The purpose of this assignment is to use the concept of rigid body dynamics and Lagrangian...

Transcribed Image Text:

The purpose of this assignment is to use the concept of rigid body dynamics and Lagrangian mechanics to model the motion of a quadcopter. A quadcopter is a flying vehicle with four equally spaced rotors, usually attached at the corner of a square frame (cf. Fig. 1). With six degrees of freedom (like conventional fixed wing airplane) and only four independent motors, quadcopter is severely under-actuated. To understand their motion, let us consider WA f4 ZB f1 W1 W3 3 TM4 YB TM1 XB f2 TM2 TM3 W2 Fig. 1. A Schematic of a Quadcopter two frames to analyze the motion of a quadcopter. The inertial frame is assumed to be attached to the ground with gravity pointing in the negative z-direction. The body frame is attached to the quadcopter with the rotor axis pointed in the positive body z-direction, denoted by zB and the arms pointing in the x and y direction as shown in Fig. 1. If we define the position of the quadcopter in the inertial frame as r = {x, y, z}, then the Lagrangian formulation can be used to derive the following translational and rotational equations of motion: 0 r = -9 1 + cos sin cos + sin & sin T sin sin cos - cos sin o m cos cos r }=^{ 1 0 - sin A A = 0 cos cos sin 0 - sin cos 0 cos Iw = wx (Iw) +T (1) (2) where (), 0, 0) denote the 3 - 2 - 1 Euler angle sequence and w = (p,q,r)T denotes the body angular rates of the quadcopter. Notice that the moment of inertia matrix, I, is a diagonal matrix due to the symmetric geometric shape of the quadcopter. T is the combined forces of rotors in the direction of body z-axis, i.e., zB: 4 4 T = = i=1 i=1 (4) T consists of roll, pitch and yaw moments. Notice that the roll moment is generated by decreasing the second rotor angular velocity and increasing the fourth rotor angular velocity. Similarly, the pitch moment is generated by decreasing the first rotor angular velocity and increasing the third rotor angular velocity. The yaw moment is generated by increasing the angular velocities of two opposite rotors and decreasing the angular velocities of the other two rotors. L M T = { N kl(-N+N) kl(-N+N) b(NNNN) where, is the distance between the rotor and center of mass of the quadcopter. You are required to complete following tasks to get more insights into quadcopter motion: 1) Let us consider the following value of parameters to simulate the quadcopter motion in MATLAB while using the in-built command ODE45 to solve the quadcopter equations of motion: g = 9.81m/s, m = 0.450kg, 1 = 0.225m, k = 2.98 107, b = 1.14 10-6 Ixx = Iyy = 4.85 10 kg m, Izz = 8.80 10- kg m - - 2 (6) (7) The initial conditions corresponds to the quadcopter being at rest at the inertial frame origin with body frame aligned with the inertial frame, i.e. r(0) = {0,0,0}, (0) = {0,0,0}, &(0) = 0(0) = (0) = 0, p(0) = q(0) = r(0) = 0 The control input, i.e., angular velocities of the four rotor are computed as follows: a) For first one second, the quadcopter is ascended by increasing all of the rotor velocities from the hover thrust. Then, the ascend is stopped by decreasing the rotor velocities for the following one second. Ni Ninover 70 sin(2t/4), t <1, i = 1,2,3,4 Ni Ninover 77 sin(2t/4), 1 t2 = (9) The over is equal for all four rotors and corresponds to thrust produced being to the weight of the quadcopter, i.e. 4 T =T; T=>T = 4kn nover = mg i=1 The quadcopter should be hovering at an altitude after this thrust profile. The purpose of this assignment is to use the concept of rigid body dynamics and Lagrangian mechanics to model the motion of a quadcopter. A quadcopter is a flying vehicle with four equally spaced rotors, usually attached at the corner of a square frame (cf. Fig. 1). With six degrees of freedom (like conventional fixed wing airplane) and only four independent motors, quadcopter is severely under-actuated. To understand their motion, let us consider WA f4 ZB f1 W1 W3 3 TM4 YB TM1 XB f2 TM2 TM3 W2 Fig. 1. A Schematic of a Quadcopter two frames to analyze the motion of a quadcopter. The inertial frame is assumed to be attached to the ground with gravity pointing in the negative z-direction. The body frame is attached to the quadcopter with the rotor axis pointed in the positive body z-direction, denoted by zB and the arms pointing in the x and y direction as shown in Fig. 1. If we define the position of the quadcopter in the inertial frame as r = {x, y, z}, then the Lagrangian formulation can be used to derive the following translational and rotational equations of motion: 0 r = -9 1 + cos sin cos + sin & sin T sin sin cos - cos sin o m cos cos r }=^{ 1 0 - sin A A = 0 cos cos sin 0 - sin cos 0 cos Iw = wx (Iw) +T (1) (2) where (), 0, 0) denote the 3 - 2 - 1 Euler angle sequence and w = (p,q,r)T denotes the body angular rates of the quadcopter. Notice that the moment of inertia matrix, I, is a diagonal matrix due to the symmetric geometric shape of the quadcopter. T is the combined forces of rotors in the direction of body z-axis, i.e., zB: 4 4 T = = i=1 i=1 (4) T consists of roll, pitch and yaw moments. Notice that the roll moment is generated by decreasing the second rotor angular velocity and increasing the fourth rotor angular velocity. Similarly, the pitch moment is generated by decreasing the first rotor angular velocity and increasing the third rotor angular velocity. The yaw moment is generated by increasing the angular velocities of two opposite rotors and decreasing the angular velocities of the other two rotors. L M T = { N kl(-N+N) kl(-N+N) b(NNNN) where, is the distance between the rotor and center of mass of the quadcopter. You are required to complete following tasks to get more insights into quadcopter motion: 1) Let us consider the following value of parameters to simulate the quadcopter motion in MATLAB while using the in-built command ODE45 to solve the quadcopter equations of motion: g = 9.81m/s, m = 0.450kg, 1 = 0.225m, k = 2.98 107, b = 1.14 10-6 Ixx = Iyy = 4.85 10 kg m, Izz = 8.80 10- kg m - - 2 (6) (7) The initial conditions corresponds to the quadcopter being at rest at the inertial frame origin with body frame aligned with the inertial frame, i.e. r(0) = {0,0,0}, (0) = {0,0,0}, &(0) = 0(0) = (0) = 0, p(0) = q(0) = r(0) = 0 The control input, i.e., angular velocities of the four rotor are computed as follows: a) For first one second, the quadcopter is ascended by increasing all of the rotor velocities from the hover thrust. Then, the ascend is stopped by decreasing the rotor velocities for the following one second. Ni Ninover 70 sin(2t/4), t <1, i = 1,2,3,4 Ni Ninover 77 sin(2t/4), 1 t2 = (9) The over is equal for all four rotors and corresponds to thrust produced being to the weight of the quadcopter, i.e. 4 T =T; T=>T = 4kn nover = mg i=1 The quadcopter should be hovering at an altitude after this thrust profile.