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1 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 1 Nonlinear Robust Attitude Tracking Control of a Table-Mount Experimental Helicopter Using Output-Feedback Zhan Li, Hugh H.T. Liu, Member, IEEE, Bo Zhu, Member, IEEE, Huijun Gao, Fellow, IEEE, Okyay Kaynak, Fellow, IEEE Abstract This work proposes a robust attitude tracking controller for a table-mounted experimental helicopter which has three rotational degree-of-freedom and only equips angular position sensors. The proposed controller can achieve outputfeedback attitude tracking of the pitch and elevation channels of the helicopter. The experimental platform is subjected to model uncertainties, coupling effects, and equips with an independent active disturbance system, which have effectively examined the robustness of proposed controller. The control law includes a second-order auxiliary system to generate filtered error signals, and a discontinuous uncertainty and disturbance estimation (UDE) term to compensate the model uncertainties and external disturbances. A Lyapunov based stability analysis shows the semiglobal asymptotic tracking ability of the proposed controller. The experimental results further demonstrate the propose method can achieve equivalent dynamic and static performance compare to other high-performance state-feedback methods, and even when the initial position is away from design point, it also gives more consistent responses than other linear methods. Index Terms Robust attitude tracking; Table-mount helicopter; Output-feedback; UDE. I. INTRODUCTION THE control of helicopters has long been an active topic in the field of unmanned aerial systems (UAS) because of both their wide applications and theoretical difficulties. Main challenges come from nonlinearities, model uncertainties, strong coupling, and external disturbances lay in the helicopter system dynamics. Extensive research has been done on tackling these challenges in the past decades using a large range of different control approaches [1] [1]. Among these works only a few provide experimental results because of the full-degree of freedom helicopter platforms are usually very fragile, expensive, and not practical for extensive tests of different control methods. However, past research works showed that the table-mount 3-degree of freedom (3-DOF) experimental helicopter from Quanser Consulting Inc., as shown Manuscript received October 13, 214; revised November 26, 214; accepted January, 21. Copyright (c) 21 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. Z. Li (corresponding author) is with Harbin Institute of Technology (HIT), China, and now visiting University of Toronto Institute for Aerospace Studies (UTIAS), Canada, zhanli@hit.edu.cn H. Liu is with UTIAS, Canada, liu@utias.utoronto.ca B. Zhu is with University of Electronic Science and Technology of China, and now visiting UTIAS, Canada, chubo926@gmail.com H. Gao is with HIT, China, hjgao@hit.edu.cn O. Kaynak is with Bogazici University, Turkey, okyay.kaynak@boun.edu.tr in Figure 1, can serve as an ideal experimental platform for helicopter controller design. Although simplified, the platform possesses essential dynamic features of helicopter control, such as nonlinearities, high order model, strong coupling, and underacturated, as well as durable structure and Simulink interface for easy implementation of different controllers. Besides, by using the active disturbance system (ADS) provided with the platform, which can independently change the effective mass of helicopter body on-line, robustness of the employed control strategy can be quantitatively studied by apply disturbance or uncertainty to the system through ADS. Representative works using this platform include Zhong et al. [16], [17], which gave promising results by designing a novel linear robust compensator; Xian et al. [18], which gave the numerical result of a nonlinear robust control law for 3- DOF attitude tracking control problem; and Rosales et al. [19], [2], which presented robust control strategies based on sliding mode observer. Pitch Helicopter Body ADS Back Motor Front Motor Base Travel Elevation Counterweight Arm Fig. 1. Photograph of Quanser table-mount 3-DOF helicopter with ADS. For Quanser table-mount helicopter platform, past works show that state-feedback methods can achieve good dynamic and static performances even in aggressive maneuvers [17], while experimental results on output-feedback approaches are very limited. Kutay et al. [21] gave the first experimental results of an adaptive output feedback approach for pitch axis only of the platform, but the coupling and high nonlinearity in multi-axis cases were not discussed. Rosales et al. [2] presented results for three axes control but only set-point regulation problem was considered and the performance was limited. However, the output-feedback control of helicopters is also of great importance in the field of UAS. It not only can (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

2 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 2 avoid the effects of usually inaccurate velocity/angular velocity measurements provided by on board low cost sensors, but also can provide an alternative control method when the UAS is not equipped with such sensors. Therefore, in this paper, we are specifically concerned with the output-feedback control of the Quanser table-mount helicopter, and try to achieve equivalent performance as state-feedback approaches but without angular velocity measurements. In addition, because of the main challenge we are trying to handle in this work is not the under-actuated control problem, and the travel axis movement can be controlled by designing a cascade controller using the proposed pitch controller as the inner-loop pitch angle regulator, we consider the fully-actuated case in this work by designing the controller for pitch and elevation axes, which is also the fundamental work for future study on the underactuated cases. Inspired by the novel scheme called the robust integral of the sign of the error (RISE) developed recently, see Xian et al. [22], [23], Patre et al. [24], [2], and the references therein, also borrowed some ideas from the related robust tracking control methods in Makkar et al. [26], Li et al. [27], [28], Lam et al. [29], an robust attitude output-feedback tracking controller for the table-mount laboratory helicopter is proposed in this paper. An auxiliary system is built first to generate filtered error signals. Then a control law is constructed accordingly, which includes both feed-forward terms and a discontinuous uncertainty and disturbance estimation (UDE) term that estimates the effect of unknown model uncertainties and disturbances. With position measurements only and very limited knowledge of system dynamics, the semiglobal asymptotic tracking ability of the control law is proved by Lyapunov-based method. It is further verified through experimental demonstration. Compared with previous results introduced before and to the best of the authors knowledge, our work yields the first experimentally verified results of a multi-channel robust output-feedback tracking controller for the table-mount 3-DOF helicopter. Experimental results show that the control law proposed in this paper can achieve equivalent good attitude tracking ability even in aggressive large angle step maneuvers comparing to other methods that use angular velocity measurements, and better performance than previous output-feedback controller designs both in dynamic and steady-state responses. We also observed more consistent dynamic performance of our method, comparing to a linear robust controller, when the initial position is far from design point. The remaining part of this paper is organized as follows. Section 2 introduces the table-mount helicopter system and presents the problem statement. In Section 3, the controller design procedure and the control law are proposed. The stability analysis of the closed-loop system is given in Section 4. Section deals with experimental results and Section 6 states the conclusion. II. PROBLEM STATEMENT The table-mount helicopter body is composed of a rectangular body frame and two propeller assemblies, as shown in Figure 1. Two DC motors, defined as front motor and back motor, are used to drive the propellers, and generate control forces to regulate the 3-DOF motions of the helicopter. The body frame is suspended from an instrumented joint which connects the center of the body frame and the end of a long arm, and is free to pitch about its center. The arm is installed on the base through a 2-DOF instrumented joint, which allows the helicopter body to elevate and travel. A counterweight is located at the other end of the arm such that the effective mass of the helicopter can be adjusted to a proper value. In addition, an active disturbance system (ADS) is equipped on the arm, which includes a ADS motor that can drive an adjustable weight moving along the arm. The ADS motor is controlled independently, which therefore can act as an external disturbance or model uncertainty. The elevation motion can be generated by applying a positive voltage on each motor, and positive pitch by applying greater voltage on the front motor. The travel motion is the result of tilted thrust vectors when the body pitches. The three attitude angles are measured by encoders mounted on the instrumented joints. Therefore, only position information can be directly measured. l h l a Pitch Axis Travel Axis Elevation Axis Fig. 2. Schematic diagram of the table-mount 3-DOF helicopter. Based on the simplified structure diagram in Figure 2, it shows the helicopter system is under-actuated, only two of the three degree of freedoms can be controlled to follow arbitrary trajectories in configuration space. The control strategy proposed in this paper intends to track the references of elevation and pitch angles, the travel axis is set to free move in this case. The 2-DOF motions of helicopter elevation, pitch can be formulated as follows. Elevation axis: J e α = K f l a cos(β)(v f + V b ) mgl a cos(α) (1) = K f l a cos(β)v s mgl a cos(α) where α is the elevation angle, β is the pitch angle, J e is the moment of inertia of the system about elevation axis, K f is the force constant of the motor-propeller combination, l a is the distance from elevation axis to the center of helicopter body, V f and V b are the voltages applied to front motor and back motor respectively, V s is the sum of V f and V b, m is the effective mass of the helicopter body, g is the gravitational acceleration constant. Pitch axis: J p β = Kf l h (V f V b ) = K f l h V d (2) (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

3 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 3 where J p is the moment of inertia about pitch axis, l h is the distance from pitch axis to either motor, V d is the difference of V f and V b. If V d is positive, the helicopter body will pitch up (positive). The pitch angle is limited to within ( π/2, π/2) mechanically. In order to take into account of model uncertainties and disturbances, elevation and pitch dynamics can be rewritten in vector form as q = g(q)u G(q) f(q, q), (3) where q R 2 = [α β] T is the attitude vector, g(q) R 2 2 = diag[k f l a cos(β)/j e K f l h /J p ] is the input matrix, u R 2 = [V s V d ] T is control voltage vector, G(q) R 2 = [mgl a cos(α)/j e ] T is the attitude related nonlinear term, and f(q, q) = [f 1 (q, q) f 2 (q, q)] T R 2 is an unknown nonlinear function that represents model uncertainty and disturbances. The following assumptions are made. Assumption 1. The model uncertainty and disturbances function f(q, q) and its first order derivative ḟ(q, q) are L 1 functions. Assumption 2. The system output is the attitude vector q only, the angular velocity vector of elevation and pitch q = [ α β] T is not available for feedback. Assumption 3. The reference trajectory given by q d (t) R 2 = [α d (t) β d (t)] T is continuously differentiable up to its third derivative such that d i q d (t) dt i L for i =, 1, 2, 3. (4) An error signal e(t) R 2 is defined as: e ė + k e e, () ė q d q, (6) where e is introduced to improve steady-state performance of the system, ė is the output tracking error, and k e R 2 2 = diag[k e1 k e2] denotes a matrix with both k e1 and k e2 are positive constants. Our control objective is then described as to design a control law u(t) such that e(t) as t without angular velocity measurements (only q is measurable). satisfy a 4 > and a 2 a 3 < a 4 ; r f, r R 2 are filtered signals defined accordingly, and k R 2 2 = diag[k 1 k 2 ] denotes a matrix with both k 1 and k 2 are positive constants. Notice that r f is measurable whereas r is not since the expression in (9) depends on q(t). After taking the derivative of (8) and using (9), we can obtain ṙ f = r f + b 1 r [(k I 2 )b 1 a 2 a 3 ]z 1 + [(k b 1 )a 2 a 2 a 4 ]z 2 + (a 2 b 2 b 2 1)e, (1) where I 2 denotes the 2 by 2 identity matrix. Then taking the derivative of (9) and using (1), we have ṙ = [ (k e + k + b 1 )]r (k e + k + I 2 )r f [(k I 2 )b 1 a 2 a 3 ]z 1 + [(k b 1 )a 2 a 2 a 4 ]z 2 + (a 2 b 2 b 2 1 k e k k 2 )e k 2 e (q d q) + q d q. (11) In order to facilitate the following analysis, and without losing generality, we assume the outputs of auxiliary system are only ż 1 and z 1, which makes r f and z 1 available to use in the control law. We also introduce another parameter matrix k 2 to denote the positive sum term before r in (11). Therefore, we have the following expressions: (k e + k + b 1 ) = k 2, (12) (k b 1 )a 2 a 2 a 4 =, (13) where k 2 R 2 2 = diag[k 21 k 22 ] denotes a matrix with both k 21 and k 22 are positive constants. Then, the expression of ṙ f and ṙ can be rewritten as ṙ f = r f + b 1 r [(k I 2 )b 1 a 2 a 3 ]z 1 ṙ + (a 2 b 2 b 2 1)e, (14) = k 2 r (k e + k + I 2 )r f [(k I 2 )b 1 a 2 a 3 ]z 1 + (a 2 b 2 b 2 1 k e k k 2 )e k 2 e (q d q) + q d q. (1) Based on the expressions in (3) and (1), the control law is designed as follows: III. CONTROLLER DESIGN Motivated by Xian et al. [23] and Makkar et al. [26], the following auxiliary system and filtered signals are defined to facilitate the subsequent design and analysis: { ż1 = z 1 + a 2 z 2 + b 1 e (7) ż 2 = a 3 z 1 a 4 z 2 + b 2 e r f ż 1 + k z 1 (8) r ė + k e + r f, (9) where z 1, z 2 R 2 are states of the auxiliary system and the initial conditions are set to be zero, a i R 2 2 = diag[a i1 a i2 ], i = 2, 3, 4 and b i R 2 2 = diag[b i1 b i2 ], i = 1, 2 are auxiliary system parameters to be designed, which u =g 1 (q)[ q d k 2 e (q d q) (k e + k + I 2 )r f [(k I 2 )b 1 a 2 a 3 ]z 1 + (a 2 b 2 b 2 1 k e k k 2 )e + G(q) + ˆf(t)] (16) ˆf(t) =k 1 sgn(e + z 1 ) + (a 2 b 2 b 2 1)e + b 1 r f, (17) where ˆf(t) R 2 is the UDE term which represents the estimation of unknown nonlinear function f(q, q) in (3), k 1 R 2 2 = diag[k 11 k 12 ] is a matrix with both k 11 and k 12 are positive constants, and function sgn( ) is defined as follows: sgn(ξ) k s [sign(ξ 1 ) sign(ξ 2 )] T, ξ = [ξ 1 ξ 2 ] T, (18) with sign( ) being the standard signum function (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

4 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 4 By substituting (16) and (17) into (1), the following closedloop error system can be obtained: ṙ = k 2 r + f(q, q) ˆf(t) = k 2 r + N(q, q) k 1 sgn(e + z 1 ) (a 2 b 2 b 2 1)e b 1 r f, (19) where N(q, q) R 2 f(q, q) denotes the unmeasurable auxiliary term. Here we introduce another unmeasurable auxiliary term N d (t) R 2 N(q d, q d ), and we can conclude N d, Ṅ d L according to Assumption 1 and 3. Finally, the closed-loop error system (19) can be rewritten as ṙ = k 2 r + Ñ + N d k 1 sgn(e + z 1 ) (a 2 b 2 b 2 1)e b 1 r f, (2) where Ñ N(q, q) N d(q d, q d ). Remark 1. Since N( ) defined here is continuously differentiable and by exploiting the mean value theorem, it can be shown that Ñ( ) in (2) can be upper bounded as follows: Ñ ρ( z ) z, (21) where denotes the Euclidean norm, z R 2 [e T e T z T 1 r T f r T ] T, and the positive function ρ( z ) is nondecreasing in z. Remark 2. Similar to discontinuous sliding mode control methods, the chattering may occur due to the discontinuous signum function used in the control law. However, high-slope saturation function can be used as a common replacement to attenuate the chattering effect when implementing the control law. IV. STABILITY ANALYSIS The main result of stability analysis can be stated as the following theorem. Theorem 1. The control law given in (16) and (17) ensures that all closed-loop signals are bounded and e(t), ė(t) as t provided the auxiliary system parameters and k 1 are selected according to (12), (13), along with the following sufficient conditions: (a 2 b 2 b 2 1) = I 2 (22) (k I 2 )b 1 a 2 a 3 = I 2 (23) k 1i > N di (t) + 1 Ṅdi(t), k i i = 1, 2, (24) where the subscript i denotes the i-th element of the diagonal matrix or vector, denotes the infinity norm; and k 2 is selected to let k 2 be sufficiently large to yield a semi-global asymptotic result. The closed loop system represented by (16) and (17) also ensures the uncertainty and disturbances in the system can be identified in the sense that f(q, q) ˆf(t) as t. Proof. We define a domain D R 2+1 which contains y(t) =, where y(t) is defined as y(t) R 2+1 [z T (t) P (t)] T, in which z(t) is defined as in (21) and P (t) R is an auxiliary function defined as P (t) k 11 e 1 () + k 12 e 2 () e T ()N d () t L(τ)dτ, (2) where k 11, k 12 are diagonal elements of k 1, and e 1, e 2 denote the elements of error vector e. L(t) R is another auxiliary function defined as L(t) r T (N d k 1 sgn(e + z 1 )). (26) Then the derivative of P (t) can be given as P (t) = L(t) = r T (N d k 1 sgn(e + z 1 )). (27) According to Lemma 1 of Xian et al. [23], page 697, along with the sufficient condition introduced in (24), we can obtain the following inequality: t L(τ)dτ k 11 e 1 () + k 12 e 2 () e T ()N d (). (28) Therefore, we can conclude that P (t) accordingly. We now let V (t, y) : R + D R + be a continuously differentiable positive-definite function defined as V = 1 2 et e et e zt 1 z rt f r f rt r + P (29) that can be bounded as 1 2 y 2 V y 2, (3) given that the sufficient condition (24) is satisfied. After taking the time derivative of (29), we have V = e T ė + e T ė + z T 1 ż1 + r T f ṙf + r T ṙ + P. (31) Here (), (14), (2), and (27) are first applied and then (8), (9), then the following simplified relation can be obtained: V =r T Ñ e T k e e e T k (a 2 b 2 b 2 1)e z T 1 k [(k I 2 )b 1 a 2 a 3 ]z 1 r T f r f r T k 2 r + e T ė e T (a 2 b 2 b 2 1)ė + z T 1 ż1 z T 1 [(k I 2 )b 1 a 2 a 3 ]ż 1 + e T e. (32) In order to cancel crossing terms in (32), we choose design parameters according to the sufficient conditions (22) and (23) introduced in Theorem 1. Then we obtain V =r T Ñ e T k e e e T k e z T 1 k z 1 r T f r f r T k 2 r + e T e. (33) Because e T e can be upper bounded as and after applying (21), we have e T e 1 2 e e 2 (34) V r ρ( z ) z ( k 2 1) r 2 r 2 ( k e 1 2 ) e 2 ( k 1 2 ) e 2 k z 1 2 r f 2 r ρ( z ) z ( k 2 1) r 2 λ z 2 (3) (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

5 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS where λ min{ k e 1 2, k 1 2, k, 1}; therefore, k e, k must be chosen as k e > 1 2, k > 1 2. (36) Then after completing the squares for the first two terms of (3), the following upper bound of V can be obtained: V (λ ρ2 ( z ) 4( k 2 1) ) z 2. (37) From (37), the following expression can be obtained: V γ z 2, k 2 > ρ2 ( z ) + 1, (38) 4λ where γ R is some positive constant. According to (3) and (38), we can state the lower and upper bounds of (29) are W 1 (y) = 1 2 y 2, and W 2 (y) = y 2 (39) respectively, and the upper bound of the derivative of (29) is W (y) = γ z 2, (4) where W 1 (y), W 2 (y) are continuous positive definite functions and W (y) = γ z 2 is a continuous positive semidefinite function defined on the following domain: D {y R 2+1 y < ρ 1 [2 λ( k 2 1)]}. (41) The bounds given by (39) and (4) show that V L in D; thus we have e, e, z 1, r f, and r L in D. Given that e, e, z 1, r f, and r L in D, it can be easily proved that ė, ė, ż 1 L in D from (), (8), and (9), using standard linear analysis methods. Since ė, ė L in D, by using Assumption 3 along with (), (6), we can conclude q, q L in D. Given q, q L in D and Assumption 1, (3) can be used to conclude q L in D. Given e, z 1, r f, r, q, and q L in D, (14), (1) can be used to conclude ṙ f, ṙ L in D. Since ė, ė, ż 1, ṙ f, ṙ L in D, along with the definitions of W (y) and z, it can be concluded that W (y) is uniformly continuous in D. Also, by taking time derivative on both sides of (3), along with Assumption 1 and 3, we can conclude... q L in D, which can be used to further conclude r L in D according to (1). Then let S D denote a set defined as S {y(t) D W 2 (y) < W 1 (ρ 1 [2 λ( k 2 1)])}. (42) Substitute the expressions of W 1 (y) and W 2 (y), we have S = {y(t) D y 2 < 1 2 (ρ 1 [2 λ( k 2 1)]) 2 }. (43) Now Theorem 8.4 of [3] can be invoked to conclude that γ z 2 as t, y() S. (44) Then, Based on the definition of z, we can state that e (t), e(t), z 1 (t), r f (t), r(t) as t, y() S. At last, (9) can be used to conclude ė(t) as t, y() S. Note that by increasing the control gain k 2, the region of attraction in (43) can be made sufficiently large to include any initial conditions. which proves the semi-global claim in Theorem 1. Further, the right most inequality in (43) can be used to calculate the region of attraction as 2 y() < 2 ρ 1 [2 λ( k 2 1)] (4) for each initial condition y(). At last, since r(t) as t, y() S according to the result in (44), and r L in D, Barbălat s lemma can be invoked to conclude ṙ(t) as t, y() S. Then the last claim of Theorem 1 can be concluded from (19) that f(q, q) ˆf(t) as t, y() S, (46) which completes the proof of Theorem 1. Remark 3. According to (12), (13), (22), and (23), the parameter matrices of auxiliary system can be chosen according to the following expressions that associated with design parameters k e, k and k 2. a 2 = 1 b 2 [I 2 + (k e + k + k 2 ) 2 ] (47) a 3 = b 2[I 2 + (k I 2 )(k e + k + k 2 )] [I 2 + (k e + k + k 2 ) 2 ] (48) a 4 = k e + 2k + k 2 (49) b 1 = (k e + k + k 2 ) () Therefore, by each choice of b 2, the auxiliary system can be determined by k e, k and k 2. Remark 4. From the proof one can tell that k 1 and k 2 need to be sufficiently large to guarantee the semi-global asymptotic result for the closed loop system. The minimum values of k 1 and k 2 are related to the bounds of lumped uncertainties and disturbances. However, these bounds are normally unknown and one can not simply choose a very large value because that will cause serious chattering. In our experiments, we choose 1 as the initial values of k 1 and k 2 and tune the values according to the experimental results. V. EXPERIMENTAL RESULTS AND DISCUSSIONS To examine its effectiveness, the control law given by (16) and (17) is implemented using QuaRC within Matlab Simulink, where the control law is constructed by Simulink blocks and then compiled, downloaded, and ran on a realtime windows target. The sampling time for all the following experiments is set to.1 second. The elevation and pitch angles are measured by two encoders installed on the instrumented joints. The resolution of that two encoders is 496 counts per revolution, which yields a resolution of.879 for both angles. Table I gives all the parameters associated with Quanser table-mount helicopter. Table II gives the auxiliary system parameters and controller gains used through out the experiments. According to Remark 3, a 2, a 3, a 3, b 1 are calculated based on the choice of k e, k, k 2, and b 2. In addition, note that in order to attenuate chattering effect introduced by discontinuous signum function, we use standard saturation functions with a same slope of 1 to replace the signum functions in the control law (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

6 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 6 TABLE I PARAMETERS OF THE QUANSER 3-DOF HELICOPTER Parameter Value Parameter Value K f.1188 N/V J e 1.34 kg m 2 l a.66 m J p.4 kg m 2 l h.178 m g 9.81 m/s 2 m.94 kg TABLE II CONTROLLER PARAMETERS Parameter Value Parameter Value k e 1.6 k e 2.7 k 1.9 k k 11.1 k 12.1 k 21. k a a a 31.8 a 32.4 a a b b b b Remark. According to Remark 3, only k e, k, k 1, and k 2 need to be tuned. Because of the pitch channel and elevation channel are decoupled in terms of controller parameters, one can firstly fix the helicopter arm to a certain elevation angle and tune the pitch channel parameters as the first step. Then based on the feasible parameters for pitch channel, one can tune the elevation channel parameters accordingly. A. Experiment 1: Set-point Tracking In this experiment, the helicopter starts from the initial position of elevation angle -27 and pitch angle. The position reference signals are two constants of and 2 for elevation and pitch channels respectively, and starts from time t =. The first and second order derivatives of the position reference signal are set to. Therefore, the helicopter is expected to take a large 27 step elevation while driving the pitch angle to 2. Notice that due to the mass difference between the two propeller assemblies, the pitch motion is actually coupled with elevation movement, especially in aggressive step motions as in this case. Besides, the effect of unmodeled dynamics, such as the nonlinear changing of effective mass with respect to elevation angle, the varying of K f with respect to voltage, and the saturation of motor voltages, will be enlarged in such aggressive attitude changes. Three sets of experimental results are given in this subsection. The first set is free of ADS, that is, the ADS moving weight is fixed to its initial position during the experiment. The second set is with ADS on, the ADS moving weight is moving from initial position to the middle of arm toward helicopter body, and then moving back and forth around the middle position with a frequency of.2 Hz and amplitude of. m. This is equivalent to add an unknown constant mass plus a changing mass to the system. In the last set, the UDE term ˆf(t) in (17), which is the estimation of uncertainty and disturbance, is set to zero, and all other settings are the same as the second set. This gives the system response without the compensation of uncertainty and disturbance. Figure 3 gives the first set of step response with ADS off. It shows that, for an aggressive 27 step motion and use position measurements only, the control law achieves settling time of 2.61 s with % criterion and overshoot of 4.48% for elevation channel. For 2 step motion in pitch channel, the settling time and overshoot are 2.49 s and 4.1%, respectively. In addition, the steady state errors of elevation and pitch angles are maintained less than ±.2 and ±., respectively. This result shows that both the dynamic and steady-state responses of the proposed output feed back control law can achieve equivalent performance comparing to other state-feedback control strategies, such as in [17] and the references therein. Figure 4 gives the second set of step response with same settings as in the first set, except we have turned on the ADS. It shows that, for same reference signals, the controller maintains similar settling time of 3.34 s with % criterion for elevation channel and 2.49 s for pitch channel. The steadystate errors of elevation and pitch angles are maintained less than ±. and ±., respectively. The steady-state errors in this set show that the performance of elevation channel is slightly weakened by the disturbance of ADS, and the pitch performance is maintained the same because the ADS can only effects elevation channel. In addition, Figure shows the outputs of UDE term ˆf(t) and the changing position of ADS weight during the experiment. As we can see in Figure c, although we can not expect an exact match between the ADS weight position and the UDE output for elevation channel because of the unit difference and the fact that UDE output also includes uncertainties other than ADS weight, it still can be claimed that the UDE matches the trend of disturbance introduced by ADS. To further illustrate the effect of UDE term ˆf(t), Figure 6 gives the third set of step response with same settings as in the second set but with UDE term ˆf(t) disconnected in the controller. It shows that, for both channels, the UDE term greatly improves the performance of the controller. B. Experiment 2: Sine-signal Tracking During this experiment, the helicopter starts from the same initial position of elevation angle -27 and pitch angle. The reference signals of elevation angle and pitch angle are given by π α d (t) = 1 18 sin(.3πt π 2 ), β π d(t) = 1 18 sin(.2πt), with which both the elevation channel and pitch channel of helicopter are expected to move sinusoidally with frequency of.1 Hz,.1 Hz, and amplitude of ±1, ±1, respectively. The first and second order derivatives of position reference s signal are generated using two differentiators of.1s+1. Again, we also give three sets of experimental results as shown in Figure 7 to 9. Figure 7 shows the tracking responses with ADS off. The tracking errors for elevation and pitch angles are within ±.3 and ±1.1, respectively. Figure 8 gives the tracking responses with ADS on. The corresponding errors for two channels are within ±.6 and ±1.3, respectively. Finally, Figure (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

7 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Step reference Elevation angle Step reference Elevation angle Pitch angle (deg) 1 1 Pitch angle (deg) 1 1 Step reference Pitch angle (c) Step reference Pitch angle (c) Voltage (V) 1 1 Voltage (V) 1 1 Front motor voltage Back motor voltage (d) Fig. 3. Step response with ADS off. Step response of elevation angle; steady-state response of elevation angle; (c) response of pitch angle; (d) control inputs of two motors. Front motor voltage Back motor voltage (d) Fig. 4. Step response with ADS on. Step response of elevation angle; steady-state response of elevation angle; (c) response of pitch angle; (d) control inputs of two motors (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

8 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 8 ADS weight position (m) Step reference ADS on and UDE off 2 ADS off and UDE on ADS on and UDE on UDE output (N.m) UDE output of elevation channel UDE output of pitch channel Pitch angle (deg) Step reference ADS on and UDE off ADS off and UDE on ADS on and UDE on Fig. 6. Step response with ADS on and UDE term ˆf(t) off. Step response of elevation angle; response of pitch angle. UDE output (N.m) UDE output (N.m) (c) (d) Fig.. ADS weight position and the outputs of UDE (Uncertainty and Disturbance Estimation) term ˆf(t) during the step response with ADS on. ADS weight position; UDE term outputs; (c) enlarged display of UDE term output for elevation channel; (d) enlarged display of UDE term output for pitch channel. presents the tracking responses with UDE term set to zero, from witch the effect of UDE term is obviously. The results demonstrate the effectiveness and tracking performance of proposed control law. C. Experiment 3: Comparison with robust LQR method In order to qualitatively investigate the difference between our output-feedback method and other methods using angular velocity measurements, we constructed a linear robust LQR controller for our platform, based on Zhong et al. [17] s work. Note that we are not trying to duplicate the same performance of robust LQR controller because of the structure and hardware differences between platforms used in the two works, but rather to observe the key differences in step responses using two types of controllers. Also, since our platform is only equipped with angular position sensors, the angular velocity measurements required by robust LQR controller are obtained by a differentiator block provided by Quanser. The parameters of our proposed controller are the same as in the previous two experiments, and the parameters of robust LQR controller are tuned to best match Zhong s result. All parameters for both controllers are maintained the same throughout this experiment. During first part of this experiment, for both controllers, the helicopter first starts from level position, that is elevation angle and pitch angle. The reference signals for elevation angle are given to command the helicopter to do four sets of step up movement separately, with step angles are, 1 1, and 2, respectively. The reference signals for pitch angle are all (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

9 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Reference angle Elevation angle Reference angle Elevation angle Pitch angle (deg) Reference angle Pitch angle Pitch angle (deg) Reference angle Pitch angle Tracking error (deg) Error of elevation angle Error of pitch angle (c) Tracking error (deg) (c) Error of elevation angle Error of pitch angle Voltage (V) Front motor voltage Back motor voltage (d) Fig. 7. Tracking response with ADS off. Tracking response of elevation angle; Tracking response of pitch angle; (c) Tracking error of both channels; (d) control inputs of two motors. ADS weight position (m) (d) Fig. 8. Tracking response with ADS on. Tracking response of elevation angle; Tracking response of pitch angle; (c) Tracking error of both channels; (d) ADS weight position (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

10 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Tracking reference ADS on and UDE off ADS off and UDE on ADS on and UDE on Reference signals degree case 1 degree case 1 1 degree case 2 degree case Pitch angle (deg) Tracking reference ADS on and UDE off ADS off and UDE on ADS on and UDE on Fig. 9. Tracking response with ADS on and UDE term ˆf(t) off. Tracking response of elevation angle; Tracking response of pitch angle. set to. For the second part of this experiment, the helicopter starts from mechanical initial position, that is elevation angle -27 and pitch angle. Again the reference signals are set to command the helicopter to do another four sets of step up movement separately with the same step angle settings as in the first part. Results are shown in Figure 1 to 12. As we can see from Figure 1, when the helicopter starts from the level position, which is the design point of robust LQR controller, the proposed output-feedback controller can achieve equivalent dynamic and static performances in all four different step cases in terms of overshoot, adjust time, and steady state error, compare to the robust LQR controller. The less than one percent overshoot in 1-degree case is usually acceptable in flight control Figure 11 gives the control voltages of both controllers. The relatively large spikes in control voltages when using robust LQR controller come from the differentiator used in our implementation, since the outputs of low resolution encoders are discontinuous. This phenomenon gives another reason of using output-feedback methods in systems that only have encoders to get position measurements. Figure 12 shows the elevation angle step responses when the helicopter starts from mechanical initial position. Note that at this initial position, the modelling error for both controllers is relatively large, since not only the linearisation error at this position, but also the fact that the effective mass of helicopter body is actually changing with respect to the elevation angle. From this figure, we can tell the proposed output-feedback controller can achieve similar well-damped performance in all step cases Reference signals degree case 1 degree case 1 1 degree case 2 degree case Fig. 1. Compare of elevation responses for four cases, starting from level position. Responses of proposed controller; Responses of robust LQR controller. Voltage (V) Voltage (V) degree case 1 degree case 1 degree case 2 degree case degree case 1 degree case 1 degree case 2 degree case Fig. 11. Compare of control voltages for four cases. Control voltages of proposed controller; Control voltages of robust LQR controller (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

11 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Reference signals degree case 1 degree case 1 1 degree case 2 degree case demonstrated more consistent performance when the initial position of the system is far from design point. Our future work is to extend the result to multi-helicopter cases, and examine its effectiveness on applications such as formation control and attitude synchronization [31] [33]. ACKNOWLEDGEMENT The authors would like to acknowledge that this work is supported in part by the National Natural Science Foundation of China, Grant Number , 61233, , , and in part by China Postdoctoral Science Foundation, Grant number 212M Reference signals degree case 1 degree case 1 1 degree case 2 degree case Fig. 12. Compare of elevation responses for four cases, starting from mechanical initial position. Responses of proposed controller; Responses of robust LQR controller. even the initial position is far from design point. On the other hand, the step responses of linear robust LQR controller have shown obvious over-damping results compare to the cases when the helicopter starts from level position. Remark 6. The results of experiment 3 reflect one of the key differences in the two approaches. The robust LQR controller gives simple linear structure that convenient for implementation. But because of the controller is designed based on the linearised model at level position, the controller need to handle large modelling error when the initial position is far from the design point. On the other hand, our outputfeedback controller uses non-linear terms to compensate part of nonlinearities and therefore reduces effort needed for the UDE term, which is more complex in structure but gives larger practical working region. VI. CONCLUSION In this paper, a robust attitude output-feedback tracking control method has been proposed. The semi-globally asymptotically tracking ability for any bounded, third-order differentiable reference trajectory of the resulting output feedback control law was proved by Lyapunov-based stability analysis. The convergence of UDE term to the actual uncertainty and disturbance was also proved. Finally, the proposed control law was validated on a table-mount experimental helicopter. Experimental results showed that the proposed output-feedback controller can achieve equivalent performance with respect to other methods using angular velocity measurements, and also REFERENCES [1] R. Mahony and T. Hamel, Robust trajectory tracking for a scale model autonomous helicopter, International Journal of Robust and Nonlinear Control, vol. 14, no. 12, pp , 24. [2] B. Kadmiry and D. Driankov, A fuzzy gain-scheduler for the attitude control of an unmanned helicopter, IEEE Trans. Fuzzy Syst., vol. 12, no. 4, pp. 2 1, 24. [3] R. Xu and U. Ozguner, Sliding mode control of a quadrotor helicopter, in Decision and Control (CDC), 26 4th IEEE Conference on. IEEE, 26, pp [4] A. Bogdanov and E. A. Wan, State-dependent Riccati equation control for small autonomous helicopters, Journal of guidance, control, and dynamics, vol. 3, no. 1, pp. 47 6, 27. [] M. López-Martínez, M. G. Ortega, C. Vivas, and F. R. Rubio, Nonlinear L 2 control of a laboratory helicopter with variable speed rotors, Automatica, vol. 43, no. 4, pp , 27. [6] L. Marconi and R. Naldi, Robust full degree-of-freedom tracking control of a helicopter, Automatica, vol. 43, no. 11, pp , 27. [7] J. Gadewadikar, F. Lewis, K. Subbarao, and B. M. Chen, Structured H-infinity command and control-loop design for unmanned helicopters, Journal of guidance, control, and dynamics, vol. 31, no. 4, pp , 28. [8] K. Starkov, L. Aguilar, and Y. Orlov, Sliding mode control synthesis of a 3-DOF helicopter prototype using position feedback, in Variable Structure Systems, 28. VSS 8. International Workshop on, 28, pp [9] G. Cai, B. Chen, K. Peng, M. Dong, and T. Lee, Modeling and control of the yaw channel of a UAV helicopter, IEEE Trans. Ind. Electron., vol., no. 9, pp , Sept 28. [1] K. A. Danapalasingam, J.-J. Leth, A. la Cour-Harbo, and M. Bisgaard, Robust helicopter stabilization in the face of wind disturbance, in Decision and Control (CDC), 21 49th IEEE Conference on. IEEE, 21, pp [11] O. Špinka, O. Holub, and Z. Hanzálek, Low-cost reconfigurable control system for small UAVs, IEEE Trans. Ind. Electron., vol. 8, no. 3, pp , March 211. [12] J. Ferruz, V. Vega, A. Ollero, and V. Blanco, Reconfigurable control architecture for distributed systems in the HERO autonomous helicopter, IEEE Trans. Ind. Electron., vol. 8, no. 12, pp , Dec 211. [13] S. Yin, H. Luo, and S. X. Ding, Real-Time Implementation of Fault- Tolerant Control Systems With Performance Optimization, IEEE Trans. Ind. Electron., vol. 61, no., pp , May 214. [14] I. Bayezit and B. Fidan, Distributed cohesive motion control of flight vehicle formations, IEEE Trans. Ind. Electron., vol. 6, no. 12, pp , Dec 213. [1] F. Li, P. Shi, L. Wu, M. Basin, and C. Lim, Quantized control design for cognitive radio networks modeled as nonlinear semi-markovian jump systems, IEEE Trans. Ind. Electron., vol. PP, no. 99, pp. 1 1, 214. [16] B. Zheng and Y. Zhong, Robust attitude regulation of a 3-DOF helicopter benchmark: theory and experiments, IEEE Trans. Ind. Electron., vol. 8, no. 2, pp , 211. [17] H. Liu, G. Lu, and Y. Zhong, Robust LQR attitude control of a 3- DOF laboratory helicopter for aggressive maneuvers, IEEE Trans. Ind. Electron., vol. 6, no. 1, pp , 213. [18] X. Liu, B. Xian, J. Guo, B. Zhao, and C. Diao, Nonlinear robust attitude tracking control of an underactuated 3 degree-of-freedom helicopter, in Control Conference (CCC), st Chinese. IEEE, 212, pp (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See

12 1.119/TIE , IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS 12 [19] H. Ríos, A. Rosales, A. Ferreira de Loza, and A. Davilay, Robust regulation for a 3-DOF helicopter via sliding-modes control and observation techniques, in American Control Conference (ACC), 21, 21, pp [2] A. Ferreira de Loza, H. Ríos, and A. Rosales, Robust regulation for a 3-DOF helicopter via sliding-mode observation and identification, Journal of the Franklin Institute, vol. 349, no. 2, pp , 212. [21] A. Kutay, A. Calise, M. Idan, and N. Hovakimyan, Experimental results on adaptive output feedback control using a laboratory model helicopter, IEEE Trans. Control Syst. Technol., vol. 13, no. 2, pp , 2. [22] B. Xian, D. Dawson, M. De Queiroz, and J. Chen, A continuous asymptotic tracking control strategy for uncertain nonlinear systems, IEEE Trans. Autom. Control, vol. 49, no. 7, pp , July 24. [23] B. Xian, M. S. de Queiroz, D. M. Dawson, and M. L. McIntyre, A discontinuous output feedback controller and velocity observer for nonlinear mechanical systems, Automatica, vol. 4, no. 4, pp. 69 7, 24. [24] P. Patre, W. MacKunis, C. Makkar, and W. Dixon, Asymptotic tracking for systems with structured and unstructured uncertainties, IEEE Trans. Control Syst. Technol., vol. 16, no. 2, pp , March 28. [2] P. Patre, W. MacKunis, K. Kaiser, and W. Dixon, Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure, IEEE Trans. Autom. Control, vol. 3, no. 9, pp , Oct 28. [26] C. Makkar, G. Hu, W. G. Sawyer, and W. E. Dixon, Lyapunov-based tracking control in the presence of uncertain nonlinear parameterizable friction, IEEE Trans. Autom. Control, vol. 2, no. 1, pp , 27. [27] H. Li, J. Yu, C. Hilton, and H. Liu, Adaptive sliding-mode control for nonlinear active suspension vehicle systems using T-S fuzzy approach, IEEE Trans. Ind. Electron., vol. 6, no. 8, pp , Aug 213. [28] H. Li, X. Jing, and H. Karimi, Output-feedback-based H control for vehicle suspension systems with control delay, IEEE Trans. Ind. Electron., vol. 61, no. 1, pp , Jan 214. [29] H. Lam and H. Li, Output-feedback tracking control for polynomial fuzzy-model-based control systems, IEEE Trans. Ind. Electron., vol. 6, no. 12, pp , Dec 213. [3] H. K. Khalil, Nonlinear systems. Prentice hall Upper Saddle River, 22, vol. 3. [31] Z. Li, X. Xing, and J. Yu, Decentralized output-feedback formation control of multiple 3-DOF laboratory helicopters, Journal of the Franklin Institute, 214. [Online]. Available: [32] B. Zhu, Z. Li, H. H.-T. Liu, and H. Gao, Robust second-order tracking of multi-vehicle systems without velocity measurements on directed communication topologies, in American Control Conference (ACC), 214, June 214, pp [33] B. Zhu, H. H.-T. Liu, and Z. Li, Robust distributed attitude synchronization of multiple three-dof experimental helicopters, Control Engineering Practice, vol. 36, no., pp , 21. Hugh H.T. Liu (M ) received his Ph.D. degree in mechanical engineering of the University of Toronto in He is currently a full professor at the University of Toronto Institute for Aerospace Studies (UTIAS), Toronto, Canada, where he also serves as the Associate Director, Graduate Studies. His research work over the past several years has included a number of aircraft systems and control related areas, and he leads the Flight Systems and Control (FSC) Research Laboratory. Bo Zhu (M 14) received his B.Sc. degree in vehicle engineering and the Ph.D. degree in guidance, navigation and control, from Beihang University (previously known as Beijing University of Aeronautics and Astronautics), Beijing, China, in 24 and 21, respectively. He is currently an associate professor with the School of Aeronautics and Astronautics, University of Electronic Science and Technology of China, Chengdu, China. His research interests include flight control of small UAVs, the transient performance of complex systems, and cooperative control of multi-vehicle systems. Huijun Gao (SM 9-F 14) received the Ph.D. degree in control science and engineering from Harbin Institute of Technology, Harbin, China, in 2. From 2 to 27, he carried out his postdoctoral research in the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada. Since 24, he has been with Harbin Institute of Technology, where he is currently a Professor and the Director of the Research Institute of Intelligent Control and Systems. His research interests include network-based control, robust control/filter theory, time-delay systems, and their engineering applications. Dr. Gao is an Associate Editor of Automatica, IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, IEEE TRANSACTIONS ON CYBER- NETICS, IEEE TRANSACTIONS ON FUZZY SYSTEMS, IEEE/ASME TRANSACTIONS ON MECHATRONICS, and IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY. He is a member of the Administrative Committee of the IEEE Industrial Electronics Society. Zhan Li received his B.Sc. degree in automation, and the Master s degree in pattern recognition and intelligent system, from Harbin Engineering University, Harbin, China, in 28 and 211, respectively. He is currently a Ph.D. candidate at the School of Astronautics, Harbin Institute of Technology, Harbin, China. His research interests include robust control of small UAVs and cooperative control of multi-vehicle systems. Okyay Kaynak (SM 9-F 3) received the Ph.D. degree from the Department of Electronic and Electrical Engineering, University of Birmingham, Birmingham, U.K., in In 1979, he joined the Department of Electrical and Electronics Engineering, Bogazici University, Istanbul, Turkey, where he is currently a Full Professor. He has served as the Chairman of the Computer Engineering and the Electrical and Electronic Engineering Departments and as the Director of the Biomedical Engineering Institute, Bogazici University. He is currently the UNESCO Chair on Mechatronics and the Director of the Mechatronics Research and Application Center. Dr. Kaynak is currently an Associate Editor of the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS and the IEEE SENSORS JOURNAL, and the Editor-in-Chief of the IEEE/ASME TRANSACTIONS ON MECHA- TRONICS. He is active in international organizations. He was the President of the IEEE Industrial Electronics Society in and the Vice President (for Conferences) of the IEEE Computational Intelligence Society in (c) 21 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See