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quadrotor.py
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quadrotor.py
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import numpy as np
from numpy.linalg import inv, norm
import scipy.integrate
from scipy.spatial.transform import Rotation
import control
def quat_dot(quat, omega):
"""
Parameters:
quat, [i,j,k,w]
omega, angular velocity of body in body axes
Returns
quat_dot, [i,j,k,w]
"""
# Adapted from "Quaternions And Dynamics" by Basile Graf.
(q0, q1, q2, q3) = (quat[0], quat[1], quat[2], quat[3])
G = np.array([[q3, q2, -q1, -q0],
[-q2, q3, q0, -q1],
[q1, -q0, q3, -q2]])
quat_dot = 0.5 * G.T @ omega
# Augment to maintain unit quaternion.
quat_err = np.sum(quat**2) - 1
quat_err_grad = 2 * quat
quat_dot = quat_dot - quat_err * quat_err_grad
return quat_dot
class Quadrotor():
"""
Quadrotor forward dynamics model.
"""
def __init__(self):
self.mass = 0.030 # kg
self.Ixx = 1.43e-5 # kg*m^2
self.Iyy = 1.43e-5 # kg*m^2
self.Izz = 2.89e-5 # kg*m^2
self.arm_length = 0.046 # meters
self.rotor_speed_min = 0 # rad/s
self.rotor_speed_max = 2500 # rad/s
self.k_thrust = 2.3e-08 # N/(rad/s)**2
self.k_drag = 7.8e-11 # Nm/(rad/s)**2
# Additional constants.
self.inertia = np.diag(
np.array([self.Ixx, self.Iyy, self.Izz])) # kg*m^2
self.g = 9.81 # m/s^2
# Precomputes
k = self.k_drag/self.k_thrust
L = self.arm_length
self.to_TM = np.array([[1, 1, 1, 1],
[0, L, 0, -L],
[-L, 0, L, 0],
[k, -k, k, -k]])
self.inv_inertia = inv(self.inertia)
self.weight = np.array([0, 0, -self.mass*self.g])
self.t_step = 0.01
# Initialize state
self.state = _unpack_state(np.zeros(13))
def reset(self, position=[0, 0, 0], yaw=0, pitch=0, roll=0):
'''
state is a 13 dimensional vector
postion*3 velocity*3 attitude(quaternion)*4 angular velocity*3
state = [x y z dx dy dz qw qx qy qz r p q]
dot_state = [dx dy dz ddx ddy ddz dqw dqx dqy dqz dr dp dq]
'''
s = np.zeros(13)
s[0] = position[0]
s[1] = position[1]
s[2] = position[2]
r = Rotation.from_euler('zxy', [yaw, roll, pitch], degrees=True)
quat = r.as_quat()
s[6] = quat[0]
s[7] = quat[1]
s[8] = quat[2]
s[9] = quat[3]
# the unassigned values of s are zeros
self.state = _unpack_state(s)
return self.state
def step(self, cmd_rotor_speeds):
'''
Considering the max and min of rotor speeds
action is a 4 dimensional vector: conmmand rotor speeds
action = [w1, w2, w3, w4]
'''
rotor_speeds = np.clip(
cmd_rotor_speeds, self.rotor_speed_min, self.rotor_speed_max)
rotor_thrusts = self.k_thrust * rotor_speeds**2
'''
Next, [w1, w2, w3, w4] into [F Mx My Mz]
'''
TM = self.to_TM @ rotor_thrusts
T = TM[0] # u1
M = TM[1:] # u2
# Form autonomous ODE for constant inputs and integrate one time step.
def s_dot_fn(t, s):
return self._s_dot_fn(t, s, T, M)
'''
The next state can be obtained through integration （Runge-Kutta）
'''
s = _pack_state(self.state)
sol = scipy.integrate.solve_ivp(
s_dot_fn, (0, self.t_step), s, first_step=self.t_step)
s = sol['y'][:, -1]
# turn state back to dict
self.state = _unpack_state(s)
# Re-normalize unit quaternion.
reward = 0
done = 0
info = {}
return self.state, reward, done, info
def _s_dot_fn(self, t, s, u1, u2):
"""
Compute derivative of state for quadrotor given fixed control inputs as
an autonomous ODE.
"""
state = _unpack_state(s)
# page 73
# Position derivative.
x_dot = state['v']
# Velocity derivative.
F = u1 * Quadrotor.rotate_k(state['q'])
v_dot = (self.weight + F) / self.mass
# Orientation derivative.
q_dot = quat_dot(state['q'], state['w'])
# Angular velocity derivative. page 26 Equation 4
omega = state['w']
omega_hat = Quadrotor.hat_map(omega)
w_dot = self.inv_inertia @ (u2 - omega_hat @ (self.inertia @ omega))
# Pack into vector of derivatives.
s_dot = np.zeros((13,))
s_dot[0:3] = x_dot
s_dot[3:6] = v_dot
s_dot[6:10] = q_dot
s_dot[10:13] = w_dot
return s_dot
def get_dLTI(self, dt):
num_x = 12
Ac = np.array([[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, self.g, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, (-self.g), 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], ])
Bc = np.array([[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[1/self.mass, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 0, 0, 0],
[0, 1/self.Ixx, 0, 0],
[0, 0, 1/self.Iyy, 0],
[0, 0, 0, 1/self.Izz]])
Cc = np.eye(num_x)
Dc = np.zeros((num_x, 4))
sysc = control.ss(Ac, Bc, Cc, Dc)
# Discretization
sysd = control.sample_system(sysc, dt, method='bilinear')
return sysd.A, sysd.B
@classmethod
def rotate_k(cls, q):
"""
Rotate the unit vector k by quaternion q. This is the third column of
the rotation matrix associated with a rotation by q.
"""
return np.array([2 * (q[0] * q[2] + q[1] * q[3]),
2 * (q[1] * q[2] - q[0] * q[3]),
1 - 2 * (q[0] ** 2 + q[1] ** 2)])
@classmethod
def hat_map(cls, s):
"""
Given vector s in R^3, return associate skew symmetric matrix S in R^3x3
"""
return np.array([[0, -s[2], s[1]],
[s[2], 0, -s[0]],
[-s[1], s[0], 0]])
def _pack_state(state):
"""
Convert a state dict to Quadrotor's private internal vector representation.
"""
s = np.zeros((13,))
s[0:3] = state['x'].squeeze()
s[3:6] = state['v'].squeeze()
s[6:10] = state['q'].squeeze()
s[10:13] = state['w'].squeeze()
return s
def _unpack_state(s):
"""
Convert Quadrotor's private internal vector representation to a state dict.
"""
state = {'x': s[0:3], 'v': s[3:6], 'q': s[6:10], 'w': s[10:13]}
return state