]. Theorem 1: Given (A, B) a stabilizable and A such that there exists a P > 0 : AT P + P A ≤ 0, given the good saturated linear controller u = σλ (−Kx + d1 ) T
where K = B P , the system x˙ = Ax + Bσλ (−Kx + d1 ) + d2 is ISS with no restrictions on the initial state and with restrictions |d1 | ≤ ∆λ and |d2 | ≤ ∆λ where the coefficient ∆ is strictly positive. C. Nested Saturation Controller As proven in [16], a system in the feedforward form η˙ n = An ηn + γn (η1 , . . . , ηn−1 , u) .. . η ˙ = A2 η2 + γ2 (η1 , u) 2 η˙ 1 = A1 η1 + γ1 (u) controlled with a nested saturation controller has a globally asymptotically stable origin if all the matrices Ai are stable. The idea behind this control strategy is the following: Step 1: Subsystem ζ1 = η1 controlled with the good saturated controller u = σλ1 (−K1 η1 + v)
(2)
is ISS with no restrictions on the initial state and with restrictions ∆1 λ1 on the auxiliary input v. The saturation
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value λ1 is chosen in order to limit the nonlinearities of the function γ1 (u). η2 Step 2: Subsystem ζ2 = controlled with (2) and ζ1 v = σλ2 (−K2 ζ2 + w) is ISS with no restrictions on the initial state and with restrictions ∆2 λ2 on the auxiliary input w. The saturation value λ2 is chosen in order to guarantee the stability of the interconnected systems ζ1 and ζ2 using the small gain theorem. ηi Step 3: Define ζi = for i = 3 . . . n and iterate ζi−1 the second step. The ISS properties of ζi are proven for suitable saturation values λi . IV. SYSTEM DECOUPLING To attain the control objective, the first step will be to decouple the control into an inner loop, which uses u2 as a control input for the attitude θ(t), and an outer loop, which uses u1 and θ as control inputs for the remaining states. For the sake of simplicity, the inner loop dynamics will be neglected in this paper. Assumption 2: The attitude of the UAV is a control input of the system1 . The resulting system will be decoupled further by choosing u1 to control the vertical dynamics and θ to control the horizontal and pendulum dynamics. To do so, equation (1) can be rewritten as z¨ u1 cos θ−mL cos αα˙ 2 −(M +m)g (3) ¨ = u1 sin θ−mL sin αα˙ 2 M x α ¨ −mgL sin α where the mass matrix M is given by: M +m 0 mL sin α 0 M +m −mL cos α . M= mL sin α −mL cos α mL2
where fz (α, θ) = −g −
m fx (α, θ) = − L sin αα˙ 2 M +m m 1 cos θ+ (cos θ−cos (θ − 2α)) gz (α, θ) = M +m 2M 1 m gx (α, θ) = sin θ+ (sin θ+sin (θ − 2α)) M +m 2M 1 gα (α, θ) = sin (θ − α) . ML
which is always verified for |θ| < θmax = arccos
m . 2M + m
(5) (6) (7)
(10)
Therefore, the lower bound of gz (α, θ) is m m 1 1+ cos θmax − min {|g (α, θ)|} = M +m 2M 2M whereas the upper bound can be calculated as max {|g (α, θ)|} =
1 . M
V. VERTICAL DYNAMICS As discussed previously, the vertical dynamics z(t) will be controlled independently from the horizontal and pendulum dynamics x(t), α(t). The following proposition provides a control law for the vertical dynamics. Proposition 1: Let θ ∈ R such that |θ| < arccos 2Mm+m and let the control input u1 be u1 = −
fz (α, θ) + Rz (z) gz (α, θ)
(11)
with Rz (z) = −σµ1 (Kd z˙ + σµ2 (Kp (z − zR ))) ,
By multiplying both sides of equation (3) for the inverse2 of M, the following dynamic model is obtained z¨ = fz (α, θ) + gz (α, θ) u1 x ¨ = fx (α, θ) + gx (α, θ) u1 (4) α ¨ = gα (α, θ) u1 m ˙2 M +m L cos αα
The following result concerning gz (α, θ) can be proven: Lemma 1: For all θ ∈ R such that |θ| < arccos 2Mm+m , the function gz (α, θ) is bounded and strictly positive for any α. Proof: Because of (7), condition gz (α, θ) > 0 holds true if m m 1+ cos θ > cos (θ − 2α) 2M 2M
(12)
1 and µ1 > 0, µ2 > 2µ Kd , Kd > 0, and Kp > 0. The system z¨ = fz (α, θ) + gz (α, θ) u1 has a Globally Asymptotically Stable (GAS) equilibrium point in [z, z] ˙ = [zR , 0]. Proof: Define z1 = z − zR and z2 = z. ˙ By substituting the control law (11), the system z¨ = fz (α, θ) + gz (α, θ) u1 can be rewritten as z˙1 = z2 z˙2 = gz (t) Rz (z1 , z2 )
where gz (t) is a time-varying parameter which is bounded and strictly positive as a result of Lemma 1. The rest of the proof follows directly from [18]. Remark 2: Note that the proposed control law for the vertical dynamics is valid only if the control input θ (t) verifies (10). This condition will be accounted for in the following section.
(8) VI. HORIZONTAL AND PENDULUM DYNAMICS (9)
1 This assumption can be removed by studying the interconnection between the inner and outer loop dynamics, following the same lines of [17]. 2 The mass matrix is always symmetric and positive definite as a result of the Euler-Lagrange Theorem.
By substituting the control law (11) in the system (4), the remaining dynamics become ( (α,θ) + Rz (z) gx (α, θ) x ¨ = fx (α, θ) − fz (α, θ) ggxz (α,θ) , gα (α,θ) α ¨ = −fz (α, θ) gz (α,θ) + Rz (z) gα (α, θ)
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where Rz (z) is a bounded and time-vanishing exogenous signal. The system can be reformulated as x˙ 1 = A1 x1 + x2 x˙ 2 = A2 x2 + γx2 (α, θ, Rz ) (13) ˙ = γα (α, θ, Rz ) α α1 α2 where x1 = x − xR , x2 = x, ˙ αT = = α α˙ , A1 = A2 = 0, and
lα (α) = γα (α, 0) − Aα α hα (α, θ) = γα (α, θ) − γα (α, 0) − Bα θ. Since Aα is a critically stable matrix and (Aα , Bα ) are stabilizable, there exists a good saturated controller in the form θ = σλ1 (−K1 α + v) .
Following from Theorem 4, the system is ISS with restric(α,θ) +Rz (z)gx (α, θ) tions |v| ≤ ∆ λ and |l + h + g R | ≤ ∆ λ where the γx2 (α, θ, Rz ) = fx (α, θ)−fz (α, θ) ggxz (α,θ) 1 1 α α α z 1 1 " # α2 coefficient ∆1 is strictly positive. By defining q1 > 0, q2 > 0 γα (α, θ, Rz ) = . (α,θ) and q3 > 0 such that q1 + q2 + q3 = 1, the second restriction −fz (α, θ) ggαz (α,θ) + Rz (z) gα (α, θ) may be rewritten as As proven in Theorems 3 and 5 of reference [16], system |lα | + |hα | + |gα Rz | ≤ (q1 + q2 + q3 ) ∆1 λ1 (13) is asymptotically stable with restrictions on α (t0 ) if the following conditions are met: which, in turn, enables the expression of three separate 1) The origin is an equilibrium point of the system for inequalities θ = 0 and Rz = 0. |lα | ≤ q1 ∆1 λ1 2) A1 and A2 are marginally stable. |hα | ≤ q2 ∆1 λ1 3) The linearized system with θ as an input and Rz = 0 |gα Rz | ≤ q3 ∆1 λ1 . is stabilizable. 4) Given the inputs Rz and v, subsystem α is ISS Given (9) and (12), it follows that |gα | ≤ M1L and |Rz | ≤ µ1 . with restrictions on the initial state α (t0 ) and with Therefore, the restriction on Rz can be enforced by choosing restrictions on the input v. µ1 ≤ q3 M L∆1 λ1 . (15) The first two conditions follow from the fact that [z (t) , x (t) , α (t) , θ (t)] = [zR , xR , 0, 0] is an equilibrium Furthermore, lα (α) and hα (α, θ) are higher-order terms, point for system (1) and Ai = 0 is stable. The third condition meaning is verified due to the properties of the linearized system ∀l > 0, ∃δl > 0 : kαk ≤ δl ⇒ |lα (α)| ≤ l kαk (16) 0 0 1 0 0 ∀h > 0, ∃δh > 0 : |θ| ≤ δh ⇒ |hα (α, θ)| ≤ h |θ| . m M +m 0 0 0 −M g M g B= A= Since |θ| ≤ λ1 , the restriction on the higher-order term hα 0 0 0 1 0 g M +m g M +m is automatically satisfied by choosing 0 0 −L M 0 L M which is stabilizable for m > 0. As for the final condition, it can be proven that: ˙ = γα (α, θ, Rz ) and let θ = Proposition 2: Consider α σλ1 (−K1 α + v) with K1 = 0 β P1 , where P1 > 0 is such that 0 −β 0 P1 + P1 1 0 −β
λ1 ≤ δh (h ) .
(17)
As for the restriction on lα , they can be satisfied given l kαk < q1 ∆1 λ1 , kαk ≤ δl (l ) .
1 0
≤0
+m g ¯ and β = MM L . Given Rz bounded and 0 < λ1 ≤ λ where ¯ λ > 0 is computable, subsystem α is ISS with nonzero restrictions on the input v and nonzero restrictions on the initial state. ˙ = γα (α, θ, Rz ) can also be Proof: The subsystem α written as
˙ = Aα α+Bα θ+lα (α)+hα (α, θ)+gα (α, θ) Rz (z) (14) α where
h < q2 ∆1 ,
The objective is to find a forward invariant level set wholly contained in a ball of radius δl (l ). Theorem 4 proves the 2 existence of an ISS Lyapunov function k kαk ≤ V (α) ≤ 2 ¯ k kαk , where k and k are two positive values, and the existence of χ > 0 such that kαk ≥ χ∆1 λ1 ⇒ V˙ (α) ≤ 0. Therefore, there exists a forward invariant level set Ωα such that χ∆1 λ1 ⊆ Ωα ⊆ k/k χ∆1 λ1 . The restrictions on lα can then be enforced by choosing
∂γα 0 1 = Aα = −β 0 ∂α 0 ∂γα 0 Bα = = β ∂θ 0
l < λ1 ≤ 3588
kq1 , kχ
kδl (l ) . kχ∆1
(18)
By taking ¯ = min δh (h ) , kδl (l ) , λ kχ∆1
(19)
conditions (17) and (18) are verified. Therefore, given Rz ≤ ¯ subsystem α is ISS with restrictions on µ1 and 0 < λ1 ≤ λ, the initial state α ∈ Ωα and with restrictions v ≤ ∆1 λ1 . Remark 3: The proof of Proposition 9 is constructive ¯ Additionally, since equation (19) can be used to compute λ. since θ (t) must also verify the conditions discussed in Remark 8, the saturation value λ1 should be chosen as m ¯ arccos λ1 = min λ, . 2M + m The stability of the overall systems can then be guaranteed by choosing µ1 in accordance with (15). Remark 4: Please note that, although the results are only local in α, the nested saturation formalism guarantees global stability with respect to the states x1 and x2 . In addition, it is worth mentioning that the nonlinearities of system (16) are reasonably mild (i.e. δ () ). In practice, this implies that the restrictions on the initial conditions α ∈ Ωα ⊆ δl (l ), as also verified in simulation, are “sufficiently big” for the purpose of the application.
Fig. 2. Closed-loop behavior of the vertical dynamics in the first scenario. Subscripts A and B refer to the employed controller.
VII. SIMULATIONS Consider an UAV of mass M = 2 [kg] carrying a payload m = 0.5 [kg] attached to a cable of length L = 0.1 [m]. Define Controller A as the control law developed in this paper and Controller B as an alternative control law which compensates the effects of the pendulum dynamics and then proceeds on controlling the position of the UAV. Controller A: Following from the results obtained in this paper, Controller A is defined as follows x1 x θ = −σλ1 K1 α+σλ2 K2 2 +σλ3 K3 x2 α α u1 = −
Fig. 3. Closed-loop behavior of the horizontal dynamics in the first scenario.
fz (α, θ) + Rz (z) gz (α, θ)
where fz (α, θ) and gz (α, θ) are given in (5) and (7) whereas Rz (z) = −σµ1 (Kd z2 + σµ2 (Kp z1 )) . The control parameters have been chosen as K1 = −0.1030 0.1713 K2 = 0.0026 −0.4633 −0.0557 K3 = 0.0322 0.1374 1.7246 0.0670 , λ1 = 1.4595, λ2 = 0.3649, λ3 = 0.1216, Kd = 15, Kp = 56, µ1 = 12.2625, and µ2 = 18.3938. Controller B: Based on equation (1), the alternative control law imposes u1 and θ such that u1 cos θ = mL cos αα˙ 2 + sin αα ¨ + (M + m) (g + Rz (z)) u1 sin θ = mL sin αα˙ 2 − cos αα ¨ + (M + m) Rx (x) . This compensates the effects of the suspended load dynamics to obtain z¨ = Rz (z) and x ¨ = Rx (x). The regulators Rz (z) and Rx (x) have the same parameters as Controller A.
Fig. 4. Closed-loop behavior of the pendulum dynamics in the first scenario.
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VIII. CONCLUSIONS This paper describes a novel control law for stabilizing the dynamics of an UAV carrying a suspended load. The control law is based on the nested saturation approach and guarantees asymptotic stability without any restrictions on the position and velocity of the aerial vehicle. Restrictions on the initial state of the suspended load are present. However, simulations have shown that the basin of attraction is satisfactory for application purposes. Future works will aim at the rigorous characterization of the region of attractivity as well as the extension to the three-dimensional case. Experimental activity is foreseen to demonstrate the ability of the closedloop system to navigate in cluttered environments. R EFERENCES Fig. 5. Closed-loop behavior of the horizontal dynamics in the second scenario.
Fig. 6. Closed-loop behavior of the pendulum dynamics in the second scenario.
The pair of control laws are compared in two different scenarios. In the first scenario, the system is given a step variation of the reference coordinate xR followed by a step variation of zR . Figures 2-4 show the closed-loop dynamics obtained with the two control laws. Figure 2 shows that the vertical dynamics of the two systems are very similar as could be expected from the decoupling strategy. The horizontal dynamics are represented in Figure 3 where Controller B performs slightly better than Controller A. However, Figure 3 shows that Controller A outperforms Controller B for what regards the limitation of the suspended load sway. In the second scenario, the reference of the UAV remains stationary and the suspended mass is subject to an impact (approximated with a step variation of the angular velocity). Figures 5-6 provide the resulting closed-loop response. The vertical dynamics were omitted due to their limited interest. Controller B maintains the UAV stationary but does not dampen the pendulum swing. On the contrary, Controller A momentarily steers the UAV away from the reference point but is successful in dissipating the mechanical energy of the pendulum. This shows that the proposed control law is robust with respect to external disturbances acting on the pendulum.
[1] P. E. I. Pounds, D. R. Bersak, and A. M. Dollar, “Grasping from the air: Hovering capture and load stability,” in IEEE International Conference on Robotics and Automation (ICRA), May 2011. [2] D. Zamerosky, G. Starr, J. Wood, and R. Lumia, “Rapid swing-free transport of nonlinear payloads using dynamic programming,” Journal of Dynamic Systems, Measurement, and Control, vol. 130, no. 4, p. 041001, 2008. [3] J. Shultz and T. Murphey, “Trajectory generation for underactuated control of a suspended mass,” in IEEE International Conference on Robotics and Automation (ICRA), May 2012, pp. 123–129. [4] J. Yu, F. L. Lewis, and T. Huang, “Nonlinear feedback control of a gantry crane,” in American Control Conference (ACC), June 1995, pp. 4310–4315. [5] N. Yanai, M. Yamamoto, and A. Mohri, “Feedback control for wiresuspended mechanism with exact linearization,” in IEEE International Conference on Intelligent Robots and Systems, Oct. 2002, pp. 2213– 2218. [6] Y. Fang, W. E. Dixon, D. M. Dawson, and E. Zergeroglu, “Nonlinear coupling control laws for an underactuated overhead crane system,” IEEE/ASME Transactions on Mechatronics, vol. 8, no. 3, pp. 418–423, 2003. [7] R. Liu, S. Li, and S. Ding, “Nested saturation control for overhead crane systems,” Transactions of the Institute of Measurement and Control, vol. 34, no. 7, p. 862, 2012. [8] J. Potter, W. Singhose, and M. Costello, “Reducing swing of model helicopter sling load using input shaping,” in IEEE International Conference on Control and Automation (ICCA), Dec. 2011. [9] M. Bernard and K. Kondak, “Generic slung load transportation system using small size helicopters,” in IEEE International Conference on Robotics and Automation (ICRA), May 2009, pp. 3258–3264. [10] I. Palunko, P. Cruz, and R. Fierro, “Agile load transportation: Safe and efficient load manipulation with aerial robots,” IEEE Robotics & Automation Magazine, vol. 19, no. 3, pp. 69–79, 2012. [11] K. Sreenath, T. Lee, and V. Kumar, “Geometric control and differential flatness of a quadrotor uav with a cable-suspended load,” in IEEE Conference on Decision and Control (CDC), Dec. 2013. [12] M.-D. Hua, T. Hamel, P. Morin, and C. Samson, “Introduction to feedback control of underactuated vtol vehicles,” IEEE Control Systems Magazine(CSM), vol. 33, no. 1, pp. 61–75, 2013. [13] M. Hehn and R. D’Andrea, “A flying inverted pendulum,” in IEEE International Conference on Robotics and Automation (ICRA), 2011. [14] A. Isidori, Nonlinear Control Systems. Springer, 1995. [15] W. Liu, Y. Chitour, and E. Sontag, “On finite-gain stabilizability of linear systems subject to input saturation,” SIAM Journal on Control and Optimization, vol. 34, no. 4, pp. 1190–1219, 1996. [16] A. R. Teel, “A nonlinear small gain theorem for the analysis of control systems with saturation,” IEEE Transactions on Automatic Control, vol. 41, no. 9, pp. 1256–1270, 1996. [17] M. M. Nicotra, R. Naldi, and E. Garone, “Taut cable control of a tethered uav,” in International Federation of Automatic Control (IFAC) World Congress, Aug. 2014. [18] L. Marconi and A. Isidori, “Robust global stabilization of a class of uncertain feedforward nonlinear systems,” Systems and Control Letters, vol. 41, no. 4, pp. 281–290, 2000.
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Nested Saturation Control of an UAV Carrying a Suspended Load Marco M. Nicotra1 , Emanuele Garone1 , Roberto Naldi2 , Lorenzo Marconi2
Abstract— This paper addresses the problem of using unmanned aerial vehicles for the transportation of suspended loads. The proposed solution introduces a novel control law capable of steering the aerial robot to a desired reference while simultaneously limiting the sway of the payload. The stability of the equilibrium is proven rigorously through the application of the nested saturation formalism. Numerical simulations demonstrating the effectiveness of the controller are provided.
I. INTRODUCTION Recent advancements in the field of Unmanned Aerial Vehicles (UAVs) have lead to the availability of inexpensive aerial robots which can be deployed in a wide range of applications. The ability to fly over obstacles and maneuver in confined spaces makes quadrotors ideal candidates for the rapid transportation of light payloads. Early works in this field propose UAVs equipped with graspers [1]. However, the main drawback of this solution is that it lows down the attitude dynamics due to the increased inertia of the vehicle. An alternative solution is to suspend the load with the aid of a cable. This solution preserves the attitude response of the vehicle, but has the disadvantage of introducing an additional degree of freedom represented by the swinging motion of the cargo. This paper focuses on this second configuration and presents a novel control law capable of controlling the UAV while reducing the sway of the suspended load. This control problem is somewhat similar to the control of an overhead crane, which has been widely investigated in literature. Proposed control strategies for the overhead crane can be split in two categories. The first focuses on the generation of swing-free trajectories [2], [3]. The second is centered on the stabilization of the swinging dynamics. Possible schemes for this second category include cascade controllers [4], feedback linearization [5], Lyapunov-based controllers [6], and nested saturation controllers [7]. As for the specific case of controlling an UAV carrying a suspended mass, some solutions have been proposed. In [8], the load oscillations are dampened by pre-filtering the reference of the UAV. In [9], the oscillating load is decoupled from the UAV dynamics with the aid of a force sensor and then dampened using a linearized state-feedback controller. In [10] the authors present two different approaches. The first stabilizes the UAV through feedback linearization and adaptive control: the feedback linearization is designed using a model without swinging mass, the effects of which are *This work is supported by a FRIA scholarship grant and the FP7 European project SHERPA. 1 Universit´ e Libre de Bruxelles (ULB), Brussels, Belgium
(mnicotra,egarone)@ulb.ac.be 2 Univerisity
of
Bologna,
Bologne,
(roberto.naldi,lorenzo.marconi)@ulb.ac.be 978-1-4799-3271-9/$31.00 ©2014 AACC
Italy
Fig. 1.
Two-dimensional model of an UAV carrying a suspended mass
compensated by the adaptive part of the controller. The second approach is based on the idea of generating swingfree trajectories by means of dynamic programming. Finally, in [11], the authors take advantage of the differential flatness of the system to impose arbitrary load trajectories. In this paper, a new control law for a quadrotor carrying a suspended mass is proposed based on the idea of nested saturation approach. The stability of the approach is proven rigorously by means of ISS arguments. The paper is organized as follows. First the nonlinear model of the system will be derived and its properties emphasized. Then the control strategy is described. The main idea is that the system can be decoupled into independent dynamic systems which are stabilized using a nested saturation controller. The stability properties of the controlled system will be detailed. Numerical simulations end the paper and show the practical effectiveness of the proposed control law. II. PROBLEM STATEMENT Consider an UAV carrying a suspended mass, as shown T in Figure 1. Let pM = [x, z] ∈ R2 describe the position of the center of mass of the UAV in the inertial reference frame and θ ∈ R be the angle between the horizon and the UAV. Moreover, let α ∈ R be the rotation angle between the gravity vector and the cable of length L connecting pM to the suspended mass. The vehicle is actuated by two propellers generating the forces f1 ∈ R≥ and f2 ∈ R≥ where R≥ = {R ≥ 0}. To simplify the notation, the system inputs are
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defined as u1 = f1 + f2 and u2 = (f1 − f2 ) b where b is the distance between the motors and the center of mass. Assumption 1: The cable is rigid, inextensible and massless. The effects of air viscosity are negligible. Under this assumption, the position and velocity of the suspended mass are sin α cos α pm = pM − L p˙m = p˙M − Lα˙ cos α − sin α
For the reader’s convenience, existing results that are necessary to this paper will be recalled.
and the total kinetic and potential energy are
A. Input to State Stability with Restrictions
K = 21 M p˙2M + 21 mp˙2m + 12 J θ˙2 P = (M + m) gz − mgL cos α. Due to the absence of dissipative forces, it is possible to define the Lagrangian function L = K − P. At this point, the dynamic model of the system can be obtained via the Euler-Lagrange theorem ∂L d ∂L − = Fi dt ∂ q˙i ∂qi
i = x, z, α, θ
1) lim [x (t) , z (t)] = [xR , zR ] t→∞ 2) lim α (t) = 0 t→∞
for all [z (t0 ) , z˙ (t0 ) , x (t0 ) , x˙ (t0 )] and for [α (t0 ) , α˙ (t0 )] belonging to a (possibly large) compact set Ωα . III. PRELIMINARIES
The following definition can be found in [14]. Definition 1: Consider a system x˙ = f (x, v) where x ∈ Rn denotes the state and v ∈ Rm is an exogenous input. Let the origin be an equilibrium point for v = 0 (i.e. f (0, 0) = 0). The system is Input to State Stable (ISS) with restrictions S ⊆ Rn on the initial state and restrictions vmax on the input if the exists a class-K function γ : R≥ ⇒ R≥ and a classKL function β : R≥ × R≥ ⇒ R≥ such that for all inputs kv (t)k < vmax and for all initial states x (0) ∈ S kx (t)k ≤ β (kx (0)k , t) + γ (kv (t)k)
where Fz = u1 cos θ
Fx = u1 sin θ
∀t ≥ 0.
B. Good Saturated Linear Controller
Fα = 0 Fθ = u2 .
Let σλ (t) be the saturation function λ if t > λ t if |t| ≤ λ . σλ (t) = −λ if t < −λ
Finally, the following model is obtained (M +m)(¨ z +g)+mL cos αα˙ 2 +sin αα ¨ = u1 cos θ (M +m)¨ x +mL sin αα˙ 2 −cos αα ¨ = u1 sin θ . 2 mL α ¨ +mL sin α¨ z −mL cos α¨ x +mgL sin α = 0 ¨ J θ = u2 (1) Remark 1: Note that, compared to the standard UAV model [12], the suspended mass introduces an additional pair of state variables (α, α). ˙ These variables are governed by the third equation of system (1), which can be re-written as g + z¨ 1 α ¨=− sin α + x ¨ cos α. L L Therefore, whenever the UAV is moving at constant speed, the dynamics of α(t) become g α ¨ = − sin α L meaning that the suspended mass behaves like an undampened pendulum. The equilibrium point α = π, addressed in [13], is not of interest in this paper. As for the equilibrium α = 0, note that 1 ¨. lim α ¨= x α→0 L Therefore, in proximity of the equilibrium, the pendulum dynamics are not affected by the vertical dynamics, but remain coupled with the horizontal ones. The objective of this paper is to develop a control law able to steer the UAV to a desired reference point while simultaneously limiting the oscillations of the suspended mass. More formally, this may be expressed as: Control Objectives: Given the system (1) and given a constant reference [xR , zR ], design a control law such that
The following theorem follows from [15] and [16]. Theorem 1: Given (A, B) a stabilizable and A such that there exists a P > 0 : AT P + P A ≤ 0, given the good saturated linear controller u = σλ (−Kx + d1 ) T
where K = B P , the system x˙ = Ax + Bσλ (−Kx + d1 ) + d2 is ISS with no restrictions on the initial state and with restrictions |d1 | ≤ ∆λ and |d2 | ≤ ∆λ where the coefficient ∆ is strictly positive. C. Nested Saturation Controller As proven in [16], a system in the feedforward form η˙ n = An ηn + γn (η1 , . . . , ηn−1 , u) .. . η ˙ = A2 η2 + γ2 (η1 , u) 2 η˙ 1 = A1 η1 + γ1 (u) controlled with a nested saturation controller has a globally asymptotically stable origin if all the matrices Ai are stable. The idea behind this control strategy is the following: Step 1: Subsystem ζ1 = η1 controlled with the good saturated controller u = σλ1 (−K1 η1 + v)
(2)
is ISS with no restrictions on the initial state and with restrictions ∆1 λ1 on the auxiliary input v. The saturation
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value λ1 is chosen in order to limit the nonlinearities of the function γ1 (u). η2 Step 2: Subsystem ζ2 = controlled with (2) and ζ1 v = σλ2 (−K2 ζ2 + w) is ISS with no restrictions on the initial state and with restrictions ∆2 λ2 on the auxiliary input w. The saturation value λ2 is chosen in order to guarantee the stability of the interconnected systems ζ1 and ζ2 using the small gain theorem. ηi Step 3: Define ζi = for i = 3 . . . n and iterate ζi−1 the second step. The ISS properties of ζi are proven for suitable saturation values λi . IV. SYSTEM DECOUPLING To attain the control objective, the first step will be to decouple the control into an inner loop, which uses u2 as a control input for the attitude θ(t), and an outer loop, which uses u1 and θ as control inputs for the remaining states. For the sake of simplicity, the inner loop dynamics will be neglected in this paper. Assumption 2: The attitude of the UAV is a control input of the system1 . The resulting system will be decoupled further by choosing u1 to control the vertical dynamics and θ to control the horizontal and pendulum dynamics. To do so, equation (1) can be rewritten as z¨ u1 cos θ−mL cos αα˙ 2 −(M +m)g (3) ¨ = u1 sin θ−mL sin αα˙ 2 M x α ¨ −mgL sin α where the mass matrix M is given by: M +m 0 mL sin α 0 M +m −mL cos α . M= mL sin α −mL cos α mL2
where fz (α, θ) = −g −
m fx (α, θ) = − L sin αα˙ 2 M +m m 1 cos θ+ (cos θ−cos (θ − 2α)) gz (α, θ) = M +m 2M 1 m gx (α, θ) = sin θ+ (sin θ+sin (θ − 2α)) M +m 2M 1 gα (α, θ) = sin (θ − α) . ML
which is always verified for |θ| < θmax = arccos
m . 2M + m
(5) (6) (7)
(10)
Therefore, the lower bound of gz (α, θ) is m m 1 1+ cos θmax − min {|g (α, θ)|} = M +m 2M 2M whereas the upper bound can be calculated as max {|g (α, θ)|} =
1 . M
V. VERTICAL DYNAMICS As discussed previously, the vertical dynamics z(t) will be controlled independently from the horizontal and pendulum dynamics x(t), α(t). The following proposition provides a control law for the vertical dynamics. Proposition 1: Let θ ∈ R such that |θ| < arccos 2Mm+m and let the control input u1 be u1 = −
fz (α, θ) + Rz (z) gz (α, θ)
(11)
with Rz (z) = −σµ1 (Kd z˙ + σµ2 (Kp (z − zR ))) ,
By multiplying both sides of equation (3) for the inverse2 of M, the following dynamic model is obtained z¨ = fz (α, θ) + gz (α, θ) u1 x ¨ = fx (α, θ) + gx (α, θ) u1 (4) α ¨ = gα (α, θ) u1 m ˙2 M +m L cos αα
The following result concerning gz (α, θ) can be proven: Lemma 1: For all θ ∈ R such that |θ| < arccos 2Mm+m , the function gz (α, θ) is bounded and strictly positive for any α. Proof: Because of (7), condition gz (α, θ) > 0 holds true if m m 1+ cos θ > cos (θ − 2α) 2M 2M
(12)
1 and µ1 > 0, µ2 > 2µ Kd , Kd > 0, and Kp > 0. The system z¨ = fz (α, θ) + gz (α, θ) u1 has a Globally Asymptotically Stable (GAS) equilibrium point in [z, z] ˙ = [zR , 0]. Proof: Define z1 = z − zR and z2 = z. ˙ By substituting the control law (11), the system z¨ = fz (α, θ) + gz (α, θ) u1 can be rewritten as z˙1 = z2 z˙2 = gz (t) Rz (z1 , z2 )
where gz (t) is a time-varying parameter which is bounded and strictly positive as a result of Lemma 1. The rest of the proof follows directly from [18]. Remark 2: Note that the proposed control law for the vertical dynamics is valid only if the control input θ (t) verifies (10). This condition will be accounted for in the following section.
(8) VI. HORIZONTAL AND PENDULUM DYNAMICS (9)
1 This assumption can be removed by studying the interconnection between the inner and outer loop dynamics, following the same lines of [17]. 2 The mass matrix is always symmetric and positive definite as a result of the Euler-Lagrange Theorem.
By substituting the control law (11) in the system (4), the remaining dynamics become ( (α,θ) + Rz (z) gx (α, θ) x ¨ = fx (α, θ) − fz (α, θ) ggxz (α,θ) , gα (α,θ) α ¨ = −fz (α, θ) gz (α,θ) + Rz (z) gα (α, θ)
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where Rz (z) is a bounded and time-vanishing exogenous signal. The system can be reformulated as x˙ 1 = A1 x1 + x2 x˙ 2 = A2 x2 + γx2 (α, θ, Rz ) (13) ˙ = γα (α, θ, Rz ) α α1 α2 where x1 = x − xR , x2 = x, ˙ αT = = α α˙ , A1 = A2 = 0, and
lα (α) = γα (α, 0) − Aα α hα (α, θ) = γα (α, θ) − γα (α, 0) − Bα θ. Since Aα is a critically stable matrix and (Aα , Bα ) are stabilizable, there exists a good saturated controller in the form θ = σλ1 (−K1 α + v) .
Following from Theorem 4, the system is ISS with restric(α,θ) +Rz (z)gx (α, θ) tions |v| ≤ ∆ λ and |l + h + g R | ≤ ∆ λ where the γx2 (α, θ, Rz ) = fx (α, θ)−fz (α, θ) ggxz (α,θ) 1 1 α α α z 1 1 " # α2 coefficient ∆1 is strictly positive. By defining q1 > 0, q2 > 0 γα (α, θ, Rz ) = . (α,θ) and q3 > 0 such that q1 + q2 + q3 = 1, the second restriction −fz (α, θ) ggαz (α,θ) + Rz (z) gα (α, θ) may be rewritten as As proven in Theorems 3 and 5 of reference [16], system |lα | + |hα | + |gα Rz | ≤ (q1 + q2 + q3 ) ∆1 λ1 (13) is asymptotically stable with restrictions on α (t0 ) if the following conditions are met: which, in turn, enables the expression of three separate 1) The origin is an equilibrium point of the system for inequalities θ = 0 and Rz = 0. |lα | ≤ q1 ∆1 λ1 2) A1 and A2 are marginally stable. |hα | ≤ q2 ∆1 λ1 3) The linearized system with θ as an input and Rz = 0 |gα Rz | ≤ q3 ∆1 λ1 . is stabilizable. 4) Given the inputs Rz and v, subsystem α is ISS Given (9) and (12), it follows that |gα | ≤ M1L and |Rz | ≤ µ1 . with restrictions on the initial state α (t0 ) and with Therefore, the restriction on Rz can be enforced by choosing restrictions on the input v. µ1 ≤ q3 M L∆1 λ1 . (15) The first two conditions follow from the fact that [z (t) , x (t) , α (t) , θ (t)] = [zR , xR , 0, 0] is an equilibrium Furthermore, lα (α) and hα (α, θ) are higher-order terms, point for system (1) and Ai = 0 is stable. The third condition meaning is verified due to the properties of the linearized system ∀l > 0, ∃δl > 0 : kαk ≤ δl ⇒ |lα (α)| ≤ l kαk (16) 0 0 1 0 0 ∀h > 0, ∃δh > 0 : |θ| ≤ δh ⇒ |hα (α, θ)| ≤ h |θ| . m M +m 0 0 0 −M g M g B= A= Since |θ| ≤ λ1 , the restriction on the higher-order term hα 0 0 0 1 0 g M +m g M +m is automatically satisfied by choosing 0 0 −L M 0 L M which is stabilizable for m > 0. As for the final condition, it can be proven that: ˙ = γα (α, θ, Rz ) and let θ = Proposition 2: Consider α σλ1 (−K1 α + v) with K1 = 0 β P1 , where P1 > 0 is such that 0 −β 0 P1 + P1 1 0 −β
λ1 ≤ δh (h ) .
(17)
As for the restriction on lα , they can be satisfied given l kαk < q1 ∆1 λ1 , kαk ≤ δl (l ) .
1 0
≤0
+m g ¯ and β = MM L . Given Rz bounded and 0 < λ1 ≤ λ where ¯ λ > 0 is computable, subsystem α is ISS with nonzero restrictions on the input v and nonzero restrictions on the initial state. ˙ = γα (α, θ, Rz ) can also be Proof: The subsystem α written as
˙ = Aα α+Bα θ+lα (α)+hα (α, θ)+gα (α, θ) Rz (z) (14) α where
h < q2 ∆1 ,
The objective is to find a forward invariant level set wholly contained in a ball of radius δl (l ). Theorem 4 proves the 2 existence of an ISS Lyapunov function k kαk ≤ V (α) ≤ 2 ¯ k kαk , where k and k are two positive values, and the existence of χ > 0 such that kαk ≥ χ∆1 λ1 ⇒ V˙ (α) ≤ 0. Therefore, there exists a forward invariant level set Ωα such that χ∆1 λ1 ⊆ Ωα ⊆ k/k χ∆1 λ1 . The restrictions on lα can then be enforced by choosing
∂γα 0 1 = Aα = −β 0 ∂α 0 ∂γα 0 Bα = = β ∂θ 0
l < λ1 ≤ 3588
kq1 , kχ
kδl (l ) . kχ∆1
(18)
By taking ¯ = min δh (h ) , kδl (l ) , λ kχ∆1
(19)
conditions (17) and (18) are verified. Therefore, given Rz ≤ ¯ subsystem α is ISS with restrictions on µ1 and 0 < λ1 ≤ λ, the initial state α ∈ Ωα and with restrictions v ≤ ∆1 λ1 . Remark 3: The proof of Proposition 9 is constructive ¯ Additionally, since equation (19) can be used to compute λ. since θ (t) must also verify the conditions discussed in Remark 8, the saturation value λ1 should be chosen as m ¯ arccos λ1 = min λ, . 2M + m The stability of the overall systems can then be guaranteed by choosing µ1 in accordance with (15). Remark 4: Please note that, although the results are only local in α, the nested saturation formalism guarantees global stability with respect to the states x1 and x2 . In addition, it is worth mentioning that the nonlinearities of system (16) are reasonably mild (i.e. δ () ). In practice, this implies that the restrictions on the initial conditions α ∈ Ωα ⊆ δl (l ), as also verified in simulation, are “sufficiently big” for the purpose of the application.
Fig. 2. Closed-loop behavior of the vertical dynamics in the first scenario. Subscripts A and B refer to the employed controller.
VII. SIMULATIONS Consider an UAV of mass M = 2 [kg] carrying a payload m = 0.5 [kg] attached to a cable of length L = 0.1 [m]. Define Controller A as the control law developed in this paper and Controller B as an alternative control law which compensates the effects of the pendulum dynamics and then proceeds on controlling the position of the UAV. Controller A: Following from the results obtained in this paper, Controller A is defined as follows x1 x θ = −σλ1 K1 α+σλ2 K2 2 +σλ3 K3 x2 α α u1 = −
Fig. 3. Closed-loop behavior of the horizontal dynamics in the first scenario.
fz (α, θ) + Rz (z) gz (α, θ)
where fz (α, θ) and gz (α, θ) are given in (5) and (7) whereas Rz (z) = −σµ1 (Kd z2 + σµ2 (Kp z1 )) . The control parameters have been chosen as K1 = −0.1030 0.1713 K2 = 0.0026 −0.4633 −0.0557 K3 = 0.0322 0.1374 1.7246 0.0670 , λ1 = 1.4595, λ2 = 0.3649, λ3 = 0.1216, Kd = 15, Kp = 56, µ1 = 12.2625, and µ2 = 18.3938. Controller B: Based on equation (1), the alternative control law imposes u1 and θ such that u1 cos θ = mL cos αα˙ 2 + sin αα ¨ + (M + m) (g + Rz (z)) u1 sin θ = mL sin αα˙ 2 − cos αα ¨ + (M + m) Rx (x) . This compensates the effects of the suspended load dynamics to obtain z¨ = Rz (z) and x ¨ = Rx (x). The regulators Rz (z) and Rx (x) have the same parameters as Controller A.
Fig. 4. Closed-loop behavior of the pendulum dynamics in the first scenario.
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VIII. CONCLUSIONS This paper describes a novel control law for stabilizing the dynamics of an UAV carrying a suspended load. The control law is based on the nested saturation approach and guarantees asymptotic stability without any restrictions on the position and velocity of the aerial vehicle. Restrictions on the initial state of the suspended load are present. However, simulations have shown that the basin of attraction is satisfactory for application purposes. Future works will aim at the rigorous characterization of the region of attractivity as well as the extension to the three-dimensional case. Experimental activity is foreseen to demonstrate the ability of the closedloop system to navigate in cluttered environments. R EFERENCES Fig. 5. Closed-loop behavior of the horizontal dynamics in the second scenario.
Fig. 6. Closed-loop behavior of the pendulum dynamics in the second scenario.
The pair of control laws are compared in two different scenarios. In the first scenario, the system is given a step variation of the reference coordinate xR followed by a step variation of zR . Figures 2-4 show the closed-loop dynamics obtained with the two control laws. Figure 2 shows that the vertical dynamics of the two systems are very similar as could be expected from the decoupling strategy. The horizontal dynamics are represented in Figure 3 where Controller B performs slightly better than Controller A. However, Figure 3 shows that Controller A outperforms Controller B for what regards the limitation of the suspended load sway. In the second scenario, the reference of the UAV remains stationary and the suspended mass is subject to an impact (approximated with a step variation of the angular velocity). Figures 5-6 provide the resulting closed-loop response. The vertical dynamics were omitted due to their limited interest. Controller B maintains the UAV stationary but does not dampen the pendulum swing. On the contrary, Controller A momentarily steers the UAV away from the reference point but is successful in dissipating the mechanical energy of the pendulum. This shows that the proposed control law is robust with respect to external disturbances acting on the pendulum.
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