Advances in Flight Control Systems Part 12 potx

0 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 V [m/s] ν−gap metric between P lti and P lpv P lti−1 P lti−2 P lti−3 (a) LTI models 0 10 20 30 40 50 0 0.1 0.2 0.3 0.4 0.5 V [m/s] ν−gap metric between P poly and P lpv P poly−1 P poly−2 P poly−3 (b) polytopic models Fig. 5. ν-gap metric 207 Autonomous Flight Control System for Longitudinal Motion of a Helicopter 0 10 20 30 40 50 1.7 1.8 1.9 2 2.1 2.2 2.3 2.4 H 2 cost V [m/s] F fix−3 F gs−1 F gs−2 F gs−3 Fig. 6. H 2 cost 5.1 Evaluation of design models According to Section 4.1, three linear interpolative polytopic models were obtained. Table 2 shows the operating points chosen for the models. While, the design points V d of three LTI models are shown in Table 3. The ν-gap metric is one of criteria measuring the model error in the frequency domain. It had been introduced in robust control theories associated with the stability margin (Vinnicombe, 2001). The ν-gap metric between two LTI models, P 1 (s) and P 2 (s), is defined as δ ν (P 1 , P 2 )  = (I + P 2 P ∗ 2 ) −1/2 (P 1 − P 2 )( I + P 1 P ∗ 1 ) −1/2  ∞ (40) The range is δ ν ∈ [0, 1]. A large δ ν means that the model error is large. The ν-gap metric is used for evaluating the model P pol y (V) and P lti (V d ). Figure 5 shows ν-gap metric between P lti (V d ) and P lpv (V) and between P pol y (V) and P lpv (V). δ ν (P lti (V), P lpv (V d )) was rapidly increased when V was shifted from V d . On the other hand, the maximum of δ ν (P pol y (V), P lpv (V)) was reduced according to the number of the operating points. It was seen that P pol y (V) appropriately approximated P lpv (V) over the entire range of the flight velocity. 5.2 Design of F and H 2 cost The design parameters for designing F in Eq. (30) were given as follows. B 1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −6.039 10.977 −154.03 49.188 3.954 −7.187 00 0.38495 −0.12395 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , C 1 = ⎡ ⎣ 0.001I 5 0 2×5 ⎤ ⎦ , D 1 = ⎡ ⎣ 0 5×2 I 2 ⎤ ⎦ 208 Advances in Flight Control Systems 0 50 100 150 −20 0 20 40 60 u [m/s] Fixed−SF 1 0 50 100 150 −30 −20 −10 0 10 w [m/s] 0 50 100 150 5 10 15 20 25 t [s] θ 0 [deg] 0 50 100 150 −2 0 2 4 6 t [s] θ c [deg] u u r w w r (a) Controlled variables and inputs 0 20 40 60 80 100 120 140 0 1000 2000 3000 Fixed−SF 1 t [s] x e [m] 0 20 40 60 80 100 120 140 −300 −200 −100 0 100 t [s] h e [m] (b) Positions Fig. 7. Time responses using F fix−1 209 Autonomous Flight Control System for Longitudinal Motion of a Helicopter 0 50 100 150 −20 0 20 40 60 u [m/s] Fixed−SF 2 0 50 100 150 −30 −20 −10 0 10 w [m/s] 0 50 100 150 5 10 15 20 25 t [s] θ 0 [deg] 0 50 100 150 −2 0 2 4 6 t [s] θ c [deg] u u r w w r (a) Controlled variables and inputs 0 20 40 60 80 100 120 140 0 1000 2000 3000 Fixed−SF 2 t [s] x e [m] 0 20 40 60 80 100 120 140 −300 −200 −100 0 100 t [s] h e [m] (b) Positions Fig. 8. Time responses using F fix−2 210 Advances in Flight Control Systems 0 50 100 150 −20 0 20 40 60 u [m/s] Fixed−SF 3 0 50 100 150 −30 −20 −10 0 10 w [m/s] 0 50 100 150 5 10 15 20 25 t [s] θ 0 [deg] 0 50 100 150 −2 0 2 4 6 t [s] θ c [deg] u u r w w r (a) Controlled variables and inputs 0 20 40 60 80 100 120 140 0 1000 2000 3000 Fixed−SF 3 t [s] x e [m] 0 20 40 60 80 100 120 140 −300 −200 −100 0 100 t [s] h e [m] (b) Positions Fig. 9. Time responses using F fix−3 211 Autonomous Flight Control System for Longitudinal Motion of a Helicopter 0 50 100 150 −20 0 20 40 60 u [m/s] GS−SF 1 0 50 100 150 −30 −20 −10 0 10 w [m/s] 0 50 100 150 5 10 15 20 25 t [s] θ 0 [deg] 0 50 100 150 −2 0 2 4 6 t [s] θ c [deg] u u r w w r (a) Controlled variables and inputs 0 20 40 60 80 100 120 140 0 1000 2000 3000 GS−SF 1 t [s] x e [m] 0 20 40 60 80 100 120 140 −300 −200 −100 0 100 t [s] h e [m] (b) Positions Fig. 10. Time responses using F gs−1 212 Advances in Flight Control Systems 0 50 100 150 −20 0 20 40 60 u [m/s] GS−SF 2 0 50 100 150 −30 −20 −10 0 10 w [m/s] 0 50 100 150 5 10 15 20 25 t [s] θ 0 [deg] 0 50 100 150 −2 0 2 4 6 t [s] θ c [deg] u u r w w r (a) Controlled variables and inputs 0 20 40 60 80 100 120 140 0 1000 2000 3000 GS−SF 2 t [s] x e [m] 0 20 40 60 80 100 120 140 −300 −200 −100 0 100 t [s] h e [m] (b) Positions Fig. 11. Time responses using F gs−2 213 Autonomous Flight Control System for Longitudinal Motion of a Helicopter 0 50 100 150 −20 0 20 40 60 u [m/s] GS−SF 3 0 50 100 150 −30 −20 −10 0 10 w [m/s] 0 50 100 150 5 10 15 20 25 t [s] θ 0 [deg] 0 50 100 150 −2 0 2 4 6 t [s] θ c [deg] u u r w w r (a) Controlled variables and inputs 0 20 40 60 80 100 120 140 0 1000 2000 3000 GS−SF 3 t [s] x e [m] 0 20 40 60 80 100 120 140 −300 −200 −100 0 100 t [s] h e [m] (b) Positions Fig. 12. Time responses using F gs−3 214 Advances in Flight Control Systems They were used for both of GS-SF and Fixed-SF. Three GS-SF gains denoted as F gs−i (i = 1, 2, 3) were designed according to Section 4.2, while three Fixed-SF gains denoted as F fix−i (i = 1, 2, 3) were designed by LQR technique in which the weights of the quadratic index were given by C T 1 C 1 and D T 1 D 1 . Figure 6 shows the H 2 cost of the closed-loop system which the designed F is combined with Eq. (30). The H 2 cost by F fix−3 was minimized at V = 40 [m/s] which was near the design point V d = 50 [m/s], but was increased in the low flight velocity region. The H 2 cost by F fix−1 and F fix−2 showed the similar result. On the other hand, the H 2 cost by F gs−2 and F gs−3 was kept small over the entire flight region. The H 2 cost by F gs−1 was small in the middle flight velocity region but was increased in the low and the high flight velocity regions. 5.3 Tracking evaluation The flight mission given in Fig. 3 was performed in Simulink. Figures 7 - 12 show the time histories of the closed-loop system with the three GS-SF and three Fixed-SF gains. In the case of F fix−1 shown in Fig. 7, the controlled variables u and w tracked their references until the acceleration phase (5 ≤ t < 30 [s]) but they were diverged in the cruise phase (30 ≤ t < 60 [s]). In the deceleration phase (60 ≤ t < 80 [s]), the closed-loop system was stabilized again but it was de-stabilized in the approach phase (t ≥ 100 [s]). Although the closed-loop system remained stable for the entire flight region in the case of F fix−2 shown in Fig. 8, oscillatory responses were observed in the cruise and approach phases. The responses using F fix−3 shown in Fig. 9 were better than those using F fix−2 . On the other hand, the three GS-SF gains provided stable responses as shown in Figs. 10 - 12, In particular, The responses by F gs−3 showed improved tracking and settling properties compared to other cases. Summarizing the simulation in MATLAB/Simulink, polytopic model P pol y−3 made the ν-gap metric smaller than other models for the entire flight region. F gs−3 designed by using P pol y−3 showed better control performance. 6. Concluding remarks This paper has presented an autonomous flight control design for the longitudinal motion of helicopter to give insights for developing autopilot techniques of helicopter-type UAVs. The characteristics of the equation of helicopter was changed during a specified flight mission because the trim values of the equation were widely varied. In this paper, gain scheduling state feedback (GS-SF) was included in the double loop flight control system to keep the vehicle stable for the entire flight region. The effectiveness of the proposed flight control system was evaluated by computer simulation in MATLAB/Simulink. The model error of the polytopic model was smaller than that of LTI models which were obtained at specified flight velocity. Flight control systems with GS-SF showed better control performances than those with fixed-gain state feedback. The double loop flight control structure was useful for accomplishing flight mission considered in this paper. 7. References [1] Boyd, S.; Ghaoui, L. E.; Feron, E. & Balakrishnan, V. (1994). Linear Matrix Inequalities in System and Control Theory, SIAM, Vol. 15, Philadelphia. [2] Bramwell, A. R. S. (1976). Helicopter Dynamics, Edward Arnold, London, 1976. 215 Autonomous Flight Control System for Longitudinal Motion of a Helicopter [3] Cho, S J.; Jang, D S. & Tahk, M L. (2005). Application of TCAS-II for Unmanned Aerial Vehicles, Proc. CD-ROM of JSASS 19th International Sessions in 43rd Aircraft Symposium, Nagoya, 2005 [4] Fujimori, A.; Kurozumi, M.; Nikiforuk, P. N. & Gupta, M. M. (1999). A Flight Control Design of ALFLEX Using Double Loop Control System, AIAA Paper, 99-4057-CP, Guidance, Navigation and Control Conference, 1999, pp. 583-592. [5] Fujimori, A.; Nikiforuk, P. N. & Gupta, M. M. (2001). A Flight Control Design of a Reentry Vehicle Using Double Loop Control System with Fuzzy Gain-Scheduling, IMechE Journal of Aerospace Engineering, Vol. 215, No. G1, 2001, pp. 1-12. [6] Fujimori, A.; Miura, K. & Matsushita, H. (2007). Active Flutter Suppression of a High-Aspect-Ratio Aeroelastic Using Gain Scheduling, Transactions of The Japan Society for Aeronautical and Space Sciences, Vol. 55, No. 636, 2007, pp. 34-42. [7] Johnson, E. N. & Kannan, S. K. (2005). Adaptive Trajectory Control for Autonomous Helicopters, Journal of Guidance, Control and Dynamics, Vol. 28, No. 3, 2005, pp. 524-538. [8] Langelaan, J. & Rock, S. (2005). Navigation of Small UAVs Operating in Forests, Proc. CD-ROM of AIAA Guidance, Navigation, and Control Conference, San Francisco, 2005. [9] Padfield, G. D. (1996). Helicopter Dynamics: The Theory and Application of Flying Qualities and Simulation Modeling, AIAA, Reston, 1996. [10] Van Hoydonck, W. R. M. (2003). Report of the Helicopter Performance, Stability and Control Practical AE4-213, Faculty of Aerospace Engineering, Delft University of Technology, 2003. [11] Vinnicombe, G. (2001). Uncertainty and Feedback ( H ∞ loop-shaping and the ν-gap metric), Imperial College Press, Berlin. [12] Wilson, J. R. (2007). UAV Worldwide Roundup 2007, Aerospace America, May, 2007, pp. 30-38. 216 Advances in Flight Control Systems [...]... d1 (5) (6) 226 Advances in Flight Control Systems When the RC-helicopter is in moving, the radius of the marker in the image after moving is defined as D2 , the center coordinates of the marker after moving are defined as ( xC 2 , yC 2 ) , and the center coordinates of actual marker is defined as ( x2 , y 2 , z2 ) Then, the following equation is acquired z2 : d1 = f : D2 (7) Here, since the focal... defined as the major axis PO , and the segment of G2 is defined as a minor axis PQ The position and posture of the micro RC helicopter are calculated by the method shown in Section 4 224 Advances in Flight Control Systems Y ymax ymax-ymin 2 P (xc,yc) ymin xmax-xmin 2 xmin X xmax Fig 10 The marker center Y O I(x,y) G1 G2 PI P (xc,yc) Q X Fig 11 The calculation method of the major axis and the minor... point detection using Harris operator for photographed image The Harris operator detects the corner point mainly from the image as a feature point The position of feature point is estimate able by related position information of an artificial marker to feature point from camera image after coordinate transformation The flight control area of RC helicopter can be expanded(see Fig.8) by using the information... using the fuzzy control system which consists of IF-Then control rules is satisfying the requirements for the flight control performance of an unmanned helicopter like hovering, takeoff, rotating, and landing It is necessary to presume three dimensional position and posture of micro RC helicopter for the autonomous flight control A position and posture presumption method for the autonomous flight control. .. taken by the camera will be a blurred image resulting from an interlace like Fig.6 We devised a method skipping the odd number line (or, even number line) of input image to acquire an clear input image while the RC helicopter is flying Fig 6 A blurring image acquired by wireless camera 3.2 Feature point extraction Feature point detection is defined in terms of local neighborhood operations applied... axis and a minor axis is shown in Fig.11 When the center coordinate is defined as P( xC , yC ) , and the coordinate of the pixel which is on the outline is defined as I(x,y), the distance PI from the center to the pixel of the outline is calculated by equation (4) PI = ( x − xc )2 + ( y − yc )2 (4) For obtaining the maximum value G1 of PI and the minimum value G2 of PI, all pixels on the outline are calculated... Autonomous Flight Control for RC Helicopter Using a Wireless Camera Fig 3 RC helicopter equipped with a micro wireless camera Fig 4 An artificial marker Fig 5 Coordinate axes and attitude angles 219 220 Advances in Flight Control Systems helicopter are computed with image processing by the computer set on the ground, and, this image processing used the position and shape of an artificial marker in the... artificial marker, the position of the helicopter is estimated by template matching between the area of natural feature points and the template area 222 Fig 7 Feature point extraction by a Harris operator Fig 8 The expansion of flight area Advances in Flight Control Systems Autonomous Flight Control for RC Helicopter Using a Wireless Camera 223 3.3 Detection of a marker Fig 9 The flow chart of marker...11 Autonomous Flight Control for RC Helicopter Using a Wireless Camera Yue Bao1, Syuhei Saito2 and Yutaro Koya1 1Tokyo City University, 2Canon Inc Japan 1 Introduction In recent years, there are a lot of researches on the subject of autonomous flight control of a micro radio control helicopter Some of them are about flight control of unmanned helicopter (Sugeno et al.,... posture are shown in Fig.9 First, a binarization processing is performed at the input image from the camera installed on the micro RC helicopter Then, the marker in the image is searched, and the outline of the marker is extracted If the outline cannot be extracted, an image is acquired from the camera again and a marker is searched again A marker center is searched after the outline of a marker is . in Flight Control Systems 226 When the RC-helicopter is in moving, the radius of the marker in the image after moving is defined as 2 D , the center coordinates of the marker after moving. College Press, Berlin. [12] Wilson, J. R. (2007). UAV Worldwide Roundup 2007, Aerospace America, May, 2007, pp. 30-38. 216 Advances in Flight Control Systems 11 Autonomous Flight Control for RC. shown in Table 3. The ν-gap metric is one of criteria measuring the model error in the frequency domain. It had been introduced in robust control theories associated with the stability margin (Vinnicombe,