Geometric Tracking Control of a Quadrotor UAV on SE(3)

Transcription

1 Geometric Tracking Control o a Quadrotor UAV on SE(3) Taeyoung Lee, Melvin Leok, and N. Harris McClamroch Abstract This paper provides new results or the tracking control o a quadrotor unmanned aerial vehicle (UAV). The UAV has our input degrees o reedom, namely the magnitudes o the our rotor thrusts, that are used to control the six translational and rotational degrees o reedom, and to achieve asymptotic tracking o our outputs, namely, three position variables or the vehicle center o mass and the direction o one vehicle body-ixed axis. A globally deined model o the quadrotor UAV rigid body dynamics is introduced as a basis or the analysis. A nonlinear tracking controller is developed on the special Euclidean group SE(3) and it is shown to have desirable closed loop properties that are almost global. Several numerical examples, including an example in which the quadrotor recovers rom being initially upside down, illustrate the versatility o the controller. I. INTRODUCTION A quadrotor unmanned aerial vehicle (UAV) consists o two pairs o counter-rotating rotors and propellers, located at the vertices o a square rame. It is capable o vertical take-o and landing (VTOL), but it does not require complex mechanical linkages, such as swash plates or teeter hinges, that commonly appear in typical helicopters. Due to its simple mechanical structure, it has been envisaged or various applications such as surveillance or mobile sensor networks as well as or educational purposes. There are several university-level projects [, [, [3, [4, and commercial products [5, [6, [7 related to the development and application o quadrotor UAVs. Despite the substantial interest in quadrotor UAVs, little attention has been paid to constructing nonlinear control systems or them, particularly to designing nonlinear tracking controllers. Linear control systems such as proportionalderivative controllers or linear quadratic regulators are widely used to enhance the stability properties o an equilibrium [, [3, [4, [8, [9. A nonlinear controller is developed or the linearized dynamics o a quadrotor UAV with saturated positions in [. Backstepping and sliding mode techniques are applied in [. Since these controllers are based on Euler angles, they exhibit singularities when representing complex rotational maneuvers o a quadrotor UAV, thereby undamentally restricting their ability to track nontrivial trajectories. Taeyoung Lee, Mechanical and Aerospace Engineering, Florida Institute o Technology, Melbourne, FL 39 taeyoung@it.edu Melvin Leok, Mathematics, University o Caliornia at San Diego, La Jolla, CA 993 mleok@math.ucsd.edu N. Harris McClamroch, Aerospace Engineering, University o Michigan, Ann Arbor, MI 489 nhm@umich.edu This research has been supported in part by NSF under grants DMS- 7663, DMS-5 and DMS-687. This research has been supported in part by NSF under grant CMS Geometric control is concerned with the development o control systems or dynamic systems evolving on nonlinear maniolds that cannot be globally identiied with Euclidean spaces [, [3, [4. By characterizing geometric properties o nonlinear maniolds intrinsically, geometric control techniques provide unique insights to control theory that cannot be obtained rom dynamic models represented using local coordinates [5. This approach has been applied to ully actuated rigid body dynamics on Lie groups to achieve almost global asymptotic stability [4, [6, [7, [8 In this paper, we develop a geometric controller or a quadrotor UAV. The dynamics o a quadrotor UAV is expressed globally on the coniguration maniold o the special Euclidean group SE(3). We construct a tracking controller to ollow prescribed trajectories or the center o mass and heading direction. It is shown that this controller exhibits almost global exponential attractiveness to the zero equilibrium o tracking errors. Since this is a coordinateree control approach, it completely avoids singularities and complexities that arise when using local coordinates. Compared to other geometric control approaches or rigid body dynamics, this is distinct in the sense that it controls an underactuated quadrotor UAV to stabilize six translational and rotational degrees o reedom using our thrust inputs, while asymptotically tracking our outputs consisting o its position and heading direction. We demonstrate that this controller is particularly useul or complex, acrobatic maneuvers o a quadrotor UAV, such as recovering rom being initially upside down. The paper is organized as ollows. We develop a globally deined model or the translational and rotational dynamics o a quadrotor UAV in Section II. Tracking control results on SE(3) are presented in Section III. Several numerical results are presented in Section IV. Proos are relegated to the Appendix. II. QUADROTOR DYNAMICS MODEL Consider a quadrotor vehicle model illustrated in Fig.. This is a system o our identical rotors and propellers located at the vertices o a square, which generate a thrust and torque normal to the plane o this square. We choose an inertial reerence rame e, e, e 3 } and a body-ixed rame b, b, b 3 }. The origin o the body-ixed rame is located at the center o mass o this vehicle. Deine m R the total mass J R 3 3 the inertia matrix with respect to the bodyixed rame R SO(3) the rotation matrix rom the body-ixed rame to the inertial rame

2 3 4 b x d bd Trajectory tracking b3d Controller Attitude tracking M x, v, R, Ω Quadrotor Dynamics x b3 R Fig.. Controller structure e b The equations o motion o this quadrotor UAV can be written as e 3 e Fig.. Quadrotor model Ω R 3 the angular velocity in the body-ixed rame x R 3 the location o the center o mass in the inertial rame v R 3 the velocity o the center o mass in the inertial rame d R the horizontal distance rom the center o mass to the center o a rotor i R the thrust generated by the i-th propeller along the b 3 axis τ i R the torque generated by the i-th propeller about the b 3 axis R the total thrust, i.e., = 4 i= i M R 3 the total moment in the body-ixed rame The coniguration o this quadrotor UAV is deined by the location o the center o mass and the attitude with respect to the inertial rame. Thereore, the coniguration maniold is the special Euclidean group SE(3). We assume that the thrust o each propeller is directly controlled, i.e., we do not consider the dynamics o rotors and propellers. The total thrust is = 4 i= i, which acts along the direction o Re 3 in the inertial rame. We also assume that the torque generated by each propeller is directly proportional to its thrust. Since it is assumed that the irst and the third propellers rotate counterclockwise, and the second and the ourth propellers rotate clockwise, the torque generated by the i-th propeller can be written as τ i = ( ) i c τ i or a ixed constant c τ. Then, the total moment in the body-ixed rame is given by M = [d( 4 ), d( 3 ), c τ ( ). These can be written in matrix orm as M M = d d d d M 3 c τ c τ c τ c τ 3 4. () The determinant o the above 4 4 matrix is 8c τ d, so it is invertible when d and c τ. Thereore, or a given, M, the thrust o each rotor i can be obtained rom (). Using this equation, the total thrust R and the moment M R 3 are considered as control inputs in this paper. ẋ = v, () m v = mge 3 Re 3, (3) Ṙ = RˆΩ, (4) J Ω + Ω JΩ = M, (5) where the hat map ˆ : R 3 so(3) is deined by the condition that ˆxy = x y or all x, y R 3, and e 3 = [; ; R 3. III. GEOMETRIC TRACKING CONTROL ON SE(3) We develop a controller to ollow a prescribed trajectory x d (t) o the location o the center o mass, and the direction o the body-ixed axis b d (t), which represents the yawing (or heading) angle o a quadrotor UAV. We develop this controller directly on the nonlinear coniguration Lie group and thereby avoid any singularities and complexities that arise in local coordinates. As a result, we are able to achieve almost global exponential attractiveness to the zero equilibrium o tracking errors. The overall controller structure is illustrated in Fig.. The equations o motion ()-(5) have a cascade structure: the rotational motion o the attitude is decoupled rom the translational motion; the translational motion is only dependent on the term Re 3 in (3). The magnitude o the total thrust is directly controlled, but the direction o the total thrust Re 3 is determined by the third bodyixed axis b 3. Thereore, in order to change the direction o the total thrust, the attitude should be changed accordingly. Here, we choose the total thrust and the desired reduced attitude R d e 3 or the third body-ixed axis b 3 such that they stabilize the zero equilibrium o the tracking error or the translational dynamics, and the remaining irst two columns o R d representing the direction o the body-ixed axes b, b, are chosen to ollow a desired direction b d. The control moment M is designed to ollow the resulting desired attitude R d obtained by b d and b 3d. A. Tracking Errors We deine the tracking errors or x, v, R, Ω as ollows. The tracking errors or the position and the velocity are given by: e x = x x d, (6) e v = v v d. (7) The attitude and angular velocity tracking error should be careully chosen as they evolve on the tangent bundle o

3 the nonlinear space SO(3). The error unction on SO(3) is chosen to be Ψ(R, R d ) = tr[ I R T d R. (8) This is locally positive-deinite about R T d R = I within the region where the rotation angle between R and R d is less than 8 [4. This set can be represented by the sublevel set o Ψ where Ψ <, namely L = R d, R SO(3) Ψ(R, R d ) < }, which almost covers SO(3). When the variation o the rotation matrix is expressed as δr = Rˆη or η R 3, the derivative o the error unction is given by D R Ψ(R, R d ) Rˆη = (RT d R R T R d ) η, (9) where the vee map : so(3) R 3 is the inverse o the hat map. We used the act that tr[ˆxŷ = xt y or any x, y R 3. From this, the attitude tracking error is chosen to be e R = (RT d R R T R d ). () The tangent vectors Ṙ T RSO(3) and Ṙd T Rd SO(3) cannot be directly compared since they lie in dierent tangent spaces. We transorm Ṙd into a vector in T R SO(3), and we compare it with Ṙ as ollows: Ṙ Ṙd(R T d R) = RˆΩ R d ˆΩd R T d R = R(Ω R T R d Ω d ). We choose the tracking error or the angular velocity as ollows: e Ω = Ω R T R d Ω d. () We can show that e Ω is the angular velocity o the rotation matrix Rd T R, represented in the body-ixed rame, since d dt (RT d R) = (RT d R) ê Ω. B. Tracking Controller For given smooth tracking commands x d (t), b d (t), and some positive constants k x, k v, k R, k Ω, the control inputs, M are chosen as ollows: = ( k x e x k v e v mge 3 + mẍ d ) Re 3, () M = k R e R k Ω e Ω + Ω JΩ J(ˆΩR T R d Ω d R T R d Ωd ), (3) where the desired attitude R d is given by R d = [ b d ; b 3d bd ; b 3d SO(3), and b3d = R d e 3 = k xe x k v e v mge 3 + mẍ d k x e x k v e v mge 3 + mẍ d. (4) Here, we assume that the denominator o (4) is non-zero, k x e x k v e v mge 3 + mẍ d, (5) and that the desired trajectory satisies mge 3 + mẍ d < B (6) or a given positive constant B. These control inputs, M are designed as ollows. The control moment M given in (3) corresponds to a tracking controller on SO(3). For the attitude dynamics o a rigid body described by (4), (5), this controller exponentially stabilizes the zero equilibrium o the attitude tracking errors. Similarly, the expression in the parentheses in () corresponds to a tracking controller or the translational dynamics on R 3. The total thrust and the desired direction b 3d o the third body-ixed axis are chosen so that i there is no attitude tracking error, the control input term Re 3 at the translational dynamics o (3) becomes this tracking controller in R 3. Thereore, the trajectory tracking error will converge to zero provided that the attitude tracking error is identically zero. Certainly, the attitude tracking error may not be zero at any instant. As the attitude tracking error increases, the direction o the control input term Re 3 o the translational dynamics deviates rom the desired direction o R d e 3. This may cause instability on the complete dynamics. In (), we careully design the total thrust so that its magnitude is reduced when there is a larger attitude tracking error. The expression o includes the dot product o the desired third body-ixed axis b 3d = R d e 3 and the current third body-ixed axis b 3 = Re 3. Thereore, the magnitude o is reduced when the angle between those two axes becomes larger. These eects are careully analyzed in the stability proo or the complete dynamics described in the appendix. In short, this control system is designed so that the position tracking error converges to zero when there is no attitude tracking error, and it is properly adjusted or nonzero attitude tracking errors to achieve asymptotic stability o the complete dynamics. C. Exponential Asymptotic Stability We irst show exponential stability o the attitude dynamics in the sublevel set L = R d, R SO(3) Ψ(R, R d ) < }, and based on this results, we show exponential stability o the complete dynamics in the smaller sublevel set L = R d, R SO(3) Ψ(R, R d ) < } Proposition : (Exponential Stability o Attitude Dynamics) Consider the control moment M deined in (3) or any positive constants k R, k Ω. Suppose that the initial condition satisies Ψ(R(), R d ()) <, (7) e Ω () < λ min (J) k R( Ψ(R(), R d ())). (8) Then, the zero equilibrium o the attitude tracking error e R, e Ω is exponentially stable. Furthermore, there exist constants α, β > such that Ψ(R(t), R d (t)) min, α e βt}. (9) Proo: The proos o all the propositions are given in the appendix. In this proposition, (7), (8) represent a region o attraction or the attitude dynamics. This requires that the initial attitude error should be less than 8. Thereore, the region o attraction or the attitude almost covers SO(3), and the region o attraction or the angular velocity can be increased by choosing a larger controller gain k R in (8).

4 Since the direction o the total thrust is ixed to the third body-ixed axis, the stability o the translational dynamics depends on the attitude tracking error. More precisely, the position tracking perormance is aected by the dierence between b 3 = Re 3 and b 3d = R d e 3. In the proceeding stability analysis, it turns out that or the stability o the complete translational and rotational dynamics, the attitude error unction Ψ should be less than, which states that the initial attitude error should be less than 9. For the stability o the complete system, we restrict the initial attitude error, and we obtain the ollowing proposition. Proposition : (Exponential Stability o the Complete Dynamics) Consider the control orce and moment M deined at (), (3). Suppose that the initial condition satisies Ψ(R(), R d ()) ψ < () or a ixed constant ψ. Deine W, W, W R to be [ c k x W = m ckv m ( + α) ckv m ( + α) k, () v( α) c [ kx e W = + c vmax m B, () B [ W = c k R λ max(j) ckω λ min(j) ckω λ min(j) k Ω c, (3) where α = B ψ ( ψ ), e vmax = max e v (), k }, v( α) c, c R. For any positive constants k x, k v, we choose positive constants c, c, k R, k Ω such that 4mk x k v ( α) c < min k v ( α), kv( + α), } k x m, + 4mk x 4k Ω k R λ min (J) c < min k Ω, kω λ max(j) + 4k R λ min (J), } kr λ min (J), k R λ max (J), ψ (4) (5) λ min (W ) > 4 W λ min (W ). (6) Then, the zero equilibrium o the tracking errors o the complete system is exponentially stable. The region o attraction is characterized by () and e Ω () < λ min (J) k R( Ψ(R(), R d ())). (7) D. Almost Global Exponential Attractiveness Proposition requires that the initial attitude error is less than 9 to achieve the exponential stability o the complete dynamics. Suppose that this is not satisied, i.e. Ψ(R(), R d ()) <. From Proposition, we are guaranteed that the attitude error unction Ψ exponentially decreases, and thereore, it enters the region o attraction o Proposition in a inite time. Thereore, by combining the results o Proposition and, we can show almost global exponential attractiveness. Deinition : (Exponential Attractiveness [9) An equilibrium point z = o a dynamic systems is exponentially attractive i, or some δ >, there exists a constant α(δ) > and β > such that z() < δ z(t) α(δ)e βt or any t >. This should be distinguished rom the stronger notion o exponential stability, in which the constant α(δ) in the above bound is replaced by α(δ) z(). Proposition 3: (Almost Global Exponential Attractiveness o the Complete Dynamics) Consider a control system designed according to Proposition. Suppose that the initial condition satisies Ψ(R(), R d ()) <, (8) e Ω () < λ min (J) k R( Ψ(R(), R d ())). (9) Then, the zero equilibrium o the tracking errors o the complete dynamics is exponentially attractive. Since the region o attraction given by (8) or the attitude almost covers SO(3), this is reerred to as almost global exponential attractiveness in this paper. The region o attraction or the angular velocity can be expanded by choosing a larger gain k R in (9). E. Properties and Extensions One o the unique properties o the presented controller is that it is directly developed on SE(3) using rotation matrices. Thereore, it avoids the complexities and singularities associated with local coordinates o SO(3), such as Euler angles. It also avoids the ambiguities that arise when using quaternions to represent the attitude dynamics. As the threesphere S 3 double covers SO(3), any attitude eedback controller designed in terms o quaternions could yield dierent control inputs depending on the choice o quaternion vectors. The corresponding stability analysis would need to careully consider the act that convergence to a single attitude implies convergence to either o the two antipodal points on S 3. The use o rotation matrices in the controller design and stability analysis completely eliminates these diiculties. Another novelty o the presented controller is the choice o the total thrust in (). This is designed to ollow position tracking commands, but it is also careully designed to guarantee the overall stability o the complete dynamics by eedback control o the direction o the third body-ixed axis. This consideration is natural as each column o a rotation matrix represents the direction o each body-ixed axis. Thereore, another advantage o using rotation matrices is that the controller has a well-deined physical interpretation. In Propositions and 3, exponential stability and exponential attractiveness are guaranteed or almost all initial attitude errors, respectively. The attitude error unction deined in (8) has the ollowing critical points: the identity matrix, and rotation matrices that can be written as exp(πˆv) or any v S. These non-identity critical points o the attitude error unction lie outside o the region o attraction. As it is a two-dimensional subspace o the three-dimensional SO(3), we claim that the presented controller exhibits almost global properties in SO(3). It is impossible to construct a smooth controller on SO(3) that has global asymptotic stability. The

5 two-dimensional amily o non-identity critical points can be reduced to our points by modiying the error unction to be tr[g(i RT d R) or a matrix G I R3 3. The presented controller can be modiied accordingly. IV. NUMERICAL EXAMPLE The parameters o the quadrotor UAV are chosen according to a quadrotor UAV developed in [. J = [.8,.845,.377 kgm, m = 4.34 kg d =.35 m, c τ = m. The controller parameters are chosen as ollows: k x = 6m, k v = 5.6m, k R = 8.8, k Ω =.54. We consider the ollowing two cases. (I) This maneuver ollows an elliptical helix while rotating the heading direction at a ixed rate. Initial conditions are chosen as x() = [,,, v() = [,,, R() = I, Ω() = [,,. The desired trajectory is as ollows. x d (t) = [.4t,.4 sin πt,.6 cos πt, b (t) = [cos πt, sin πt,. (II) This maneuver recovers rom being initially upside down. Initial conditions are chosen as x() = [,,, v() = [,,, R() = , Ω() = [,, The desired trajectory is as ollows. x d (t) = [,,, b (t) = [,,. Simulation results are presented in Figures 3 and 4. For Case (I), the initial value o the attitude error unction Ψ() is less than.5. This satisies the conditions or Proposition, and exponential asymptotic stability is guaranteed. As shown in Figure 3, the tracking errors exponentially converge to zero. This example illustrates that the proposed controlled quadrotor UAV can ollow a complex trajectory that involve large angle rotations and nontrivial translations accurately. In Case (II), the initial attitude error is 78, which yields the initial attitude error unction Ψ() =.995 >. This corresponds to Proposition 3, which implies almost global exponential attractiveness. In Figure 4(b), the attitude error unction Ψ decreases, and it becomes less than at t =.88 seconds. Ater that instant, the position tracking error and the angular velocity error converge to zero as shown in Figures 4(c) and 4(d). The region o attraction o the proposed control system almost covers SO(3), so that the corresponding controlled quadrotor UAV can recover rom being initially upside down. (a) Snapshots or t.6 (an animation is available at taeyoung) (b) Attitude error unction Ψ 5 (d) Angular velocity (Ω:solid, Ω d :dotted, (rad/sec)) (c) Position (x:solid, x d :dotted, (m)) 5 (e) Thrust o each rotor (N) Fig. 3. Case I: ollowing an elliptic helix (horizontal axes represent simulation time in seconds) V. CONCLUSION We presented a global dynamic model or a quadrotor UAV, and we developed a geometric tracking controller directly on the special Euclidean group that is intrinsic and coordinate-ree, thereby avoiding the singularities o Euler angles and the ambiguities o quaternions in representing attitude. It exhibits exponential stability when the initial attitude error is less than 9, and it yields almost global exponentially attractiveness when the initial attitude error is less than 8. These are illustrated by numerical examples. This controller can be extended as ollows. In this paper, our input degrees o reedom are used to track a threedimensional position, and a one-dimensional heading direction. But, without changing the controller structure, they can be used to ollow arbitrary three-dimensional attitude commands. The remaining one input degree o reedom can be used to maintain the altitude as much as possible. By constructing a hybrid controller based on these two tracking modes, we can generate complicated acrobatic maneuvers o a quadrotor UAV. APPENDIX Proo o Proposition : We irst ind the error dynamics or e R, e Ω, and deine a Lyapunov unction. Then, we show that under the given conditions, (R(t), R d (t)) always lie in the

6 From (3), (3), the time derivative o V is given by V = e Ω Jė Ω [ k Rtr ˆΩ d Rd T R + Rd T RˆΩ + c ė R e Ω + c e R ė Ω = k Ω e Ω k R e R e Ω k Rtr [ R T d Rê Ω.5.5 (a) Snapshots or.5 t 4 (Snapshots are shited orward to represent the evolution o time. In reality, the quadrotor is lipped at a ixed position. An animation is available at taeyoung) (b) Attitude error unction Ψ 4 6 (d) Angular velocity (Ω:solid, Ω d :dotted, (rad/sec)) (c) Position (x:solid, x d :dotted, (m)) (e) Thrust o each rotor (N) Fig. 4. Case II: recovering rom an initially upside down attitude (horizontal axes represent simulation time in seconds) sublevel set L, which guarantees the positive-deiniteness o the attitude error unction Ψ. From this, we show the exponential stability o the attitude error dynamics. a) Error Dynamics: We ind the error dynamics or e R, e Ω as ollows. From the deinition o e Ω in (), the time derivative o e R is given by ė R = (RT d Rê Ω + ê Ω R T R d ) = (tr[ R T R d I R T R d )e Ω C(R T d R)e Ω. (3) We can show that C(R T d R) or any R T d R SO(3). Thus, ė R e Ω. From (), the time derivative o e Ω is given by Jė Ω = J Ω + J(ˆΩR T R d Ω d R T R d Ωd ). Substituting the equation o motion (5) and the control moment (3), this reduces to Jė Ω = k R e R k Ω e Ω. (3) b) Lyapunov Candidate: For a positive constant c, let a Lyapunov candidate V be V = e Ω Je Ω + k R Ψ(R, R d ) + c e R e Ω. (3) + c C(R T d R)e Ω e Ω + c e R J ( k R e R k Ω e Ω ). But, the third term o the above expression can be written as tr [ R T d Rê Ω = tr[ R T d Rê Ω ê Ω R T R d = tr[ ê Ω (R T d R R T R d ) = tr[ê Ω ê R = e Ω e R. Thereore, we obtain V = k Ω e Ω c k R e R J e R + c C(R T d R)e Ω e Ω c k Ω e R J e Ω. (33) Since C(Rd T R), this is bounded by V z T W z, (34) where z = [ e R, e Ω T, and the matrix W R is given by [ W = c k R λ max(j) ckω λ min(j) ckω λ min(j) k Ω c. (35) c) Boundedness o e R : Suppose that c =, then rom (3), (34), we have V c= = e Ω Je Ω + k R Ψ(R, R d ), V c= = k Ω e Ω. This implies that V c= is non-increasing. Thereore, using (8), the attitude error unction is bounded as ollows: k R Ψ(R(t), R d (t)) V c= (t) V () < k c= R. (36) This guarantees that there exists a constant ψ such that Ψ(R(t), R d (t)) ψ <, or any t. (37) Thereore (R(t), R d (t)) always lies in the sublevel set L ψ L. d) Exponential Stability: Within the sublevel set L ψ, the attitude error unction is positive-deinite, and we obtain e R Ψ e R. (38) ψ Thereore, the Lyapunov unction V is bounded by z T M z V z T M z, (39) where M = [ kr c, M c λ min (J) = [ kr ψ c c. λ max (J) (4)

7 We choose the positive constant c such that 4k Ω k R λ min (J) c < min k Ω, kω λ max(j) + 4k R λ min (J), k R λ min (J), } k R λ max (J), ψ which makes the matrices W, M, M positive-deinite. Then, the Lyapunov candidate V and V are bounded by λ min (M ) z V λ max (M ) z, (4) V λ min (W ) z. (4) Let β = λmin(w) λ max(m ). Then, we have V β V. (43) Thereore the zero equilibrium o the tracking error e R, e Ω is exponentially stable. Using (38), this implies that ( ψ )λ min (M )Ψ λ min (M ) e R λ min (M ) z V (t) V ()e βt. So, Ψ exponentially decreases. But, rom (37), it is also guaranteed that Ψ <. This yields (9). Proo o Proposition : We irst derive the tracking error dynamics. In particular, the velocity error is careully expressed according to the deinition o b 3d. Using a Lyapunov analysis, we show that the velocity tracking error is uniormly bounded, rom which we establish the exponential stability o the complete dynamics. a) Attitude Dynamics Error: The assumptions o Proposition, (), (7) imply (7), (8). The results o Proposition can be directly applied throughout this proo. From (7), equation (36) can be replaced by k R Ψ(R(t), R d (t)) V c= (t) V () < k c= R. (44) This guarantees that there exist a constant ψ such that Ψ(R(t), R d (t)) ψ < (45) or all t. This implies that or the given conditions, the attitude error always lies in the sublevel set L, i.e. the attitude error is less than 9. b) Position & Velocity Error Dynamics: Consider the error dynamics o the translational dynamics. The derivative o e v is given by mė v = mẍ mẍ d = mge 3 Re 3 mẍ d. (46) Consider a quantity e T 3 Rd T Re 3, which represents the cosine o the angle between b 3 and e 3. Since Ψ represents the cosine o the eigen-axis rotation angle between R d and R, we have > e T 3 Rd T Re 3 > Ψ >. Thereore, the quantity e T 3 RT d e Re3 v is well-deined. To rewrite this error dynamics o in terms o the attitude error e R, we add and subtract e R de T 3 to the right hand side o (46) to obtain 3 RT d Re3 mė v = mge 3 mẍ d e T 3 RT d Re R d e 3 X, (47) 3 where X R 3 is deined by X = e T 3 RT d Re ((e T 3 Rd T Re 3 )Re 3 R d e 3 ). (48) 3 Let A = k x e x k v e v mge 3 +mẍ d be the desired control orce or the translational dynamics. Then, rom (), (4), we obtain = A Re 3 = ( A R d e 3 ) Re 3. Thereore, e T 3 RT d Re R d e 3 = ( A R de 3 ) Re 3 3 e T 3 RT d Re 3 A A = A. Substituting this into (47), the error dynamics o e v can be written as mė v = k x e x k v e v X. (49) c) Lyapunov Candidate or the Translation Dynamics: For a positive constant c, let a Lyapunov candidate V be V = k x e x + m e v + c e x e v. (5) The derivative o V along the solution o (49) is given by V = (k v c ) e v c k x m e x c k v m e x e v c } + X m e x + e v. (5) We ind the bound o X at (48) as ollows. Since = A (e T 3 R T d Re 3), we have X A (e T 3 R T d Re 3 )Re 3 R d e 3 (k x e x + k v e v + B) (e T 3 R T d Re 3 )Re 3 R d e 3. Since the last term (e T 3 Rd T Re 3)Re 3 R d e 3 represents the sine o the angle between Re 3 and R d e 3, and e R represents the sine o the eigen-axis rotation angle between R d and R, we have (e T 3 R T d Re 3 )Re 3 R d e 3 e R = Ψ( Ψ) Thereore, X is bounded by ψ ( ψ ) <. X (k x e x + k v e v + B)α, (5) where α = ψ ( ψ ) <. Substituting this into (5), V (k v ( α) c ) e v c k x m ( α) e x + c k v m ( + α) e x e v + e R k x e x e v + c } m B e x + B e v ). (53) d) Boundedness o e v : In the above expression or V, there is a third-order error term, namely k x e R e x e v. We ind a bound on e v to change this term into a second-order error term or the preceding Lyapunov analysis. Suppose c =, k x =, then rom (5), (53), we have V c=k x= = m e v, V c=k x= k v( α) e v + B e v.

8 B This implies that when e v > k v( α), the time derivative o e v is negative, and e v monotonically decreases. Thereore, e v is uniormly bounded by } B e v (t) < max e v (), e vmax. (54) k v ( α) e) Lyapunov Candidate or the Complete System:: Let V = V + V be the Lyapunov candidate o the complete system. V = k x e x + m e v + c e x e v + e Ω Je Ω + k R Ψ(R, R d ) + c e R e Ω. (55) Using the results o Proposition, namely, (33), (39), we can show that the Lyapunov candidate V is bounded by z T M z + z T M z V z T M z + z T M z, (56) where z = [ e x, e v T, z = [ e R, e Ω T R, and the matrices M, M, M, M are given by M = [ kx c, M c m = [ kx c, c m M = [ kr c, M c λ min (J) = [ kr ψ c. c λ max (J) Using (34), (53), the time-derivative o V is given by V z T W z + z T W z z T W z, (57) where W, W, W R are deined in ()-(3). ) Exponential Stability: Under the given conditions (4), (5) o the proposition, all o the matrices M, M, W, M, M, W, and the Lyapunov candidate V become positive-deinite. The condition given by (6) guarantees that V becomes negative-deinite. Thereore, the zero equilibrium o the tracking errors is exponentially stable. Proo o Proposition 3: The given assumptions (8), (9) satisy the assumption o Proposition, rom which the tracking error z = [ e R, e Ω is guaranteed to exponentially decreases, and to enter the region o attraction o Proposition, given by (), (7), in a inite time t. Thereore, i we show that the tracking error z = [ e x, e v is bounded in t [, t, then the total tracking error z = [z, z is uniormly bounded or any t >, and it exponentially decreases or t > t. This yields exponential attractiveness. The boundedness o z is shown as ollows. The error dynamics or e v can be written as mė v = mge 3 Re 3 mẍ d. Let V 3 be a positive-deinite unction o e x and e v : V 3 = e x + m e v. Then, we have e x V 3, e v m V 3. The timederivative o V 3 is given by V 3 e x e v + e v B + e v (k x e x + k v e v + B) = k v e v + (B + (k x + ) e x ) e v d V 3 + d V3, where d = k v m + (k x + ) m, d = B m. Suppose that V 3 or a time interval [t a, t b [, t. In this time interval, we have V 3 V 3. 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