From Coursera, State Estimation and Localization for Self-Driving Cars by University of Toronto
https://www.coursera.org/learn/state-estimation-localization-self-driving-cars
GNSS/INS Sensing for Pose Estimation
3D Geometry and Reference Frames
Reference frame and vector coordinates
- Vectors can be expressed in different coordinate frames
- The coordinates of the vector are related through a rotation matrix: $r_b = C_{ba}r_a$, $C_{ba}$ is the rotation matrix takes coordinates in frame a and rotates them into frame b
Rotation representations
- rotation matrix (direction cosine matrix):
- $C_{ba}=[b_1 \;b_2\; b_3]^T[a_1 \; a_2\; a_3] \in R^{3\times 3}$
- $r_b = C_{ba}r_a$
- $C_{ba}C^T_{ba} = C_{ba}C_{ab} = 1$
- Unit quaternions
- $q=[q_w \; q_v]T = [\cos \frac \phi 2 \quad \hat u \sin \frac \phi 2]^T$
- $| q| = 1$
- $r_b = C(q_{ba}r_a)$
- $C(q) = (q^2_w-q^T_vq_v)1+2q_vq^T_v + 2q_w[q_v]_x$
- Quaternion multiplication and rotations
- Euler angles
- $C(\theta_3,\theta_2,\theta_1) = C_3(\theta_3)C_2(\theta_2)C_1(\theta_1) $
- suffer from singularity
The importance of the ECEF, ECIF and Navigation reference frames
Reference Frames
- ECIF: Earth-Centred Inertial Frame
- ECIF coordinate frame is fixed, Earth rotates about the z axis.
- ECEF: Earth-Centred Earth-Fixed Frame
- ECEF coordinate frame rotates with the Earth.
- x axis aligns the prime meridian
- Navigation
- NED frame
- ENU frame
- Sensor/Vehicle frame
The Inertial Measurement Unit (IMU)
Components
- gyroscopes
- measures a rotational rate in the sensor frame
- Microelectromechanical systems(MEMS) are much smaller and cheaper
- Measure rotational rates instead of orientation directly
- Measurements are noisy and drift over time
- accelerometers
- measures a specific force (or acceleration relative to free-fall) in the sensor frame
- Cheaper MEMS based accelerometers use a miniature cantilever beam with a proof mass attached to it. When the sensor is accelerated, the beam deflects.
- More expensive sensors may also use Piezoelectric materials
- Accelerometers measure acceleration relative to free-fall-this is also called the proper acceleration or specific force: $$a_{mean} = f = \frac{F_{non-gravity}}{m}$$
- In localization, we typically require the acceleration relative to a fixed reference frame
- ‘coordinate’acceleration
- computed using fundamental equation for accelerometers in a gravity field: $f+g=\ddot r_i$
The Global Navigation Satellite Systems (GNSS)
Global Navigation Satellite System (GNSS) is a catch-all term for a satellite system(s) that can be used to pinpoint a receiver’s position
GPS - Computing Position:
- Each GPS satellite transmits a signal that encodes
- its position (via accurate ephemeris information)
- time of signal transmission (via onboard atomic clock)
- To compute a GPS position fix in the Earth-centred frame, the receiver uses the speed of light to compute distances to each satellite based on time of signal arrival
- At least four satellites are required to solve for 3D position, three if only 2D is required
GPS I Error Sources:
- Ephemeris & clock errors
- A clock error of $lx10^{-6}$s gives a 300m position error!
- Geometric Dilution of Precision (GDOP)
- The configuration of the visible satellites affects position precision
Improvements of GPS:
- Basic GPS:
- mobile receiver
- no error correction
- ~ 10m accuracy
- Differential GPS (DGPS):
- mobile receiver + fixed base station(s)
- estimate eror caused by atmospheric effects
- ~10m accuracy
- Real-Time Kinematic (RTK) GPS
- mobile receiver + fixed base station(s)
- estimate relative position using phase of carrier signal
- ~2cm accuracy