# Control of a Quadrotor Helicopter Using Visual Feedback

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2 angle representation given in Equation 1, has been chosen for the representation of the rotations. R-- s~co s~sos~ + c~c~ s~soc~ - c~s~ ). (1) -8o co8~ coc~ where co and so represent cos and sin respectively. Each rotor produces moments as well as vertical forces. These moments have been experimentally observed to be linearly dependent on the forces for low speeds. There are four input forces and six output states (z, y, z,, ~, ) therefore the quadrotor is an under-actuated system. The rotation direction of two of the rotors are clockwise while the other two are counterclockwise, in order to balance the moments and produce yaw motions as needed. F 3.J.J F 1 t J zb, mg Figure 1: 3D Quadrotor Model. The equations of motion can be written using the force and moment balance. /~ ( 4 Ei=I fi)( cos sin cos ~ + sin sin ~) - Kl~b z 7Vt ( }-~= ~ Fz) (sin sin cos ~ - cos sin ~) - K2 ~) 7Vt 5_(~ E~=~ r~)(co~ co~ ~) -.~g- K~ (2) 7gt O = l(-f1 - F2 + F3 + F4 - K4)/J1 The K,i's given above are the drag coefficients. In the fob lowing we assume the drag is zero, since drag is negligible at low speeds. For convenience, we will define the inputs to be u~ = (F~ + F2 + F3 + F4)/m = - r. + + a)/j (3) ~ = (-f~ + f~ + f~ - f~)/j~ ua = C(F~ - F2 + F3 - Fa)/J3. z o where J,z's are the moment of inertia with respect to the axes and C is the force-to-moment scaling factor. The u l represents a total thrust on the body in the z-axis, u2 and u3 are the pitch and roll inputs and u4 is a yawing moment. Therefore the equations of motion become /~ = Ul(COS sincos~ + sinsin~) = u21 = Ul(sinsincos~ - cossin~) ~ = u31 = ~(co~co~) - g ~ = ~. (4) The center of gravity is assumed to be at the middle of the connecting link. As the center of gravity moves up (or down) d units, then the angular acceleration becomes less sensitive to the forces, therefore stability is increased. Stability can also be increased by tilting the rotor forces towards the center. This will decrease the roll and pitch moments as well as the total vertical thrust. 3 Control of a Quadrotor Our goal is to use an external camera as the primary sensor and use onboard gyros to get the tilt angles and stabilize the helicopter in an inner control loop. Due to the weight limitations we can not add GPS or other accelerometers on the system. Therefore our controller should be able to get the positions and speeds from the camera only. One other aspect of the controller selection depends on the method of control of the UAV. It can be mode-based or non-mode based. For the mode based controller, independent controllers for each state are needed, and a higher level controller decides how these interact. On the other hand for a non-mode based controller, a single controller controls all of the states together. In this section we will present the implement of feedback linearization and a backstepping controller to the quadrotor model and show that it can be stabilized and controlled. 3.1 Feedback Linearization One approach to make the quadrotor helicopter autonomous is the use of a controller that can switch between many modes such as; hover, take-off, landing, left/right, search, tilt-up, tilt-down etc. These low level control tasks can be connected to a higher level controller that sets the goal points and does the motion planning. Similarly, Koo et al. used hybrid control methodologies in [6] for autonomous helicopters. A natural starting place for this is to ask what modes can be controlled using feedback linearizing controllers. We can use exact input-output linearization and choose outputs to be z,, ~ and, in order to control the altitude, yaw and tilt angles of the quadrotor. But this controller introduces zero dynamics which results in the drift of the helicopter in the x-y plane. Therefore such a controller is unstable. The zero dynamics for this system are -- tano = -(t~ ~/co~ o). (5) 73

3 K~( ~ 2~)/tl ) ~), These zero dynamics are not desirable, and so another controller or a combination of controllers is needed. One can pick the outputs to be z,x,y,, which results in a complex equation with higher derivatives. Alternatively, we may pick two outputs, z- x, and use separate controllers for controlling and y-motion. The problem with this approach is the need of switching between controllers. To get the inputs, we differentiate the equations until the inputs appear. A fourth order derivative of the states is necessary. {/1 = sin (Vl - cos + Ul~) 2 sin )+ cos (v2-22/1 cos + Ul~) 2 sin) (6) ~ = (co~ (~ - 2~ co~ + ~O ~ ~i~ )- 22/1 cos + U l 2 sin ))/u 1, where vl and v2 are given as Vl = -K12- K2~? - x (7) v2 = -K32 - K4~ - z. We can control the y-axis motion and yaw by PD controllers. Motion along the y-axis can be related to the tilt angle by Equation 9. Therefore the tilt angle is selected based on the y position and the velocity: 7O 6 5 4O N 3 2O O % Y a% o ( ~ -5 _ ~ - ~ 5 4 ao Figure 3: Feedback linearization path. Once the inputs are obtained, one can use Equation 3 to find the forces and set the motor speeds to get the desired lift from each of the rotors. We can put together these controllers into a hybrid controller of the form shown in Figure 4. Hover mode is the central mode, where the model is stabilized by keeping the positions (x,y,z) constant and (,~,O) angles zero. The basic commands will be switched from this hover mode. lo x where ~ - K~ (~ - ~) + K~ (~ - ~) (s) ~ K~(~ - ~n -- arcsin(kpy + Kn2)) ~d - K~fj + Kd~ il - K~,y 2-2KpKdy~- K~ 2 (9) z=o Mode Take-off ~> z<z_fly Mode Landing ~< z>o... \... ~ \z z~ r~:~-~ / - y=y d 2\ ( x_d, y_d, z_d ~"( phi<phi_d / ~eta_d, ~ si_d, ph~i~ (or phi>phi_d~ ~ p t i=phi_d~ Figure 2 shows the quadrotor simulation, where it moves from (4,2,6) to the origin with initial zero yaw and tilt angles. Note the tilt-up motion of the quadrotor in order to slow down and reach the origin with zero velocity ,o t o I /#... ~-1oi/ -&-2oil - F -5O i teta vs time '\ \\\ L i:li 3O, 2 4 psi vs time _11 ~'~ 2' 4' _IL \ \ 2 4 phi vs time Figure 2: Feedback linearization simulation results. Figure 4: Hybrid Controller Model. Climb and descend modes change the z-value, while keeping the other values constant. These modes will terminate only when the desired z-value is achieved. Left/Right mode is responsible for controlling the y-axis motions. As the model tilts around the x-axis, it will start moving on the y-axis. Forward/Reverse mode tilts around y-axis by changing the ~ angle. The hover mode will switch to the landing mode when the flight is complete. 3.2 Backstepping Controller Backstepping controllers [71 are especially useful when some states are controlled through other states. As it was observed in the previous section, in order to control the x and y motion of the quadrotor, tilt angles need to be controlled. Therefore a backstepping controller has been developed in this section. Similar ideas of using 74

4 - u backstepping with visual servoing have been developed for a traditional helicopter by Hamel and Mahony [8]. The approach here is a bit simpler in implementation, and relies on only very simple estimates of pose. We will use a small angle assumption on the yaw angle,, to justify neglecting certain terms from Equation 4 to give 2-1 sin cos ~ cos (1) ~) -- - u 1 sin ~ cos. First we notice that motion in the y-direction can be controlled through changes in the roll. This leads to a backstepping controller for y- ~ control given by ~t3 1 Ul COS ~) COS (- 5y -- 1~) - 9Ul sin ~ cos --4~t COS ~) COS (~ -~- ~t sin ~ cos +2u1~ sin ~ sin + u1~ cos ~ sin cos ~ sin - U l ~2 sin ~ cos ). (11) To develop a controller for motion along the x-axis, we assume the tilt is slowly varying or cos ~ const. This leads to a backstepping controller for x- of 1 u2 Ul cos cos ~ cos (-5x - 1:b - 9Ul sin cos ~ cos N %,5 Y 3 3 J''~ ~ ~ " _ Figure 6: Backstepping controller path. controller is strong enough to handle random errors which simulate the pose estimation errors and disturbances as shown in Figure 7. The error introduced on x and y has variance of.5 cm and error on z has variance of 2 era. The yaw variance is 1.5 degrees. The helicopter moves from 1 cm to 15 cm while reducing the yaw angle from 3 degrees to zero. The mean and standard deviation are found to be 15 cm and 1.7 cm for z and 2.4 degrees and 1.1 degrees for respectively. x vs time Y vs time z vs time E 5 ~ E 5 ~ 2 x -4u 1 ~) cos cos ~ cos + Ul 2 sin cos ~ cos +2u1 sin cos ~ sin + Ul~) cos cos ~ sin o lo o t!o 2 tme (s) - u i ~) cos cos ~ sin - u i ~2 sin cos ~ cos ). (12) teta vs time psi vs time phi vs time The altitude and the yaw on the other hand, can be controlled by a P D controller. Ul g -Jr-.[~pl(Zd -- Z) -Jr-.[~dl(7]d -- Z) (13) cos cos - ) + - b) -1 2o teta vs time ~ i 15,// lo!\\\ i= -5 15,~ 5 \ -5 Y vs time psi vs time i /\ phi vs time -1-1 o 4O 2O o Figure 5: Backstepping controller simulation results. The simulation results in Figures 5 and 6 show the motion of the quadrotor from position (4, 2, 6) to origin. The o i t 1_ 2 Ime (s) ~o "I,,,, ~,~i',,~,?, ' ~5-5 ~ t! 2 t! 2 tme (s) tme (s) Figure 7: Backstepping controller simulation with random noise at x, y, z and yaw values. Moving to an arbitrary heading A backstepping controller is used in the previous section to perform tilts around the x and y axes of the helicopter, assuming the desired yaw angle is zero. Using the invariances of the system, it is straightforward to see that the same controller can be applied for an arbitrary ~d. The x t and yt axes shown in Figure 8 are x' = x cos Cd + Y sin Cd y' = --xsin Cd + y COS Cd. (14) The equation of motion relative to the initial frame can be rotated to align the body frame to the desired yaw angle. ~) - uz/~z (q) - q)d) sin ~ sin _ cos ~ (15) / 75

5 v.\, / sets the goal positions and sends them to the on-board controller via a radio link. G O A L ~ Off-board Controller y. A Images Image Compu- Processing " tation Remote Basic - Control On-board Commands Device " Controller x F4 F 1 Figure 11: Off-board controller. Figure 8: Moving along arbitrary direction to goal. ~~~2i~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~iiiii~i~ i ii ~ ~~iii~i!i~iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii~i~iiii~ii~i~!~!!~i~i~!i!!n The backstepping controller described above will be used to perform control on the new x t- yt frame and go to the desired point of that frame. Controllers used for yaw and altitude control can also be used in this part. This controller can deal with any desired yaw angle. Simulation results are given in Figure 9 for moving to origin with ~d = 6. x vs time Y vs time Z vs time Figure 12: Quadrotor tracking with a camera. ~ 6I! 1 = a!,. 6 t 16( ) 26 6 tim:(s) 26 lo 5 ~ -1 _15 l o 2o o 2o ti (s) ti (s) Figure 9: Simulation results for backstepping controller with arbitrary heading. 4 Real Time Control and Vision System To make a helicopter fully autonomous, we need a flight controller as shown in Figure 1. There is an off-board controller that receives camera images, processes them and sends control inputs to the on-board processor. The on-board processor stabilizes the model by checking the gyroscopes and listens for the commands sent from the off-board controller. The rotor speeds are set accordingly to achieve the desired positions and orientations. Helicopter pose is estimated by a pose estimation algorithm. This algorithm uses 2.5 cm radius colored blobs that are attached to the bottom of the quadrotor as shown in Figure 12. A blob tracking algorithm is used to get the positions and areas of the blobs on the image plane. Then the purpose of the pose estimation algorithm is to obtain (x,y,z) positions, pitch angles (, ~) and the yaw angle ( ) of the helicopter in real-time relative to the camera frame. The position of each blob is calculated as z~ - (fx + fy)~//(2v/a~) = Ox) /fx (16) Yi = (vi - Oy)zi/fy, where fx and fv are the focal lengths in x and y respectively, C is the number of pixels per unit area, Ox and v are the image center coordinates, Ai are the area of the blobs and ui and vi are the image coordinates. - arctan(yl - y5/xl - x5) - arcsin(z4 - z2/d) - arcsin(z5 - zl/d). (17) Off-board On-board u. Actuators forces.. Positions Controller v. Controller I I m ments Quadrotor an 'es I I tilts ] Helicopter ] gl Gyr... pes ~ E21g(e s Camera Figure 1: Control Diagram. Environment The off-board controller shown in Figure 11 is responsible for the main computation. It processes the images and 76 The position of the helicopter is estimated by averaging the five blob positions. A normalization is performed using the real center difference between blobs. Yaw angle can be obtained from blob positions and the tilt angles can be estimated from the height differences of the blobs, where d is given as the distance between blobs in Equation 17. These estimates depend on area calculations, therefore they are sensitive to noise. Other methods that do not depend on the area measurements will be implemented to lower the errors in the pitch and roll angle estimation.

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