Quadrotor Modeling and Control

Transcription

1 Introduction to Robotics Guest Lecture on Aerial Robotics Quadrotor Modeling and Control Nathan Michael February 05, 2014

2 Lecture Outline Modeling: Dynamic model from first principles Propeller model and force and moments generation Control Attitude control (inner loop) Position control (outer loop) Current research challenges

3 Lecture Objective Develop preliminary concepts required to enable autonomous flight: e 3 e 1 e 2 D. Mellinger, N. Michael, and V. Kumar. Trajectory generation and control for precise aggressive maneuvers with quadrotors. Intl. J. Robot. Research, 31(5): , Apr Vehicle model 2. Attitude and position control 3. Trajectory generation

4 Quadrotor Model Newton-Euler equations: Concept Review total force mass linear acceleration apple F = apple m I 3 apple a + apple! mv! I 3! linear velocity total torque moment of inertia angular velocity angular acceleration

5 Quadrotor Model Concept Review Rigid transformation: p e = R eb p b + r e R eb p b b 3 e 3 b 2 b 1 r e rotation translation e 1 e 2 Euler angle parameterization of rotation: R eb = R z ( )R y ( )R x ( ) ZYX (321) form

6 Quadrotor Model Concept Review Euler angle parameterization of rotation: R eb p b b 3 e 3 b 2 R eb = R z ( )R y ( )R x ( ) yaw pitch roll r e e 1 e 2 b 1 R x ( )= c s 0 s c 3 2 c 0 3 s 5 R y ( ) = R z ( ) = s 0 c 2 c s 3 0 4s c

7 Quadrotor Model Newton-Euler equations: apple F = apple m I 3 apple a + apple! mv! I 3! f 4 f 3 b 3 f 2 e 3 b 2 f 1 f 3 b 3 Total force: Body: f 2 f 4 b 2 f 1 b 1 f 1 4X f = i=1 f i f 3 COM b 3 f 2 f 4 b 2 b 1 along b 3 F b = f r e e 1 e 2 b 1 Inertial: F e = R eb F b mg gravity

8 Quadrotor Model Newton-Euler equations: F m13 = 03 f3 f4 03 I3 a! mv +! I3! b2 b3 f2 f1 f1 b1 f4 ) b2 = d (f3 f1 ) f b1 e1 induced moments b1 b2 b2 b1 d b1 = d (f2 b 3 f2 e2 b2 f4 Total torque: Recall: = r F e3 f4 re f3 b 3 f2 f3 b 3 = propeller direction of rotation 3 + 4

9 Quadrotor Model Equations of motion: apple apple apple apple m a! mv Fe 0 3 I 3 +! I 3! = = apple Reb F b mg [,, ] T b1 b2 b3 F e = R eb F 2 b 3 0 F b = 405 f mg b1 = d (f 2 f 4 ) b2 = d (f 3 f 1 ) b3 = Motor model: b 1 1 b f i = c T! 2 i i = ±c Q! 2 i f b1 b2 b = Approximate relationship between propeller speeds and generated thrusts and moments 3 2 c T c T c T c T 0 dc T 0 dc T 7 6 dc T 0 dc T c Q c Q c Q c Q w 1 2 w 2 2 w 2 3 w

10 Lecture Outline Modeling: Dynamic model from first principles Propeller model and force and moments generation Control Attitude control (inner loop) Position control (outer loop) Current research challenges

11 Control System Diagram p d Position u 1 = f d Motor! i Dynamic Model Trajectory Planner d Attitude Planner R d Attitude u 2 = d b 1, d b 2, d b 3 T Recent tutorial on quadrotor control: R. Mahony, V. Kumar, and P. Corke. Multirotor aerial vehicles: Modeling, estimation, and control of quadrotor. IEEE Robot. Autom. Mag., 19(3):20 32, Sept

12 Attitude Control Inner Loop PD control law: u 2 = k R e R k! e! Rotation error metric: nonlinear e! =!! d e R = 1 2 R d T R R T R d _

13 Attitude Control Inner Loop Linearize the nonlinear model about hover: Rotation error metric: after linearization R 0 = R ( 0 =0, 0 =0, 0 ) R d = R z ( 0 + ) R yx (, ) e R = 1 2 u 4 R d T R 0 R0 T R d _ =[,, ] T _

14 Attitude Control Inner Loop PD control law: u 2 = k R e R k! e! e R =[,, ] T e! =!! d p d Position u 1 = f d Motor! i Dynamic Model Trajectory Planner d Attitude Planner R d Attitude u 2 = d b 1, d b 2, d b 3 T

15 Position Control Outer Loop PD control law: e a + k d e v + k p e p =0 Linearize the nonlinear model about hover: Nominal input: u 1 = mg u 2 = p d Position u 1 = f d Motor! i Dynamic Model Trajectory Planner d Attitude Planner R d Attitude u 2 = d b 1, d b 2, d b 3 T

16 Position Control Outer Loop PD control law: u 1 = mb T 3 g + a d + K d e v + K p e p e p = p e v = v p d v d How do we pick the gains? p d Position u 1 = f d Motor! i Dynamic Model Trajectory Planner d Attitude Planner R d Attitude u 2 = d b 1, d b 2, d b 3 T

17 Lecture Outline Modeling: Dynamic model from first principles Propeller model and force and moments generation Control Attitude control (inner loop) Position control (outer loop) Current research challenges

18 Current Research Challenges How should we coordinate multiple robots given network and vehicle limitations?

19 Current Research Challenges How do we estimate the vehicle state and localize in an unknown environment using only onboard sensing? Camera GPS Laser IMU Barometer Cameras IMU

20 Current Research Challenges How do we estimate the vehicle state and localize in an unknown environment using only onboard sensing?

21 Lecture Summary Modeling: Dynamic model from first principles Propeller model and force and moments generation e 3 Control Attitude control (inner loop) Position control (outer loop) Current research challenges e 1 e 2 1. Vehicle model 2. Attitude and position control 3. Trajectory generation