Control of Complex Maneuvers for a Quadrotor UAV using Geometric Methods on SE(3)

Contr ol of Complex Maneuvers f or a Quadr otor U A V using Geometric Methods on SE ( 3 ) T aeyoung Lee ∗ , Melvin Leok † , and N. Harris McClamroch Abstract — This paper provides new r esults for control of complex flight maneuvers for a quadrotor unmanned aerial vehicle. The flight maneuvers are defined by a concatenation of flight modes, each of which is achieved by a nonlinear controller that solves an output tracking problem. A mathematical model of the quadrotor U A V rigid body dynamics, defined on the configuration space SE ( 3 ) , is introduced as a basis for the analysis. W e focus on three output tracking problems, namely (1) outputs given by the vehicle attitude, (2) outputs given by the three position variables for the vehicle center of mass, and (3) output given by the three velocity variables f or the vehicle center of mass. A nonlinear tracking controller is developed on the special Euclidean group SE ( 3 ) for each flight mode, and the closed loop is shown to ha ve desirable properties that are almost global in each case. Several numerical examples, including one example in which the quadrotor recov ers from being initially upside down and another example that includes switching and transitions between differ ent flight modes, illustrate the versatility and generality of the proposed approach. I . I N T R O D U C T I O N A quadrotor unmanned aerial vehicle (UA V) consists of two pairs of counter-rotating rotors and propellers, located at the vertices of a square frame. It is capable of vertical take-of f and landing (VTOL), but it does not require comple x mechanical linkages, such as swash plates or teeter hinges, that commonly appear in typical helicopters. Due to its sim- ple mechanical structure, it has been en visaged for v arious applications such as surveillance or mobile sensor networks as well as for educational purposes. Despite the substantial interest in quadrotor U A Vs, little attention has been paid to constructing nonlinear control sys- tems that can achieve complex aerobatic maneuvers. Linear control systems such as proportional-deri vati ve controllers or linear quadratic regulators are widely used to enhance the stability properties of an equilibrium [1], [2], [3], [4], [5]. The quadrotor dynamics is modeled as a collection of sim- plified hybrid dynamic modes, where each mode represents a particular local operating region. But, it is required to do complex reachability analyses to guarantees the safety and performance of such hybrid system [6]. A nonlinear controller is developed for the linearized dynamics of a quadrotor UA V in [7]. Backstepping and T aeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute of T echnology , Melbourne, FL 39201 taeyoung@fit.edu Melvin Leok, Mathematics, University of California at San Diego, La Jolla, CA 92093 mleok@math.ucsd.edu N. Harris McClamroch, Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109 nhm@umich.edu ∗ This research has been supported in part by NSF under grants CMMI- 1029551. † This research has been supported in part by NSF under grants DMS- 0726263, DMS-1001521, DMS-1010687, and CMMI-1029445. sliding mode techniques are applied in [8]. Since all of these controllers are based on Euler angles, they exhibit singular - ities when representing complex rotational maneuvers of a quadrotor UA V , thereby significantly restricting their ability to achiev e complex flight maneuvers. An attitude control system based on quaternions is ap- plied to a quadrotor U A V [9]. Quaternions do not hav e singularities, but they have ambiguities in representing an attitude, as the three-sphere, the unit-vectors in R 4 , double- cov ers the attitude configuration of the special orthogonal group, SO ( 3 ) . Therefore, a single physical attitude of a rigid body may yields two different control inputs, which causes inconsistency in the resulting control system. A specific choice between two quaternions generates discontinuity that makes the resulting control system sensitiv e to noise and disturbances [10]. It is possible to construct continuous controllers, but they may e xhibit unwinding behavior , where the controller unnecessarily rotates a rigid body through lar ge angles, ev en if the initial attitude is close to the desired attitude, thereby breaking L yapunov stability [11]. Geometric control, as utilized in this paper , is concerned with the dev elopment of control systems for dynamic sys- tems ev olving on nonlinear manifolds that cannot be globally identified with Euclidean spaces [12], [13]. By characterizing geometric properties of nonlinear manifolds intrinsically , geometric control techniques provide unique insights into control theory that cannot be obtained from dynamic models represented using local coordinates. This approach has been applied to fully actuated rigid body dynamics on Lie groups to achieve almost global asymptotic stability [13], [14], [15], [16]. In this paper, we make use of geometric methods to define and analyze controllers that can achieve complex aerobatic maneuvers for a quadrotor UA V . The dynamics of the quadrotor U A V are expressed globally on the configuration manifold, which is the special Euclidean group SE ( 3 ) . Based on a hybrid control architecture, we construct controllers that can achiev e output tracking for outputs that correspond to each of several flight modes, namely an attitude controlled flight mode, a position controlled flight mode, and a velocity controlled flight mode. The proposed controller exhibits the following unique features: (i) It guarantees almost global tracking features of a quadrotor U A V as the region of attraction almost cov ers the attitude configuration space SO ( 3 ) . Such global stability analysis on the special Euclidean group of a quadrotor U A V is unprecedented. (ii) Hybrid control structures between dif- ferent tracking mode is robust to switching conditions due to the almost global stability properties. Therefore, aggressiv e maneuvers of a quadrotor U A V can be achiev ed in a unified way , without need for complex reachability analyses. (iii) The proposed control system extends the existing geometric controls of the rigid body dynamics into an underactuated rigid body system, where its translation dynamics is cou- pled to the rotational dynamics in a unique way . (iv) It is coordinate-free. Therefore, it completely avoids singularities, complexities, discontinuities, or ambiguities that arise when using local coordinates or quaternions. The paper is organized as follows. W e de velop a globally defined model for the translational and rotational dynamics of a quadrotor U A V in Section II. The hybrid control archi- tecture and three flight modes are introduced in Section III. Section IV presents results for the attitude controlled flight mode; Sections V and VI present results for the position controlled flight mode, and the velocity controlled flight mode, respectiv ely . Sev eral numerical results that demon- strate complex aerobatic maneuvers for a typical quadrotor U A V are presented in Section VII. I I . Q UA D R OTO R DY NA M I C S M O D E L Consider a quadrotor UA V model illustrated in Figure 1. This is a system of four identical rotors and propellers located at the vertices of a square, which generate a thrust and torque normal to the plane of this square. W e choose an inertial reference frame { ~ e 1 , ~ e 2 , ~ e 3 } and a body-fixed frame { ~ b 1 , ~ b 2 , ~ b 3 } . The origin of the body-fixed frame is located at the center of mass of this vehicle. The first and the second ax es of the body-fixed frame, ~ b 1 , ~ b 2 , lie in the plane defined by the centers of the four rotors, as illustrated in Figure 1. The third body-fixed axis ~ b 3 is normal to this plane. Each of the inertial reference frame and the body-fixed reference frame consist of a triad of orthogonal vectors defined according to the right hand rule. In the subsequent de velopment, these references frames are tak en as basis sets and we use vectors in R 3 to represent physical vectors and we use 3 × 3 real matrices to represent linear transformations between the vector spaces defined by these two frames. Define m ∈ R the total mass J ∈ R 3 × 3 the inertia matrix with respect to the body-fixed frame R ∈ SO ( 3 ) the rotation matrix from the body-fixed frame to the inertial frame Ω ∈ R 3 the angular velocity in the body-fixed frame x ∈ R 3 the position v ector of the center of mass in the inertial frame v ∈ R 3 the velocity v ector of the center of mass in the inertial frame d ∈ R the distance from the center of mass to the center of each rotor in the ~ b 1 , ~ b 2 plane f i ∈ R the thrust generated by the i -th pro- peller along the − ~ b 3 axis ~ e 1 ~ e 2 ~ e 3 ~ b 1 ~ b 2 ~ b 3 f 1 f 2 f 3 f 4 x R Fig. 1. Quadrotor model τ i ∈ R the torque generated by the i -th pro- peller about the ~ b 3 axis f ∈ R the total thrust magnitude, i.e., f = P 4 i =1 f i M ∈ R 3 the total moment vector in the body- fixed frame The configuration of this quadrotor U A V is defined by the location of the center of mass and the attitude with respect to the inertial frame. Therefore, the configuration manifold is the special Euclidean group SE ( 3 ) , which is the semidirect product of R 3 and the special orthogonal group SO ( 3 ) = { R ∈ R 3 × 3 | R T R = I , det R = 1 } . The following conv entions are assumed for the rotors and propellers, and the thrust and moment that they ex ert on the quadrotor U A V . W e assume that the thrust of each propeller is directly controlled, and the direction of the thrust of each propeller is normal to the quadrotor plane. The first and third propellers are assumed to generate a thrust along the direction of − ~ b 3 when rotating clockwise; the second and fourth propellers are assumed to generate a thrust along the same direction of − ~ b 3 when rotating counterclockwise. Thus, the thrust magnitude is f = P 4 i =1 f i , and it is positiv e when the total thrust vector acts along − ~ b 3 , and it is negati ve when the total thrust vector acts along ~ b 3 . By the definition of the rotation matrix R ∈ SO ( 3 ) , the total thrust vector is gi ven by − f R e 3 ∈ R 3 in the inertial frame. W e also assume that the torque generated by each propeller is directly proportional to its thrust. Since it is assumed that the first and the third propellers rotate clockwise and the second and the fourth propellers rotate counterclockwise to generate a positive thrust along the direction of − ~ b 3 , the torque generated by the i -th propeller about ~ b 3 can be written as τ i = ( − 1) i c τ f f i for a fixed constant c τ f . All of these assumptions are common [9], [3]. The presented control system can readily be extended to include linear rotor dynamics, as studied in [8]. Under these assumptions, the moment vector in the body- fixed frame is giv en by     f M 1 M 2 M 3     =     1 1 1 1 0 − d 0 d d 0 − d 0 − c τ f c τ f − c τ f c τ f         f 1 f 2 f 3 f 4     . (1) The determinant of the above 4 × 4 matrix is 8 c τ f d 2 , so it is in vertible when d 6 = 0 and c τ f 6 = 0 . Therefore, for giv en thrust magnitude f and gi ven moment v ector M , the thrust of each propeller f 1 , f 2 , f 3 , f 4 can be obtained from (1). Using this equation, the thrust magnitude f ∈ R and the moment vector M ∈ R 3 are viewed as control inputs in this paper . The equations of motion of the quadrotor U A V can be written as ˙ x = v , (2) m ˙ v = mge 3 − f Re 3 , (3) ˙ R = R ˆ Ω , (4) J ˙ Ω + Ω × J Ω = M , (5) where the hat map ˆ · : R 3 → so (3) is defined by the condition that ˆ xy = x × y for all x, y ∈ R 3 (see Appendix A). Throughout this paper , λ m ( · ) and λ M ( · ) denote the minimum eignev alue and the maximum eigen value of a matrix, respectiv ely . I I I . G E O M E T R I C T R A C K I N G C O N T R O L S Since the quadrotor U A V has four inputs, it is possible to achiev e asymptotic output tracking for at most four quadrotor U A V outputs. The quadrotor UA V has three translational and three rotational degrees of freedom; it is not possible to achiev e asymptotic output tracking of both attitude and position of the quadrotor U A V . This motiv ates us to introduce sev eral flight modes. Each flight mode is associated with a specified set of outputs for which exact tracking of those outputs define that flight mode. The three flight modes considered in this paper are: • Attitude controlled flight mode: the outputs are the attitude of the quadrotor U A V and the controller for this flight mode achieves asymptotic attitude tracking. • Position controlled flight mode: the outputs are the position vector of the center of mass of the quadrotor U A V and the controller for this flight mode achieves asymptotic position tracking. • V elocity controlled flight mode: the outputs are the velocity vector of the center of mass of the quadrotor U A V and the controller for this flight mode achieves asymptotic velocity tracking. A complex flight maneuver can be defined by specifying a concatenation of flight modes together with conditions for switching between them; for each flight mode one also specifies the desired or commanded outputs as functions of time. For example, one might define a complex aerobatic flight maneuver for the quadrotor U A V that consists of a hov ering flight segment by specifying a constant position vector , a reorientation segment by specifying the time ev olu- tion of the vehicle attitude, and a surveillance flight segment by specifying a time-varying position vector . The controller in such a case would switch between nonlinear controllers defined for each of the flight modes. These types of complex aerobatic maneuvers, inv olving large angle transitions between flight modes, ha ve not been much studied in the literature. Such a hybrid flight control architecture has been proposed in [17], [18], [19], [6], but they are sensitive to switching conditions as the region of attraction for each flight mode is limited, and they required complicated reachability set analyses to guarantee safety and performance. The proposed control system is rob ust to switching conditions since each flight mode has almost global stability properties, and it is straightforward to design a complex maneuver of a quadrotor UA V . I V . A T T I T U D E C O N T RO L L E D F L I G H T M O D E An arbitrary smooth attitude tracking command R d ( t ) ∈ SO ( 3 ) is given as a function of time. The correspond- ing angular velocity command is obtained by the attitude kinematics equation, ˆ Ω d = R T d ˙ R d . W e first define errors associated with the attitude dynamics of the quadrotor UA V . The attitude and angular velocity tracking error should be carefully chosen as they ev olve on the tangent bundle of SO ( 3 ) . First, define the real-v alued error function on SO ( 3 ) × SO ( 3 ) : Ψ( R, R d ) = 1 2 tr  I − R T d R  . (6) This function is locally positive-definite about R = R d within the region where the rotation angle between R and R d is less than 180 ◦ [13]. For a gi ven R d , this set can be repre- sented by the sublevel set L 2 = { R ∈ SO ( 3 ) | Ψ( R , R d ) < 2 } , which almost covers SO ( 3 ) . The variation of a rotation matrix can be expressed as δ R = R ˆ η for η ∈ R 3 , so that the deriv ative of the error function is given by D R Ψ( R, R d ) · R ˆ η = − 1 2 tr  R T d R ˆ η  = e R · η (7) where the attitude tracking error e R ∈ R 3 is chosen as e R = 1 2 ( R T d R − R T R d ) ∨ . (8) The vee map ∨ : so (3) → R 3 is the in verse of the hat map. W e used a property of the hat map giv en by equation (50) in Appendix A. The tangent vectors ˙ R ∈ T R SO ( 3 ) and ˙ R d ∈ T R d SO ( 3 ) cannot be directly compared since they lie in different tangent spaces. W e transform ˙ R d into a vector in T R SO ( 3 ) , and we compare it with ˙ R as follows: ˙ R − ˙ R d ( R T d R ) = R ( ˆ Ω − R T R d ˆ Ω d R T d R ) = R ˆ e Ω , where the tracking error for the angular v elocity e Ω ∈ R 3 is defined as follows: e Ω = Ω − R T R d Ω d . (9) W e show that e Ω is the angular velocity vector of the relative rotation matrix R T d R , represented in the body-fixed frame, since d dt ( R T d R ) = ( R T d R ) ˆ e Ω . (10) W e no w introduce a nonlinear controller for the attitude controlled flight mode, described by an expression for the moment vector: M = − k R e R − k Ω e Ω + Ω × J Ω − J ( ˆ Ω R T R d Ω d − R T R d ˙ Ω d ) , (11) where k R , k Ω are positiv e constants. In this attitude controlled mode, it is possible to ignore the translational motion of the quadrotor UA V ; consequently the reduced model for the attitude dynamics are given by equations (4), (5), using the controller expression (11). W e now state the result that ( e R , e Ω ) = (0 , 0) is an exponentially stable equilibrium of the reduced closed loop dynamics. Pr oposition 1: (Exponential Stability of Attitude Con- trolled Flight Mode) Consider the control moment M defined in (11) for any positiv e constants k R , k Ω . Suppose that the initial conditions satisfy Ψ( R (0) , R d (0)) < 2 , (12) k e Ω (0) k 2 < 2 λ M ( J ) k R (2 − Ψ( R (0) , R d (0))) . (13) Then, the zero equilibrium of the closed loop tracking error ( e R , e Ω ) = (0 , 0) is exponentially stable. Furthermore, there exist constants α 2 , β 2 > 0 such that Ψ( R ( t ) , R d ( t )) ≤ min  2 , α 2 e − β 2 t  . (14) Pr oof: See Appendix B. In this proposition, equations (12), (13) describe a region of attraction for the reduced closed loop dynamics. An estimate of the domain of attraction is obtained for which the quadrotor attitude lies in the sublevel set L 2 = { R ∈ SO ( 3 ) | Ψ( R, R d ) < 2 } for a giv en R d . This requires that the initial attitude error should be less than 180 ◦ , in terms of the rotation angle about the eigenaxis between R and R d . Therefore, in Proposition 1, exponential stability is guaranteed for almost all initial attitude errors. More explicitly , the attitudes that lie outside of the region of attraction are of the form exp( π ˆ s ) R d for some s ∈ S 2 . Since the y comprise a two-dimensional manifold in the three- dimensional SO ( 3 ) , we claim that the presented controller exhibits almost global properties in SO ( 3 ) . It should be noted that topological obstructions prev ent one from constructing a smooth controller on SO ( 3 ) that has an equilibrium solution that is global asymptotically stable [20]. The size of the region of attraction can be increased by choosing a larger controller gain k R in (13). Asymptotic tracking of the quadrotor attitude does not require specification of the thrust magnitude. As an auxil- iary problem, the thrust magnitude can be chosen in many different ways to achieve an additional translational motion objectiv e. As an example of a specific selection approach, we assume that the objective is to asymptotically track a quadrotor alti- tude command. It is straightforward to obtain the following corollary of Proposition 1. Pr oposition 2: (Exponential Stability of Attitude Con- trolled Flight Mode with Altitude T racking) Consider the control moment vector M defined in (11) satisfying the as- sumptions of Proposition 1. In addition, the thrust magnitude is giv en by f = k x ( x 3 − x 3 d ) + k v ( ˙ x 3 − ˙ x 3 d ) + mg − m ¨ x 3 d e 3 · Re 3 , (15) where k x , k v are positi ve constants, x 3 d ( t ) is the quadrotor altitude command, and we assume that e 3 · Re 3 6 = 0 . (16) The conclusions of Proposition 1 hold and in addition the quadrotor altitude x 3 ( t ) asymptotically tracks the altitude command x 3 d ( t ) . Pr oof: See Appendix C. Since the translational motion of the quadrotor U A V can only be partially controlled; this flight mode is most suitable for short time periods where an attitude maneuver is to be completed. The translational equations of motion of the quadrotor U A V , during an attitude flight mode, are gi ven by equations (2), (3), and whatev er thrust magnitude controller , e.g., equation (15), is selected. V . P O S I T I O N C O N T RO L L E D F L I G H T M O D E W e now introduce a nonlinear controller for the position controlled flight mode. W e show that this controller achie ves almost global asymptotic position tracking, that is the output position vector of the quadrotor U A V asymptotically tracks the commanded position. This flight mode requires analysis of the coupled translational and rotational equations of motion; hence, we make use of the notation and analysis in the prior section to describe the properties of the closed loop system in this flight mode. An arbitrary smooth position tracking command x d ( t ) ∈ R 3 is chosen. The position tracking errors for the position and the velocity are given by: e x = x − x d , (17) e v = v − ˙ x d . (18) The nonlinear controller for the p os ition controlled flight mode, described by control expressions for the thrust mag- nitude and the moment vector , are: f = ( k x e x + k v e v + mg e 3 − m ¨ x d ) · Re 3 , (19) M = − k R e R − k Ω e Ω + Ω × J Ω − J ( ˆ Ω R T R c Ω c − R T R c ˙ Ω c ) , (20) where k x , k v , k R , k Ω are positiv e constants. Following the prior definition of the attitude error and the angular v elocity error e R = 1 2 ( R T c R − R T R c ) ∨ , e Ω = Ω − R T R c Ω c , (21) and the computed attitude R c ( t ) ∈ SO ( 3 ) and computed angular velocity Ω c ∈ R 3 are giv en by R c = [ b 1 c ; b 3 c × b 1 c ; b 3 c ] , ˆ Ω c = R T c ˙ R c , (22) Force controller Moment controller - - - - - Quadrotor Dynamics - f M b 3 c x d ( b 1 d ) x, v, R, Ω 6 - q q Controller Fig. 2. Controller structure for position controlled flight mode where b 3 c ∈ S 2 is defined by b 3 c = − − k x e x − k v e v − mg e 3 + m ¨ x d k− k x e x − k v e v − mg e 3 + m ¨ x d k , (23) and b 1 c ∈ S 2 is selected to be orthogonal to b 3 c , thereby guaranteeing that R c ∈ SO ( 3 ) . W e assume that k− k x e x − k v e v − mg e 3 + m ¨ x d k 6 = 0 , (24) and the commanded acceleration is uniformly bounded such that k − mg e 3 + m ¨ x d k < B (25) for a given positi ve constant B . The thrust magnitude controller and the moment vec- tor controller is feedback dependent on the position and translational velocity and they depend on the commanded position, translational velocity and translational acceleration. The control moment vector has a form that is similar to that for the attitude controlled flight mode. Howe ver , the attitude err or and angular velocity error are defined with respect to a computed attitude, angular velocity and angular acceleration, that are constructed according to the indicated procedure. The nonlinear controller given by equations (19), (20) can be given a backstepping interpretation. The computed attitude R c giv en in equation (22) is selected so that the thrust axis − b 3 of the quadrotor UA V tracks the computed direction given by − b 3 c in (23), which is a direction of the thrust vector that achieves position tracking. The moment expression (20) causes the attitude of the quadrotor U A V to asymptotically track R c and the thrust magnitude expression (19) achiev es asymptotic position tracking. The closed loop system for this position controlled flight mode is illustrated in Figure 2. The corresponding closed loop control system is described by equations (2), (3), (4), (5), using the controller expressions (19) and (20). W e no w state the result that ( e x , e v , e R , e Ω ) = (0 , 0 , 0 , 0) is an exponentially stable equilibrium of the closed loop dynamics. Pr oposition 3: (Exponential Stability of Position Con- trolled Flight Mode) Consider the thrust magnitude f and moment vector M defined by equations (19), (20). Suppose that the initial conditions satisfy Ψ( R (0) , R c (0)) < 1 , (26) k e x (0) k < e x max , (27) for a fixed constant e x max . Define W 1 , W 12 , W 2 ∈ R 2 × 2 to be W 1 =  c 1 k x m (1 − α ) − c 1 k v 2 m (1 + α ) − c 1 k v 2 m (1 + α ) k v (1 − α ) − c 1  , (28) W 12 =  c 1 m B 0 B + k x e x max 0  , (29) W 2 = " c 2 k R λ M ( J ) − c 2 k Ω 2 λ m ( J ) − c 2 k Ω 2 λ m ( J ) k Ω − c 2 # , (30) where Ψ( R (0) , R c (0)) < ψ 1 < 1 , and α = p ψ 1 (2 − ψ 1 ) . For positiv e constants k x , k v , we choose positiv e constants c 1 , c 2 , k R , k Ω such that c 1 < min  k v (1 − α ) , 4 mk x k v (1 − α ) 2 k 2 v (1 + α ) 2 + 4 mk x (1 − α ) , p k x m  , (31) c 2 < min  k Ω , 4 k Ω k R λ m ( J ) 2 k 2 Ω λ M ( J ) + 4 k R λ m ( J ) 2 , p k R λ m ( J )  , (32) λ m ( W 2 ) > 4 k W 12 k 2 λ m ( W 1 ) . (33) Then, the zero equilibrium of the closed loop tracking errors ( e x , e v , e R , e Ω ) = (0 , 0 , 0 , 0) is exponentially stable. A region of attraction is characterized by (26), (27), and k e Ω (0) k 2 < 2 λ M ( J ) k R ( ψ 1 − Ψ( R (0) , R c (0))) , (34) λ M ( M 12 ) k z 1 (0) k 2 + λ M ( M 0 22 ) k z 2 (0) k 2 < 1 2 k x e 2 x max , (35) where z 1 = [ k e x k , k e v k ] T , z 2 = [ k e R k , k e Ω k ] T ∈ R 2 and M 12 = 1 2  k x c 1 c 1 m  , M 0 22 = 1 2  2 k R 2 − ψ 1 c 2 c 2 λ M ( J )  . Pr oof: See Appendix D. Proposition 3 requires that the initial attitude err or is less than 90 ◦ to achiev e exponential stability for this flight mode. Suppose that this is not satisfied, i.e. 1 ≤ Ψ( R (0) , R c (0)) < 2 . W e can apply Proposition 1, which states that the attitude error function Ψ exponentially decreases, and therefore, it enters the region of attraction of Proposition 3 in a finite time. Therefore, by combining the results of Proposition 1 and 3, we can sho w almost global exponential attractiveness when Ψ( R (0) , R c (0)) < 2 . Definition 1: (Exponential Attractiv eness [21]) An equi- librium point z = 0 of a dynamic systems is exponentially attractive if, for some δ > 0 , there exists a constant α ( δ ) > 0 and β > 0 such that k z (0) k < δ implies k z ( t ) k ≤ α ( δ ) e − β t for all t > 0 . This should be distinguished from the stronger notion of exponential stability , in which the above bound is replaced by k z ( t ) k ≤ α ( δ ) k z (0) k e − β t . Pr oposition 4: (Almost Global Exponential Attractiv eness of the Position Controlled Flight Mode) Consider the thrust magnitude f and moment vector M defined in expressions (19), (20). Suppose that the initial conditions satisfy 1 ≤ Ψ( R (0) , R c (0)) < 2 , (36) k e Ω (0) k 2 < 2 λ M ( J ) k R (2 − Ψ( R (0) , R c (0))) . (37) Then, the zero equilibrium of the closed loop tracking errors ( e x , e v , e R , e Ω ) = (0 , 0 , 0 , 0) is exponentially attracti ve. Pr oof: See Appendix E. In Proposition 4, exponential attracti veness is guaranteed for almost all initial attitude errors. Since the attitudes that lie outside of the region of attraction comprise a two- dimensional manifold in the three-dimensional SO ( 3 ) , as dis- cussed in Section IV, we claim that the presented controller exhibits almost global properties in SO ( 3 ) . As described abov e, the construction of the orthogonal matrix R c in volves ha ving its third column b 3 c specified by a normalized feedback function, and its first column b 1 c is chosen to be orthogonal to the third column. The unit vector b 1 c can be arbitrarily chosen in the plane normal to b 3 c , which corresponds to a one-dimensional degree of choice. This reflects the fact that the quadrotor U A V has four control inputs that are used to track a three-dimensional position command. By choosing b 1 c properly , we constrain the asymptotic direction of the first body-fixed axis. Here, we propose to specify the pr ojection of the first body-fixed axis onto the plane normal to b 3 c . In particular , we choose a desired direction b 1 d ∈ S 2 , that is not parallel to b 3 c , and b 1 c is selected as b 1 c = Pro j[ b 1 d ] , where Pro j[ · ] denotes the normalized projection onto the plane perpendicular to b 3 c . In this case, the first body-fixed axis does not conv erge to b 1 d , but it con verges to the projection of b 1 d , i.e. b 1 → b 1 c = Pro j[ b 1 d ] as t → ∞ . In other words, the first body-fixed axis con verges to the intersection of the plane normal to b 3 c and the plane spanned by b 3 c and b 1 d (see Figure 3). From (23), we observe that b 3 c asymptotically conv erges to the direction g e 3 − ¨ x d . In short, the additional input is used to guarantee that the first body-fixed axis asymptotically lies in the plane spanned by b 1 d and g e 3 − ¨ x d . Suppose that ¨ x d = 0 , then the third body-fixed axis con verges to the gravity direction e 3 . In this case, we can choose b 1 d arbitrarily in the horizontal plane, and it follows that b 1 c → Pro j[ b 1 d ] = b 1 d as t → ∞ . Therefore, the first body-fixed axis b 1 asymptotically con ver ges to b 1 d , which can be used to specify the heading direction of the quadrotor U A V in the horizontal plane. Pr oposition 5: (Almost Global Exponential Attractiv eness of Position Controlled Flight Mode with Specified Asymp- totic Direction of First Body-Fixed Axis) Consider the mo- ment vector M defined in (20) and the thrust magnitude f defined in (19) satisfying the assumptions of Propositions 3 and 4. In addition, the first column of R c , namely b 1 c is con- structed as follows. W e choose b 1 d ( t ) ∈ S 2 , and we assume that it is not parallel to b 3 c . The unit vector b 1 c is constructed b 3 c b 1 d b 2 c = b 3 c × b 1 c b 1 c = Pro j[ b 1 d ] Plane normal to b 3 c Plane spanned by b 1 d and b 3 c Fig. 3. Con vergence property of the first body-fixed axis: b 3 c is determined by (23). W e choose an arbitrary b 1 d that is not parallel to b 3 c , and project it on to the plane normal to b 3 c to obtain b 1 c . This guarantees that the first body-fixed axis asymptotically lies in the plane spanned by b 1 d and b 3 c , which conv erges to the direction of g e 3 − ¨ x d as t → ∞ . by projecting b 1 d onto the plane normal to b 3 c , and normal- izing it: b 1 c = − 1 k b 3 c × b 1 d k ( b 3 c × ( b 3 c × b 1 d )) . (38) Then, the conclusions of Propositions 3 and 4 hold, and the first body-fixed axis asymptotically lies in the plane spanned by b 1 d and g e 3 − ¨ x d . In the special case where ¨ x d = 0 , we can choose b 1 d in the horizontal plane. Then, the first body-fixed axis asymptotically con verges to b 1 d . Expressions for Ω c and ˙ Ω c that appear in Proposition 5 are summarized in [22]. These additional properties of the closed loop can be interpreted as characterizing the asymptotic direction of the first body-fixed axis and the asymptotic direction of the third body-fixed axis as it depends on the commanded vehicle acceleration. These physical properties may be of importance in some flight maneuvers. V I . V E L O C I T Y C O N T R O L L E D F L I G H T M O D E W e now introduce a nonlinear controller for the velocity controlled flight mode. An arbitrary velocity tracking com- mand t → v d ( t ) ∈ R 3 is giv en. The velocity tracking error is giv en by: e v = v − v d . (39) The nonlinear controller for the velocity controlled flight mode is given by f = ( k v e v + mg e 3 − m ˙ v d ) · Re 3 , (40) M = − k R e R − k Ω e Ω + Ω × J Ω − J ( ˆ Ω R T R c Ω c − R T R c ˙ Ω c ) , (41) where k v , k R , k Ω are positiv e constants, and following the prior definition of the attitude error and the angular v elocity error e R = 1 2 ( R T c R − R T R c ) ∨ , e Ω = Ω − R T R c Ω c , (42) and R c ( t ) ∈ SO ( 3 ) and Ω c ∈ R 3 are constructed as: R c = [ b 1 c ; b 3 c × b 1 c ; b 3 c ] , ˆ Ω c = R T c ˙ R c , (43) where b 3 c ∈ S 2 is defined by b 3 c = − − k v e v − mg e 3 + m ˙ v d k− k v e v − mg e 3 + m ˙ v d k . (44) and b 1 c ∈ S 2 is selected to be orthogonal to b 3 c , thereby guaranteeing that R c ∈ SO ( 3 ) . W e assume that k− k v e v − mg e 3 + m ¨ x d k 6 = 0 , (45) k − mg e 3 + m ¨ x d k < B (46) for a given positi ve constant B . The overall controller structure and the corresponding stability properties are similar to the position controlled flight mode. More explicitly , the closed loop dynamics have the property that ( e v , e R , e Ω ) = (0 , 0 , 0) is an equilibrium that is exponentially stable for any initial condition satisfying Ψ( R (0) , R c (0)) < 1 , and it is exponentially attractive for any initial condition satisfying Ψ( R (0) , R c (0)) < 2 . Due to page limitations, the explicit statements of propositions and proofs for the velocity controlled flight mode are relegated to [22]. V I I . N U M E R I C A L R E S U LT S I L L U S T R A T I N G C O M P L E X F L I G H T M A N E U V E R S Numerical results are presented to demonstrate the prior approach for performing complex flight maneuv ers for a typical quadrotor U A V . The parameters are chosen to match a quadrotor UA V described in [23]. J = [0 . 0820 , 0 . 0845 , 0 . 1377] kg − m 2 , m = 4 . 34 kg d = 0 . 315 m , c τ f = 8 . 004 × 10 − 3 m . The controller parameters are chosen as follows: k x = 16 m, k v = 5 . 6 m, k R = 8 . 81 , k Ω = 2 . 54 . W e consider two complex flight maneuvers. The first case corresponds to the position controlled mode; the results in Proposition 4 are referenced. The second case in volves transitions between all of the three flight modes. Case (I): P osition Controlled Flight Mode: Consider a hov ering maneuver for which the quadrotor U A V recov- ers from being initially upside down. The desired tracking commands are as follows. x d ( t ) = [0 , 0 , 0] , b 1 d ( t ) = [1 , 0 , 0] . and it is desired to maintain the quadrotor U A V at a constant altitude. Initial conditions are chosen as x (0) = [0 , 0 , 0] , v (0) = [0 , 0 , 0] , R (0) =   1 0 0 0 − 0 . 9995 − 0 . 0314 0 0 . 0314 − 0 . 9995   , Ω(0) = [0 , 0 , 0] . This initial condition corresponds to an upside down quadro- tor U A V . The preferred direction of the total thrust vector in the con- trolled system is − b 3 . But initially , it is given by − b 3 (0) = − R (0) e 3 = [0 , 0 . 0314 , 0 . 9995] , which is almost opposite to the thrust direction [0 , 0 , − 1] required for the given hovering command. This yields a large initial attitude error , namely 178 ◦ in terms of the rotation angle about the eigen-axis between R c (0) and R (0) , and the corresponding the initial attitude err or is Ψ(0) = 1 . 995 . 0 1 2 3 4 5 6 − 0.5 0 0.5 1 1.5 2 t Ψ (a) Attitude error function Ψ 0 1 2 3 4 5 6 − 1 0 1 0 1 2 3 4 5 6 − 0.5 0 0.5 x, x d 0 1 2 3 4 5 6 − 1 0 1 t (b) Position ( x :solid, x d :dotted, ( m )) 0 1 2 3 4 5 6 − 10 0 10 0 1 2 3 4 5 6 − 1 0 1 Ω 0 1 2 3 4 5 6 − 1 0 1 t (c) Angular velocity ( rad / sec ) 0 1 2 3 4 5 6 − 50 0 50 f 1 0 1 2 3 4 5 6 − 50 0 50 f 2 0 1 2 3 4 5 6 − 50 0 50 f 3 0 1 2 3 4 5 6 − 50 0 50 f 4 t (d) Thrust of each rotor ( N ) Fig. 4. Case I: position controlled flight mode for a hovering, recovering from an initially upside down attitude Therefore, we cannot apply Proposition 3 that giv es ex- ponential stability when Ψ(0) < 1 , but by Proposition 4, we can guarantee exponential attractiv eness. From Proposition 1, the attitude err or function Ψ decreases; it ev entually becomes less than 1 at t = 0 . 88 seconds as illustrated in Figure 4(a). At that instant, the attitude err or enters the region of attraction specified in Proposition 3. Therefore, for t > 0 . 88 seconds, the position tracking error conv erges to zero exponentially as shown in Figures 4(b). The region of attraction of the proposed control system almost cov ers SO ( 3 ) , so that the controlled quadrotor U A V can recov er from being initially upside do wn. Case (II): T ransition Between Sever al Flight Modes: This flight maneuver consists of a sequence of fiv e flight modes, including a rotation by 720 ◦ (see Figure 5). (a) V elocity controlled flight mode ( t ∈ [0 , 4) ) v d ( t ) = [1 + 0 . 5 t, 0 . 2 sin(2 π t ) , − 0 . 1] , b 1 d ( t ) = [1 , 0 , 0] . (b) Attitude controlled flight mode ( t ∈ [4 , 6) ): rotation about the second body-fixed axis by 720 ◦ R d ( t ) = exp(2 π ( t − 4) ˆ e 2 ) . (c) Position controlled flight mode ( t ∈ [6 , 8) ) x d ( t ) = [14 − t, 0 , 0] , b 1 d ( t ) = [1 , 0 , 0] . (d) Attitude controlled flight mode ( t ∈ [8 , 9) ): rotation about the first body-fixed axis by 360 ◦ R d ( t ) = exp(2 π ( t − 8) ˆ e 1 ) . (e) Position controlled flight mode ( t ∈ [9 , 12] ) x d ( t ) = [20 − 5 3 t, 0 , 0] , b 1 d ( t ) = [0 , 1 , 0] . initial/terminal position (a) velocity tracking (b) attitude tracking (c) position tracking (d) attitude tracking (e) position tracking ~ e 1 ~ e 2 ~ e 3 ~ b 1 ~ b 1 ~ b 1 Fig. 5. Case II: complex maneuver of a quadrotor U A V inv olving a rotation by 720 ◦ about ~ e 2 (b), and a rotation by 360 ◦ about ~ e 1 (d), with transitions between several flight modes. The direction of the first body-fixed axis is specified for velocity/position tracking modes ((a),(c),(e)) (an animation illustrating this maneuver is av ailable at http://my .fit.edu/˜taeyoung). Initial conditions are same as the first case. The second case inv olves transitions between several flight modes. It begins with a velocity controlled flight mode. As the initial attitude error function is less than 1, the velocity tracking error exponentially conv erges as shown at Figure 6(d), and the first body-fixed axis asymptotically lies in the plane spanned by b 1 d = e 1 and g e 3 − ˙ v d . Since k ˙ v d k  g , the first body-fixed axis remains close to the plane composed of e 1 and e 3 , as illustrated in Figure 6(e). This is followed by an attitude tracking mode to rotate the quadrotor by 720 ◦ about the second body-fixed axis according to Proposition 1. As discussed in Section IV, the thrust magnitude f can be arbitrarily chosen in an attitude controlled flight mode. W e cannot apply the results of Proposition 2 for altitude tracking, since the third body- fixed axis becomes horizontal sev eral times during the giv en attitude maneuver . Here we choose the thrust magnitude giv en by f ( t ) = ( k x ( x ( t ) − x c ) + k v v ( t ) + mg e 3 ) · R ( t ) e 3 , which is equiv alent to the thrust magnitude for the position controlled flight mode given in (19), when x d ( t ) = x c = [8 , 0 , 0] . This does not guarantee asymptotic con vergence of the quadrotor U A V position to [8 , 0 , 0] since the direction of the total thrust is determined by the giv en attitude command. But, it has the effects that the position of the quadrotor UA V stays close to x c , as illustrated at Figure 6(b). Next, a position tracking mode is again engaged, and the quadrotor U A V soon follo ws a straight line. Another attitude tracking mode and a position tracking mode are repeated to rotate the quadrotor by 360 ◦ about the direction of the second body-fixed axis. The thrust magnitude is chosen as f ( t ) = ( k x ( x ( t ) − x c ) + k v v ( t ) + mg e 3 ) · R ( t ) e 3 , where x c = [6 , 0 , 0] , to make the position of the quadrotor U A V remain close to x c during this attitude maneuver , as discussed above. For the position tracking modes (c) and (e), we hav e ¨ x d = 0 , and b 1 d lies in the horizontal plane. Therefore, according to Proposition 5, the first body-fixed b 1 asymptotically conv erges to b 1 d , as shown at Figure 6(e). For example, during the last position tracking mode (e), the first body-fixed axis points to the left of the flight path since b 1 d is specified to be e 2 . These illustrate that by switching between an attitude mode and a position and heading flight mode, the quadrotor UA V can perform the prescribed complex acrobatic maneuver . V I I I . C O N C L U S I O N S W e presented a global dynamic model for a quadrotor U A V , and we developed tracking controllers for three dif fer- ent flight modes; these were developed in terms of the special Euclidean group that is intrinsic and coordinate-free, thereby av oiding the singularities of Euler angles and the ambiguities of quaternions in representing attitude. Using the proposed geometric based controllers for the three flight modes we studied, the quadrotor exhibits exponential stability when the initial attitude error is less than 90 ◦ , and it yields almost global exponentially attractiveness when the initial attitude error is less than 180 ◦ . By switching between dif ferent controllers for these flight modes, we have demonstrated that the quadrotor U A V can perform complex acrobatic maneuvers. Several different complex flight maneuvers were demonstrated in the numerical examples. A P P E N D I X A. Pr operties of the Hat Map The hat map ˆ · : R 3 → so (3) is defined as ˆ x =   0 − x 3 x 2 x 3 0 − x 1 − x 2 x 1 0   (47) for x = [ x 1 ; x 2 ; x 3 ] ∈ R 3 . This identifies the Lie algebra so (3) with R 3 using the vector cross product in R 3 . The in verse of the hat map is referred to as the vee map, ∨ : so (3) → R 3 . Sev eral properties of the hat map are summarized as follows. ˆ xy = x × y = − y × x = − ˆ y x, (48) − 1 2 tr [ ˆ x ˆ y ] = x T y , (49) 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 1.2 1.4 t Ψ (a) Attitude error function Ψ 0 2 4 6 8 10 12 0 5 10 0 2 4 6 8 10 12 − 0.5 0 0.5 x, x d 0 2 4 6 8 10 12 − 1 0 1 t (b) Position ( x :solid, x d :dotted, ( m )) 0 2 4 6 8 10 12 − 10 0 10 0 2 4 6 8 10 12 − 10 0 10 Ω , Ω d 0 2 4 6 8 10 12 − 5 0 5 t (c) Angular velocity ( Ω :solid, Ω d :dotted, ( rad / sec )) 0 2 4 6 8 10 12 − 1 0 1 0 2 4 6 8 10 12 − 1 0 1 b 1 ,b 1 d 0 2 4 6 8 10 12 − 1 0 1 t (d) V elocity ( v :solid, v d :dotted, ( m / s )) 0 2 4 6 8 10 12 − 1 0 1 0 2 4 6 8 10 12 − 1 0 1 b 1 ,b 1 d 0 2 4 6 8 10 12 − 1 0 1 t (e) First body-fixed axis ( b 1 :solid, b 1 d :dotted) Fig. 6. Case II: transitions between several flight modes for a comple x maneuver tr [ ˆ xA ] = tr [ A ˆ x ] = 1 2 tr  ˆ x ( A − A T )  = − x T ( A − A T ) ∨ , (50) ˆ xA + A T ˆ x = ( { tr [ A ] I 3 × 3 − A } x ) ∧ , (51) R ˆ xR T = ( Rx ) ∧ , (52) for any x, y ∈ R 3 , A ∈ R 3 × 3 , and R ∈ SO ( 3 ) . B. Pr oof of Pr oposition 1 W e first find the error dynamics for e R , e Ω , and define a L yapunov function. Then, we show that under the gi ven conditions, R ( t ) always lies in the suble vel set L 2 , which guarantees the positiv e-definiteness of the attitude error function Ψ . From this, we show exponential stability of the attitude error dynamics. a) Attitude Err or Dynamics: W e find the error dynam- ics for Ψ , e R , e Ω as follows. Using the attitude kinematics equations, namely ˙ R = R ˆ Ω , ˙ R d = R d ˆ Ω d , and equation (52), the time deriv ative of Ψ is giv en by ˙ Ψ( R, R d ) = − 1 2 tr h − ˆ Ω d R T d R + R T d R ˆ Ω i = − 1 2 tr h R T d R ( ˆ Ω − R T R d ˆ Ω d R T d R ) i . By (9), (50), this can be written as ˙ Ψ( R, R d ) = 1 2 e T Ω ( R T d R − R T R d ) ∨ = e R · e Ω . (53) Using equations (10) and (51), the time deriv ativ e of e R can be written as ˙ e R = 1 2 ( R T d R ˆ e Ω + ˆ e Ω R T R d ) ∨ = 1 2 ( tr  R T R d  I − R T R d ) e Ω ≡ C ( R T d R ) e Ω . (54) Now we show that k C ( R T d R ) k 2 ≤ 1 for any R T d R ∈ SO ( 3 ) . Using Rodrigues’ formula [13], we can show that the eigen values of C T (exp ˆ x ) C (exp ˆ x ) are giv en by cos 2 k x k , 1 2 (1 + cos k x k ) , and 1 2 (1 + cos k x k ) , which are less than or equal to 1 for any x ∈ R 3 . Therefore, k C ( R T d R ) k 2 ≤ 1 , and this implies that k ˙ e R k ≤ k e Ω k . (55) From equation (9), the time deriv ativ e of e Ω is giv en by J ˙ e Ω = J ˙ Ω + J ( ˆ Ω R T R d Ω d − R T R d ˙ Ω d ) , where we use a property of the hat map, ˆ xx = 0 for any x ∈ R 3 . Substituting the equation of motion (5) and the control moment (20), this reduces to J ˙ e Ω = − k R e R − k Ω e Ω . (56) In short, the attitude error dynamics are giv en by equations (53), (54), (56), and they satisfy (55). b) L yapunov Candidate: For a non-negati ve constant c 2 , let a L yapunov candidate V 2 be V 2 = 1 2 e Ω · J e Ω + k R Ψ( R, R d ) + c 2 e R · e Ω . (57) From equations (53), (54), (56), the time deri vati ve of V 2 is giv en by ˙ V 2 = e Ω · J ˙ e Ω + k R e R · e Ω + c 2 ˙ e R · e Ω + c 2 e R · ˙ e Ω = − k Ω k e Ω k 2 − c 2 k R e R · J − 1 e R + c 2 C ( R T d R ) e Ω · e Ω − c 2 k Ω e R · J − 1 e Ω . (58) Since k C ( R T d R ) k ≤ 1 , this is bounded by ˙ V 2 ≤ − z T 2 W 2 z 2 , (59) where z 2 = [ k e R k , k e Ω k ] T , and the matrix W 2 ∈ R 2 × 2 is giv en by W 2 = " c 2 k R λ M ( J ) − c 2 k Ω 2 λ m ( J ) − c 2 k Ω 2 λ m ( J ) k Ω − c 2 # . (60) c) Boundedness of Ψ : Define V 0 2 = V 2   c 2 =0 . From (57), (58), we have V 0 2 = 1 2 e Ω · J e Ω + k R Ψ( R, R d ) , ˙ V 0 2 = − k Ω k e Ω k 2 ≤ 0 . This implies that V 0 2 is non-increasing, i.e., V 0 2 ( t ) ≤ V 0 2 (0) . Using (13), the initial value of V 0 2 is bounded by V 0 2 (0) < 2 k R . Therefore, we obtain k R Ψ( R ( t ) , R d ( t )) ≤ V 0 2 ( t ) ≤ V 0 2 (0) < 2 k R . (61) Therefore, the attitude error function is bounded by Ψ( R ( t ) , R d ( t )) ≤ ψ 2 < 2 , for any t ≥ 0 , (62) and for ψ 2 = 1 k R V 0 2 (0) . Therefore, R ( t ) always lies in the sublev el set L 2 = { R ∈ SO ( 3 ) | Ψ( R, R d ) < 2 } . d) Exponential Stability: Now , we show exponential stability of the attitude dynamics by considering the general case where the constant c 2 is positiv e. Using Rodrigues’ formula, we can show that Ψ( R, R d ) = 1 − cos k x k , (63) k e R k 2 = sin 2 k x k = (1 + cos k x k )Ψ( R, R d ) = (2 − Ψ( R, R d ))Ψ( R, R d ) , (64) when R T d R = exp ˆ x for x ∈ R 3 . Therefore, from (62), the attitude error function satisfies 1 2 k e R k 2 ≤ Ψ( R, R d ) ≤ 1 2 − ψ 2 k e R k 2 . (65) This implies that Ψ is positive-definite and decrescent. It follows that the L yapunov function V 2 is bounded as z T 2 M 21 z 2 ≤ V 2 ≤ z T 2 M 22 z 2 , (66) where M 21 = 1 2  k R − c 2 − c 2 λ m ( J )  , M 22 = 1 2  2 k R 2 − ψ 2 c 2 c 2 λ M ( J )  . (67) W e choose the positiv e constant c 2 such that c 2 < min  k Ω , 4 k Ω k R λ m ( J ) 2 k 2 Ω λ M ( J ) + 4 k R λ m ( J ) 2 , p k R λ m ( J )  , which makes the matrix W 2 in (59) and the matrices M 21 , M 22 in (66) positive-definite. Therefore, we obtain λ m ( M 21 ) k z 2 k 2 ≤ V 2 ≤ λ M ( M 22 ) k z 2 k 2 , (68) ˙ V 2 ≤ − λ m ( W 2 ) k z 2 k 2 . (69) Let β 2 = λ m ( W 2 ) λ M ( M 22 ) . Then, we have ˙ V 2 ≤ − β 2 V 2 . (70) Therefore, the zero equilibrium of the attitude tracking error e R , e Ω is exponentially stable. Using (65), this implies that (2 − ψ 2 ) λ m ( M 21 )Ψ ≤ λ m ( M 21 ) k e R k 2 ≤ λ m ( M 21 ) k z 2 k 2 ≤ V 2 ( t ) ≤ V 2 (0) e − β 2 t . Thus, the attitude error function Ψ exponentially decreases. But, from (62), it is also guaranteed that Ψ < 2 . This yields (14). C. Pr oof of Pr oposition 2 The rotational dynamics (4), (5) are decoupled from the translational dynamics (2), (3). As the control moment and assumptions are identical to Proposition 1, all of the conclu- sions of Proposition 1 hold. T o sho w altitude tracking, we take the dot product of (3) with e 3 to obtain m ¨ x 3 = mg − f e 3 · Re 3 . Substituting (15) into this, we obtain the altitude error dynamics as follows: m ¨ x 3 = − k x ( x 3 − x 3 d ) − k v ( ˙ x 3 − ˙ x 3 d ) + m ¨ x 3 d . It it clear that this second-order linear system is e xponentially stable for positive k x , k v . D. Pr oof of Pr oposition 3 W e first deriv e the tracking error dynamics. Using a L yapunov analysis, we show that the v elocity tracking error is uniformly bounded, from which we establish exponential stability . a) Boundedness of e R : The assumptions of Proposition 3, namely (26), (34) imply satisfaction of the assumptions of Proposition 1, (12), (13), replacing the notation R d by R c . Therefore, the results of Proposition 1 can be directly applied throughout this proof. From (34), equation (61) can be replaced by k R Ψ( R ( t ) , R c ( t )) ≤ V 0 2 (0) < k R ψ 1 . (71) Therefore, the attitude error function is bounded by Ψ( R ( t ) , R d ( t )) ≤ ψ 1 < 1 , for any t ≥ 0 . (72) This implies that for the attitude alw ays lies in the sublev el set L 1 = { R ∈ SO ( 3 ) | Ψ( R , R c ) < 1 } . From (63), the attitude err or is less than 90 ◦ . Similar to (65), we can show that 1 2 k e R k 2 ≤ Ψ( R, R c ) ≤ 1 2 − ψ 1 k e R k 2 . (73) W e also define the following domain D D = { ( e x , e v , R, e Ω ) ∈ R 3 × R 3 × L 1 × R 3 | k e x k < e x max } , (74) for a fixed constant e x max , restricting the magnitude of the position error . The subsequent L yapunov analysis is dev eloped in this domain D . b) T ranslational Error Dynamics: The time deri vati ve of the position error is ˙ e x = e v . The time-deriv ative of the velocity error is giv en by m ˙ e v = m ¨ x − m ¨ x d = mg e 3 − f Re 3 − m ¨ x d . (75) Consider the quantity e T 3 R T c Re 3 , which represents the cosine of the angle between b 3 = R e 3 and b c 3 = R c e 3 . Since 1 − Ψ( R, R c ) represents the cosine of the eigen-axis rotation angle between R c and R , as discussed in (63), we have 1 > e T 3 R T c Re 3 > 1 − Ψ( R, R c ) > 0 . Therefore, the quantity 1 e T 3 R T c Re 3 is well-defined. T o rewrite the error dynamics of e v in terms of the attitude error e R , we add and subtract f e T 3 R T c Re 3 R c e 3 to the right hand side of (75) to obtain m ˙ e v = mg e 3 − m ¨ x d − f e T 3 R T c Re 3 R c e 3 − X , (76) where X ∈ R 3 is defined by X = f e T 3 R T c Re 3 (( e T 3 R T c Re 3 ) Re 3 − R c e 3 ) . (77) Let A = − k x e x − k v e v − mg e 3 + m ¨ x d . Then, from (19), (23), we ha ve f = − A · Re 3 and b 3 c = R c e 3 = − A/ k A k , i.e. − A = k A k R c e 3 . By combining these, we obtain f = ( k A k R c e 3 ) · Re 3 . Therefore, the third term of the right hand side of (76) can be written as − f e T 3 R T c Re 3 R c e 3 = − ( k A k R c e 3 ) · Re 3 e T 3 R T c Re 3 · − A k A k = A = − k x e x − k v e v − mg e 3 + m ¨ x d . Substituting this into (76), the error dynamics of e v can be written as m ˙ e v = − k x e x − k v e v − X . (78) c) L yapunov Candidate for T ranslation Dynamics: For a positiv e constant c 1 , let a L yapunov candidate V 1 be V 1 = 1 2 k x k e x k 2 + 1 2 m k e v k 2 + c 1 e x · e v . (79) The deriv ativ e of V 1 along the solution of (78) is giv en by ˙ V 1 = k x e x · e v + e v · {− k x e x − k v e v + X } + c 1 e v · e v + c 1 m e x · {− k x e x − k v e v + X } = − ( k v − c 1 ) k e v k 2 − c 1 k x m k e x k 2 − c 1 k v m e x · e v + X · n c 1 m e x + e v o . (80) W e find a bound on X using (77) as follows. Since f = k A k ( e T 3 R T c Re 3 ) , we have k X k ≤ k A k k ( e T 3 R T c Re 3 ) Re 3 − R c e 3 k ≤ ( k x k e x k + k v k e v k + B ) k ( e T 3 R T c Re 3 ) Re 3 − R c e 3 k . The last term k ( e T 3 R T c Re 3 ) Re 3 − R c e 3 k represents the sine of the angle between b 3 = Re 3 and b c 3 = R c e 3 , since ( b 3 c · b 3 ) b 3 − b 3 c = b 3 × ( b 3 × b 3 c ) . From (64), k e R k represents the sine of the eigen-axis rotation angle between R c and R . Therefore, we hav e k ( e T 3 R T c Re 3 ) Re 3 − R c e 3 k ≤ k e R k . From (64), (72), it follows that k ( e T 3 R T d Re 3 ) Re 3 − R d e 3 k ≤ k e R k = p Ψ(2 − Ψ) ≤ p ψ 1 (2 − ψ 1 ) ≡ α < 1 . Therefore, X is bounded by k X k ≤ ( k x k e x k + k v k e v k + B ) k e R k ≤ ( k x k e x k + k v k e v k + B ) α. (81) Substituting this into (80), ˙ V 1 ≤ − ( k v − c 1 ) k e v k 2 − c 1 k x m k e x k 2 − c 1 k v m e x · e v + ( k x k e x k + k v k e v k + B ) k e R k n c 1 m k e x k + k e v k o ≤ − ( k v (1 − α ) − c 1 ) k e v k 2 − c 1 k x m (1 − α ) k e x k 2 + c 1 k v m (1 + α ) k e x kk e v k + k e R k n c 1 m B k e x k + B k e v k + k x k e x kk e v k o . (82) In the above expression for ˙ V 1 , there is a third-order error term, namely k x k e R kk e x kk e v k . It is possible to choose its upper bound as k x α k e x kk e v k similar to other terms, b ut the corresponding stability analysis becomes complicated, and the initial attitude error should be reduced further . Instead, we restrict our analysis to the domain D defined at (74), and an upper bound is chosen as k x e x max k e R kk e v k . d) L yapunov Candidate for the Complete System:: Let V = V 1 + V 2 be the L yapunov candidate of the complete system. V = 1 2 k x k e x k 2 + 1 2 m k e v k 2 + c 1 e x · e v + 1 2 e Ω · J e Ω + k R Ψ( R, R d ) + c 2 e R · e Ω . (83) Using (73), the bound of the L yapunov candidate V can be written as z T 1 M 11 z 1 + z T 2 M 21 z 2 ≤ V ≤ z T 1 M 12 z 1 + z T 2 M 0 22 z 2 , (84) where z 1 = [ k e x k , k e v k ] T , z 2 = [ k e R k , k e Ω k ] T ∈ R 2 , and the matrices M 11 , M 12 , M 21 , M 22 are giv en by M 11 = 1 2  k x − c 1 − c 1 m  , M 12 = 1 2  k x c 1 c 1 m  , M 21 = 1 2  k R − c 2 − c 2 λ m ( J )  , M 0 22 = 1 2  2 k R 2 − ψ 1 c 2 c 2 λ M ( J )  . Using (59) and (82), the time-deriv ativ e of V is given by ˙ V ≤ − z T 1 W 1 z 1 + z T 1 W 12 z 2 − z T 2 W 2 z 2 , (85) where W 1 , W 12 , W 2 ∈ R 2 × 2 are defined as follows: W 1 =  c 1 k x m (1 − α ) − c 1 k v 2 m (1 + α ) − c 1 k v 2 m (1 + α ) k v (1 − α ) − c 1  , (86) W 12 =  c 1 m B 0 B + k x e x max 0  , (87) W 2 = " c 2 k R λ M ( J ) − c 2 k Ω 2 λ m ( J ) − c 2 k Ω 2 λ m ( J ) k Ω − c 2 # . (88) e) Exponential Stability: Under the gi ven conditions (31), (32) of the proposition, all of the matrices M 11 , M 12 , W 1 , M 21 , M 22 , W 2 , and the L yapunov candidate V become positiv e-definite, and ˙ V ≤ − λ m ( W 1 ) k z 1 k 2 + k W 12 k 2 k z 1 kk z 2 k − λ m ( W 2 ) k z 2 k 2 . The condition giv en by (33) guarantees that ˙ V becomes negati ve-definite. Therefore, the zero equilibrium of the tracking errors of the complete dynamics is exponentially stable. A (conservati ve) region of attraction is characterized by a sub-level set of V contained in the domain D , as written at (35), as well as (34) required for the boundedness of e R . E. Pr oof of Pr oposition 4 The given assumptions (36), (37) satisfy the assumption of Proposition 1, from which the tracking error z 2 = [ k e R k , k e Ω k ] is guaranteed to exponentially decreases, and to enter the region of attraction of Proposition 3, gi ven by (26), (34), in a finite time t ∗ . Therefore, if we show that the tracking error z 1 = [ k e x k , k e v k ] is bounded in t ∈ [0 , t ∗ ] , then the tracking error z = [ z 1 , z 2 ] is uniformly bounded for any t > 0 , and it exponentially decreases for t > t ∗ . This yields exponential attractiv eness. The boundedness of z 1 is shown as follows. The error dynamics or e v can be written as m ˙ e v = mg e 3 − f Re 3 − m ¨ x d . Let V 3 be a positive-definite function of k e x k and k e v k : V 3 = 1 2 k e x k 2 + 1 2 m k e v k 2 . Then, we have k e x k ≤ √ 2 V 3 , k e v k ≤ q 2 m V 3 . The time- deriv ativ e of V 3 is giv en by ˙ V 3 = e x · e v + e v · ( mg e 3 − f Re 3 − m ¨ x d ) ≤ k e x kk e v k + k e v kk mg e 3 − m ¨ x d k + k e v kk Re 3 k| f | . Using (25), (19), we obtain ˙ V 3 ≤ k e x kk e v k + k e v k B + k e v k ( k x k e x k + k v k e v k + B ) = k v k e v k 2 + (2 B + ( k x + 1) k e x k ) k e v k ≤ d 1 V 3 + d 2 p V 3 , where d 1 = k v 2 m + 2( k x + 1) 1 √ m , d 2 = 2 B q 2 m . Suppose that V 3 ≥ 1 for a time interval [ t a , t b ] ⊂ [0 , t ∗ ] . In this time interval, we ha ve √ V 3 ≤ V 3 . Therefore, ˙ V 3 ≤ ( d 1 + d 2 ) V 3 ⇒ V 3 ( t ) ≤ V 3 ( t a ) e ( d 1 + d 2 )( t − t a ) . Therefore, for any time interval in which V 3 ≥ 1 , V 3 is bounded. This implies that V 3 is bounded for 0 ≤ t ≤ t ∗ . 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