Elective in Robotics. Quadrotor Modeling (Marilena Vendittelli)

Transcription

1 Elective in Robotics Quadrotor Modeling (Marilena Vendittelli)

2 Introduction Modeling Control Problems Models for control Main control approaches Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 2

3 Hummingbird Ascending Technologies GmbH CEA Quadrotor DraganFlyer X4 Draganfly Innovations Inc. Mesicopter Stanford University applications surveying, maintenance aerial transportation, manipulation communication networks search and rescue operations microdrones GmbH STARMAC Stanford University all these activities require vertical, stationary, slow flight a quadrotor is characterized by high maneuverability vertical take-off and landing (VTOL) hovering capabilities Pelican MIT in coll. with Ascending Technologies GmbH... NanoQuad Elective in Robotics - Quadrotor Modeling (M. Vendittelli) KMel Robotics 3

4 four motors located at the extremities of a cross-shaped frame controlled by varying the angular speed of each rotor Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 4

5 actuation force distribution f2 R,2!2 f1!1 R,1 f i = b 2 i i =1,...,4 R,i = d 2 i R,3 f3!3!4 motor control a low level controller stabilizes the rotational speed of each blade f4 R,4 b thrust factor, d drag factor both depend on the rotor geometry and profile, its disk area and radius and on air density can be determined by static thrust test!d + - C(s)! Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 5

6 configuration ' SRI x y SRB Ã zb xb yb # z (x, y, z) position of SRB origin in SRI (', #, Ã) RPY angles expressing the orientation of SRB w.r.t. SRI I R B = c c c s s s c c s c + s s s c s s s + c c s s c s c s c s c c Elective in Robotics - Quadrotor Modeling (M. Vendittelli)

7 equation of motion f2 f1 R,1 R,2!2 '!1 x SRI y z f3 SRB Ã xb yb zb!3!4 R,3 R,4 dynamics of a rigid body with mass m subject to external forces applied to the center of mass according to Newton-Eulero formalism # f4 translational dynamics in SRI FI = m V I FI external force applied to the com and expressed in SRI V I =(v x,v y,v z ) velocity of the com expressed in SRI rotational dynamics in SRB MB = J + J MB, J resp. external moment around com and inertia tensor expressed in SRB =( p, q, r) rotational velocity expressed in SRB Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 7

8 inertia matrix and rotational velocity J = Ix I y I z = p q r = 1 0 s 0 c c s 0 s c c control inputs T = f1+f2+f3+f4 ' = l(f2 f4) # = l(f1 f3) R,2 f2 f3 T SRB Ã zb ' xb yb l # f1!1 f4 R,1 Ã = R,1 + R,2 R,3+ R,4 R,3 R,4 Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 8

9 applied forces and moments 0 0 F I = 0 + I R B (,, ) 0 mg T weight + F A + F D actuation aerodynamics disturbances L M B = L 0 LÃ + + A + D gyroscopic effects T ' Ã # Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 9

10 mathematical model of the system ẋ ẏ ż = v x = v y = v z = f( )+g( ) u v x = F A,x (cos( ) sin() cos( ) + sin( ) sin( )) T m v y = F A,y (sin( ) sin() cos( ) sin( ) cos( )) T m v z = F A,z + g cos() cos( ) T m = p + sin( ) tan()q + cos( ) tan()r state =(x, y, z, v x,v y,v z,,,, p, q, r) inputs u =(T,,, ) = cos( )q sin( )r ṗ = sin( ) sec()q + cos( ) sec()r = A,x + I r q r + I y I z qr + I x I x I x r I r r average blades rotation velocity blades inertia q = A,y + I r I y p r + I z I y I x pr + I y ṙ = A,z + I x I y I z pq + I z Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 10

11 simplified model for control design negligible aerodynamics gyroscopic effects ẍ = assuming small ' and # ) symmetric shape negligible disturbances (cos( ) sin( ) cos() + sin( ) sin()) T m (,, ) (p, q, r) ÿ = (sin( ) sin( ) cos() sin() cos( )) T m z = = = I x I y cos( ) cos() T m + g = Iz Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 11

12 control and planning problems attitude control eight control position control trajectory planning trajectory tracking sensor-based control Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 12

13 control system attitude control ', #, Ã T» 'd, #d, Ãd position control xd, yd, zd Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 13

14 attitude control determine the torques ', #, Ã necessary to obtain a stable desired attitude 'd, #d, Ãd =[K p ( d )+K d ( d )] =[K p ( d )+K d ( d )] =[K p ( d )+K d ( d )] height control determine the thrust T necessary to bring and keep the quadrotor to a desired height z d ) from the z dynamics: T = m cos(#) cos(') [g + z d + K zp (z d z)+k zd (ż d ż)] Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 14

15 simulation results: error ' error # error à error z error Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 15 16

16 simulation results: control inputs T ' # Ã Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 16

17 advanced nonlinear control techniques guarantee better convergence and robustness performance exteroceptive sensors (camera, laser, sonar) allow indoor flight and can be used to obtain an accurate estimation of the system state References R. Mahony, V. Kumar, P. Corke, "Multirotor Aerial Vehicles: Modeling, Estimation, and Control of Quadrotor," IEEE Robotics & Automation Magazine, vol.19, no.3, pp , Sept Elective in Robotics - Quadrotor Modeling (M. Vendittelli) 17