Merge pull request #2054 from jenniferoum/feature/equivariant-filter
Implement Equivariant Filter Bias example in Pythonrelease/4.3a0
Frank Dellaert
GitHub
commit
7435127fc8
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"""
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Implementation of Attitude-Bias-Calibration EqF form:
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"Overcoming Bias: Equivariant Filter Design for Biased Attitude Estimation with Online Calibration"
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https://ieeexplore.ieee.org/document/9905914
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This module is Alessandro Fornasier's equivariant filter code (https://github.com/aau-cns/ABC-EqF)
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converted to use GTSAM's libraries.
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Authors: Jennifer Oum & Darshan Rajasekaran
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"""
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import numpy as np
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import gtsam
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from gtsam import Rot3, Unit3
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from dataclasses import dataclass
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from typing import List
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coordinate = "EXPONENTIAL"
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def checkNorm(x: np.ndarray, tol: float = 1e-3):
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"""Check norm of a vector being 1 or nan
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:param x: A numpy array
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:param tol: tollerance, default 1e-3
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:return: Boolean true if norm is 1 or nan
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"""
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return abs(np.linalg.norm(x) - 1) < tol or np.isnan(np.linalg.norm(x))
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class State:
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"""Define the state of the Biased Attitude System
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----------
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R is a rotation matrix representing the attitude of the body
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b is a 3-vector representing the gyroscope bias
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S is a list of rotation matrix, each representing the calibration of the corresponding direction sensor
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----------
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Let's assume we want to use three known direction a, b, and c, where only the sensor that measure b is
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uncalibrated (we'd like to estimate the calibration states). Therefore, the system's d list looks like
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d = [b, a, c], and the S list should look like S = [Sb]. The association between d and S is done via indeces.
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In general S[i] correspond to the calibration state of the sensor that measure the direcion d[i]
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----------
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"""
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# Attitude rotation matrix R
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R: Rot3
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# Gyroscope bias b
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b: np.ndarray
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# Sensor calibrations S
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S: List[Rot3]
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def __init__(
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self,
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R: Rot3 = Rot3.Identity(),
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b: np.ndarray = np.zeros(3),
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S: List[Rot3] = None,
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):
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"""Initialize State
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:param R: A SO3 element representing the attitude of the system as a rotation matrix
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:param b: A numpy array with size 3 representing the gyroscope bias
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:param S: A list of SO3 elements representing the calibration states for "uncalibrated" sensors,
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if no sensor require a calibration state, than S will be initialized as an empty list
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"""
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if not isinstance(R, gtsam.Rot3):
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raise TypeError(
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"the attitude rotation matrix R has to be of type SO3 but type is",
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type(R),
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)
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self.R = R
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if not (isinstance(b, np.ndarray) and b.size == 3):
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raise TypeError(
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"The gyroscope bias has to be probvided as numpy array with size 3"
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)
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self.b = b
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if S is None:
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self.S = []
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else:
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if not isinstance(S, list):
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raise TypeError("Calibration states has to be provided as a list")
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for calibration in S:
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if not isinstance(calibration, Rot3):
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raise TypeError(
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"Elements of the list of calibration states have to be of type SO3"
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)
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self.S = S
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@staticmethod
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def identity(n: int):
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"""Return a identity state with n calibration states
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:param n: number of elements in list B associated with calibration states
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:return: The identity element of the State
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"""
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return State(Rot3.Identity(), np.zeros(3), [Rot3.Identity() for _ in range(n)])
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class Input:
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"""Define the input space of the Biased Attitude System
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----------
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w is a 3-vector representing the angular velocity measured by a gyroscope
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----------
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"""
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# Angular velocity
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w: np.ndarray
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# Noise covariance of angular velocity
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Sigma: np.ndarray
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def __init__(self, w: np.ndarray, Sigma: np.ndarray):
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"""Initialize Input
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:param w: A numpy array with size 3 representing the angular velocity measurement from a gyroscope
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:param Sigma: A numpy array with shape (6, 6) representing the noise covariance of the
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angular velocity measurement and gyro bias random walk
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"""
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if not (isinstance(w, np.ndarray) and w.size == 3):
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raise TypeError(
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"Angular velocity has to be provided as a numpy array with size 3"
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)
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if not (
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isinstance(Sigma, np.ndarray) and Sigma.shape[0] == Sigma.shape[1] == 6
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):
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raise TypeError(
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"Input measurement noise covariance has to be provided as a numpy array with shape (6. 6)"
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)
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if not np.all(np.linalg.eigvals(Sigma) >= 0):
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raise TypeError("Covariance matrix has to be semi-positive definite")
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self.w = w
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self.Sigma = Sigma
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@staticmethod
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def random() -> "Input":
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"""Return a random angular velocity
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:return: A random angular velocity as a Input element
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"""
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return Input(np.random.randn(3), np.eye(6))
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def W(self) -> np.ndarray:
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"""Return the Input as a skew-symmetric matrix
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:return: self.w as a skew-symmetric matrix
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"""
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return Rot3.Hat(self.w)
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class G:
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"""Symmetry group (SO(3) |x so(3)) x SO(3) x ... x SO(3)
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----------
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Each element of the B list is associated with a calibration states in State's S list where the association is done
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via corresponding index. In general B[i] is the SO(3) element of the symmetry group that correspond to the
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state's calibration state S[i]. For example, let's assume we want to use three known direction a, b, and c, where
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only the sensor that measure b is uncalibrated (we'd like to estimate the calibration states). Therefore,
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the system's d list is defined as d = [b, a, c], and the state's S list is defined as S = [Sb]. The symmetry group
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B list should be defined as B = [Bb] where Ba is the SO(3) element of the symmetry group that is related to Sb
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----------
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"""
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A: Rot3
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a: np.ndarray
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B: List[Rot3]
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def __init__(
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self,
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A: Rot3 = Rot3.Identity(),
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a: np.ndarray = np.zeros((3, 3)),
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B: List[Rot3] = None,
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):
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"""Initialize the symmetry group G
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:param A: SO3 element
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:param a: np.ndarray with shape (3, 3) corresponding to a skew symmetric matrix
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:param B: list of SO3 elements
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"""
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if not isinstance(A, Rot3):
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raise TypeError("A has to be of type SO3")
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self.A = A
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if not (isinstance(a, np.ndarray) and a.shape == (3, 3)):
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raise TypeError("a has to be a numpy array with shape (3, 3)")
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self.a = a
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if B is None:
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self.B = []
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else:
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for b in B:
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if not isinstance(b, Rot3):
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raise TypeError("Elements of B have to be of type SO3")
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self.B = B
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def __mul__(self, other: "G") -> "G":
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"""Define the group operation
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:param other: G
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:return: A element of the group G given by the "multiplication" of self and other
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"""
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assert isinstance(other, G)
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assert len(self.B) == len(other.B)
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return G(
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self.A * other.A,
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self.a + Rot3.Hat(self.A.matrix() @ Rot3.Vee(other.a)),
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[self.B[i] * other.B[i] for i in range(len(self.B))],
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)
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@staticmethod
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def identity(n: int):
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"""Return the identity of the symmetry group with n elements of SO3 related to sensor calibration states
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:param n: number of elements in list B associated with calibration states
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:return: The identity of the group G
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"""
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return G(Rot3.Identity(), np.zeros((3, 3)), [Rot3.Identity() for _ in range(n)])
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@staticmethod
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def Rot3LeftJacobian(arr: np.ndarray) -> np.ndarray:
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"""Return the SO(3) Left Jacobian
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:param arr: A numpy array with size 3
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:return: The left Jacobian of SO(3)
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"""
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if not (isinstance(arr, np.ndarray) and arr.size == 3):
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raise ValueError("A numpy array with size 3 has to be provided")
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angle = np.linalg.norm(arr)
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# Near |phi|==0, use first order Taylor expansion
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if np.isclose(angle, 0.0):
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return np.eye(3) + 0.5 * Rot3.Hat(arr)
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axis = arr / angle
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s = np.sin(angle)
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c = np.cos(angle)
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return (
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(s / angle) * np.eye(3)
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+ (1 - (s / angle)) * np.outer(axis, axis)
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+ ((1 - c) / angle) * Rot3.Hat(axis)
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)
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def exp(x: np.ndarray) -> "G":
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"""Return a group element X given by X = exp(x) where x is a numpy array
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:param x: A numpy array
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:return: A element of the group G given by the exponential of x
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"""
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if not (isinstance(x, np.ndarray) and x.size >= 6):
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raise ValueError(
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"Wrong shape, a numpy array with size 3n has to be provided"
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)
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if (x.size % 3) != 0:
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raise ValueError(
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"Wrong size, a numpy array with size multiple of 3 has to be provided"
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)
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n = int((x.size - 6) / 3)
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A = Rot3.Expmap(x[0:3])
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a = Rot3.Hat(G.Rot3LeftJacobian(x[0:3]) @ x[3:6])
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B = [Rot3.Expmap(x[(6 + 3 * i) : (9 + 3 * i)]) for i in range(n)]
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return G(A, a, B)
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def inv(self) -> "G":
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"""Return the inverse element of the symmetry group
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:return: A element of the group G given by the inverse of self
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"""
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return G(
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self.A.inverse(),
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-Rot3.Hat(self.A.inverse().matrix() @ Rot3.Vee(self.a)),
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[B.inverse() for B in self.B],
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)
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class Direction:
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"""Define a direction as a S2 element"""
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# Direction
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d: Unit3
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def __init__(self, d: np.ndarray):
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"""Initialize direction
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:param d: A numpy array with size 3 and norm 1 representing the direction
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"""
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if not (isinstance(d, np.ndarray) and d.size == 3 and checkNorm(d)):
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raise TypeError("Direction has to be provided as a 3 vector")
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self.d = Unit3(d)
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def blockDiag(A: np.ndarray, B: np.ndarray) -> np.ndarray:
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"""Create a lock diagonal matrix from blocks A and B
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:param A: numpy array
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:param B: numpy array
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:return: numpy array representing a block diagonal matrix composed of blocks A and B
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"""
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if A is None:
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return B
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elif B is None:
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return A
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else:
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return np.block(
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[
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[A, np.zeros((A.shape[0], B.shape[1]))],
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[np.zeros((B.shape[0], A.shape[1])), B],
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]
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)
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def repBlock(A: np.ndarray, n: int) -> np.ndarray:
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"""Create a block diagonal matrix repeating the A block n times
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:param A: numpy array representing the block A
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:param n: number of times to repeat A
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:return: numpy array representing a block diagonal matrix composed of n-times the blocks A
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"""
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res = None
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for _ in range(n):
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res = blockDiag(res, A)
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return res
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def numericalDifferential(f, x) -> np.ndarray:
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"""Compute the numerical derivative via central difference"""
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if isinstance(x, float):
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x = np.reshape([x], (1, 1))
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h = 1e-6
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fx = f(x)
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n = fx.shape[0]
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m = x.shape[0]
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Df = np.zeros((n, m))
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for j in range(m):
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ej = np.zeros(m)
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ej[j] = 1.0
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Df[:, j : j + 1] = (f(x + h * ej) - f(x - h * ej)).reshape(m, 1) / (2 * h)
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return Df
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def lift(xi: State, u: Input) -> np.ndarray:
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"""The Lift of the system (Theorem 3.8, Equation 7)
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:param xi: A element of the State
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:param u: A element of the Input space
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:return: A numpy array representing the Lift
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"""
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n = len(xi.S)
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L = np.zeros(6 + 3 * n)
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L[0:3] = u.w - xi.b
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L[3:6] = -u.W() @ xi.b
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for i in range(n):
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L[(6 + 3 * i) : (9 + 3 * i)] = xi.S[i].inverse().matrix() @ L[0:3]
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return L
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def checkNorm(x: np.ndarray, tol: float = 1e-3):
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"""Check norm of a vector being 1 or nan
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:param x: A numpy array
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:param tol: tollerance, default 1e-3
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:return: Boolean true if norm is 1 or nan
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"""
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return abs(np.linalg.norm(x) - 1) < tol or np.isnan(np.linalg.norm(x))
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def stateAction(X: G, xi: State) -> State:
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"""Action of the symmetry group on the state space, return phi(X, xi) (Equation 4)
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:param X: A element of the group G
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:param xi: A element of the State
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:return: A new element of the state given by the action of phi of G in the State space
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"""
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if len(xi.S) != len(X.B):
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raise ValueError(
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"the number of calibration states and B elements of the symmetry group has to match"
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)
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return State(
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xi.R * X.A,
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X.A.inverse().matrix() @ (xi.b - Rot3.Vee(X.a)),
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[(X.A.inverse() * xi.S[i] * X.B[i]) for i in range(len(X.B))],
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)
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def velocityAction(X: G, u: Input) -> Input:
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"""Action of the symmetry group on the input space, return psi(X, u) (Equation 5)
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:param X: A element of the group G
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:param u: A element of the Input
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:return: A new element of the Input given by the action of psi of G in the Input space
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"""
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return Input(X.A.inverse().matrix() @ (u.w - Rot3.Vee(X.a)), u.Sigma)
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def outputAction(X: G, y: Direction, idx: int = -1) -> np.ndarray:
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"""Action of the symmetry group on the output space, return rho(X, y) (Equation 6)
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:param X: A element of the group G
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:param y: A direction measurement
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:param idx: indicate the index of the B element in the list, -1 in case no B element exist
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:return: A numpy array given by the action of rho of G in the Output space
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"""
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if idx == -1:
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return X.A.inverse().matrix() @ y.d.unitVector()
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else:
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return X.B[idx].inverse().matrix() @ y.d.unitVector()
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def local_coords(e: State) -> np.ndarray:
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"""Local coordinates assuming __xi_0 = identity (Equation 9)
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:param e: A element of the State representing the equivariant error
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:return: Local coordinates assuming __xi_0 = identity
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"""
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if coordinate == "EXPONENTIAL":
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tmp = [Rot3.Logmap(S) for S in e.S]
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eps = np.concatenate(
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(
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Rot3.Logmap(e.R),
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e.b,
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np.asarray(tmp).reshape(
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3 * len(tmp),
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),
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)
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)
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elif coordinate == "NORMAL":
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raise ValueError("Normal coordinate representation is not implemented yet")
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# X = G(e.R, -SO3.Rot3.Hat(e.R @ e.b), e.S)
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# eps = G.log(X)
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else:
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raise ValueError("Invalid coordinate representation")
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return eps
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def local_coords_inv(eps: np.ndarray) -> "State":
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"""Local coordinates inverse assuming __xi_0 = identity
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:param eps: A numpy array representing the equivariant error in local coordinates
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:return: Local coordinates inverse assuming __xi_0 = identity
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"""
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X = G.exp(eps) # G
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if coordinate == "EXPONENTIAL":
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e = State(X.A, eps[3:6, :], X.B) # State
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elif coordinate == "NORMAL":
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raise ValueError("Normal coordinate representation is not implemented yet")
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# stateAction(X, State(SO3.identity(), np.zeros(3), [SO3.identity() for _ in range(len(X.B))]))
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else:
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raise ValueError("Invalid coordinate representation")
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return e
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def stateActionDiff(xi: State) -> np.ndarray:
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"""Differential of the phi action phi(xi, E) at E = Id in local coordinates (can be found within equation 23)
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:param xi: A element of the State
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:return: (Dtheta) * (Dphi(xi, E) at E = Id)
|
||||
"""
|
||||
coordsAction = lambda U: local_coords(stateAction(G.exp(U), xi))
|
||||
differential = numericalDifferential(coordsAction, np.zeros(6 + 3 * len(xi.S)))
|
||||
return differential
|
||||
|
||||
|
||||
class Measurement:
|
||||
"""Define a measurement
|
||||
----------
|
||||
cal_idx is a index corresponding to the cal_idx-th calibration related to the measurement. Let's consider the case
|
||||
of 2 uncalibrated sensor with two associated calibration state in State.S = [S0, S1], and a single calibrated sensor.
|
||||
cal_idx = 0 indicates a measurement coming from the sensor that has calibration S0, cal_idx = 1 indicates a
|
||||
measurement coming from the sensor that has calibration S1. cal_idx = -1 indicates that the measurement is coming
|
||||
from a calibrated sensor
|
||||
----------
|
||||
"""
|
||||
|
||||
# measurement
|
||||
y: Direction
|
||||
|
||||
# Known direction in global frame
|
||||
d: Direction
|
||||
|
||||
# Covariance matrix of the measurement
|
||||
Sigma: np.ndarray
|
||||
|
||||
# Calibration index
|
||||
cal_idx: int = -1
|
||||
|
||||
def __init__(self, y: np.ndarray, d: np.ndarray, Sigma: np.ndarray, i: int = -1):
|
||||
"""Initialize measurement
|
||||
|
||||
:param y: A numpy array with size 3 and norm 1 representing the direction measurement in the sensor frame
|
||||
:param d: A numpy array with size 3 and norm 1 representing the direction in the global frame
|
||||
:param Sigma: A numpy array with shape (3, 3) representing the noise covariance of the direction measurement
|
||||
:param i: index of the corresponding calibration state
|
||||
"""
|
||||
|
||||
if not (isinstance(y, np.ndarray) and y.size == 3 and checkNorm(y)):
|
||||
raise TypeError("Measurement has to be provided as a (3, 1) vector")
|
||||
if not (isinstance(d, np.ndarray) and d.size == 3 and checkNorm(d)):
|
||||
raise TypeError("Direction has to be provided as a (3, 1) vector")
|
||||
if not (
|
||||
isinstance(Sigma, np.ndarray) and Sigma.shape[0] == Sigma.shape[1] == 3
|
||||
):
|
||||
raise TypeError(
|
||||
"Direction measurement noise covariance has to be provided as a numpy array with shape (3. 3)"
|
||||
)
|
||||
if not np.all(np.linalg.eigvals(Sigma) >= 0):
|
||||
raise TypeError("Covariance matrix has to be semi-positive definite")
|
||||
if not (isinstance(i, int) or i == -1 or i > 0):
|
||||
raise TypeError("calibration index is a positive integer or -1")
|
||||
|
||||
self.y = Direction(y)
|
||||
self.d = Direction(d)
|
||||
self.Sigma = Sigma
|
||||
self.cal_idx = i
|
||||
|
||||
|
||||
@dataclass
|
||||
class Data:
|
||||
"""Define ground-truth, input and output data"""
|
||||
|
||||
# Ground-truth state
|
||||
xi: State
|
||||
n_cal: int
|
||||
|
||||
# Input measurements
|
||||
u: Input
|
||||
|
||||
# Output measurements as a list of Measurement
|
||||
y: list
|
||||
n_meas: int
|
||||
|
||||
# Time
|
||||
t: float
|
||||
dt: float
|
||||
|
||||
|
||||
class EqF:
|
||||
def __init__(self, Sigma: np.ndarray, n: int, m: int):
|
||||
"""Initialize EqF
|
||||
|
||||
:param Sigma: Initial covariance
|
||||
:param n: Number of calibration states
|
||||
:param m: Total number of available sensor
|
||||
"""
|
||||
|
||||
self.__dof = 6 + 3 * n
|
||||
self.__n_cal = n
|
||||
self.__n_sensor = m
|
||||
|
||||
if not (
|
||||
isinstance(Sigma, np.ndarray)
|
||||
and (Sigma.shape[0] == Sigma.shape[1] == self.__dof)
|
||||
):
|
||||
raise TypeError(
|
||||
f"Initial covariance has to be provided as a numpy array with shape ({self.__dof}, {self.__dof})"
|
||||
)
|
||||
if not np.all(np.linalg.eigvals(Sigma) >= 0):
|
||||
raise TypeError("Covariance matrix has to be semi-positive definite")
|
||||
if not (isinstance(n, int) and n >= 0):
|
||||
raise TypeError("Number of calibration state has to be unsigned")
|
||||
if not (isinstance(m, int) and m > 1):
|
||||
raise TypeError("Number of direction sensor has to be grater-equal than 2")
|
||||
|
||||
self.__X_hat = G.identity(n)
|
||||
self.__Sigma = Sigma
|
||||
self.__xi_0 = State.identity(n)
|
||||
self.__Dphi0 = stateActionDiff(self.__xi_0) # Within equation 23
|
||||
self.__InnovationLift = np.linalg.pinv(self.__Dphi0) # Within equation 23
|
||||
|
||||
def stateEstimate(self) -> State:
|
||||
"""Return estimated state
|
||||
|
||||
:return: Estimated state
|
||||
"""
|
||||
return stateAction(self.__X_hat, self.__xi_0)
|
||||
|
||||
def propagation(self, u: Input, dt: float):
|
||||
"""Propagate the filter state
|
||||
|
||||
:param u: Angular velocity measurement from IMU
|
||||
:param dt: delta time between timestamp of last propagation/update and timestamp of angular velocity measurement
|
||||
"""
|
||||
|
||||
if not isinstance(u, Input):
|
||||
raise TypeError(
|
||||
"angular velocity measurement has to be provided as a Input element"
|
||||
)
|
||||
|
||||
L = lift(self.stateEstimate(), u) # Equation 7
|
||||
|
||||
Phi_DT = self.__stateTransitionMatrix(u, dt) # Equation 17
|
||||
# Equivalent
|
||||
# A0t = self.__stateMatrixA(u) # Equation 14a
|
||||
# Phi_DT = expm(A0t * dt)
|
||||
|
||||
Bt = self.__inputMatrixBt() # Equation 27
|
||||
M_DT = (
|
||||
Bt @ blockDiag(u.Sigma, repBlock(1e-9 * np.eye(3), self.__n_cal)) @ Bt.T
|
||||
) * dt
|
||||
|
||||
self.__X_hat = self.__X_hat * G.exp(L * dt) # Equation 18
|
||||
self.__Sigma = Phi_DT @ self.__Sigma @ Phi_DT.T + M_DT # Equation 19
|
||||
|
||||
def update(self, y: Measurement):
|
||||
"""Update the filter state
|
||||
|
||||
:param y: A measurement
|
||||
"""
|
||||
|
||||
# Cross-check calibration
|
||||
assert y.cal_idx <= self.__n_cal
|
||||
|
||||
Ct = self.__measurementMatrixC(y.d, y.cal_idx) # Equation 14b
|
||||
delta = Rot3.Hat(y.d.d.unitVector()) @ outputAction(
|
||||
self.__X_hat.inv(), y.y, y.cal_idx
|
||||
)
|
||||
Dt = self.__outputMatrixDt(y.cal_idx)
|
||||
S = Ct @ self.__Sigma @ Ct.T + Dt @ y.Sigma @ Dt.T # Equation 21
|
||||
K = self.__Sigma @ Ct.T @ np.linalg.inv(S) # Equation 22
|
||||
Delta = self.__InnovationLift @ K @ delta # Equation 23
|
||||
self.__X_hat = G.exp(Delta) * self.__X_hat # Equation 24
|
||||
self.__Sigma = (np.eye(self.__dof) - K @ Ct) @ self.__Sigma # Equation 25
|
||||
|
||||
def __stateMatrixA(self, u: Input) -> np.ndarray:
|
||||
"""Return the state matrix A0t (Equation 14a)
|
||||
|
||||
:param u: Input
|
||||
:return: numpy array representing the state matrix A0t
|
||||
"""
|
||||
|
||||
W0 = velocityAction(self.__X_hat.inv(), u).W()
|
||||
A1 = np.zeros((6, 6))
|
||||
|
||||
if coordinate == "EXPONENTIAL":
|
||||
A1[0:3, 3:6] = -np.eye(3)
|
||||
A1[3:6, 3:6] = W0
|
||||
A2 = repBlock(W0, self.__n_cal)
|
||||
elif coordinate == "NORMAL":
|
||||
raise ValueError("Normal coordinate representation is not implemented yet")
|
||||
else:
|
||||
raise ValueError("Invalid coordinate representation")
|
||||
|
||||
return blockDiag(A1, A2)
|
||||
|
||||
def __stateTransitionMatrix(self, u: Input, dt: float) -> np.ndarray:
|
||||
"""Return the state transition matrix Phi (Equation 17)
|
||||
|
||||
:param u: Input
|
||||
:param dt: Delta time
|
||||
:return: numpy array representing the state transition matrix Phi
|
||||
"""
|
||||
|
||||
W0 = velocityAction(self.__X_hat.inv(), u).W()
|
||||
Phi1 = np.zeros((6, 6))
|
||||
Phi12 = -dt * (np.eye(3) + (dt / 2) * W0 + ((dt**2) / 6) * W0 * W0)
|
||||
Phi22 = np.eye(3) + dt * W0 + ((dt**2) / 2) * W0 * W0
|
||||
|
||||
if coordinate == "EXPONENTIAL":
|
||||
Phi1[0:3, 0:3] = np.eye(3)
|
||||
Phi1[0:3, 3:6] = Phi12
|
||||
Phi1[3:6, 3:6] = Phi22
|
||||
Phi2 = repBlock(Phi22, self.__n_cal)
|
||||
elif coordinate == "NORMAL":
|
||||
raise ValueError("Normal coordinate representation is not implemented yet")
|
||||
else:
|
||||
raise ValueError("Invalid coordinate representation")
|
||||
|
||||
return blockDiag(Phi1, Phi2)
|
||||
|
||||
def __inputMatrixBt(self) -> np.ndarray:
|
||||
"""Return the Input matrix Bt
|
||||
|
||||
:return: numpy array representing the state matrix Bt
|
||||
"""
|
||||
|
||||
if coordinate == "EXPONENTIAL":
|
||||
B1 = blockDiag(self.__X_hat.A.matrix(), self.__X_hat.A.matrix())
|
||||
B2 = None
|
||||
for B in self.__X_hat.B:
|
||||
B2 = blockDiag(B2, B.matrix())
|
||||
elif coordinate == "NORMAL":
|
||||
raise ValueError("Normal coordinate representation is not implemented yet")
|
||||
else:
|
||||
raise ValueError("Invalid coordinate representation")
|
||||
|
||||
return blockDiag(B1, B2)
|
||||
|
||||
def __measurementMatrixC(self, d: Direction, idx: int) -> np.ndarray:
|
||||
"""Return the measurement matrix C0 (Equation 14b)
|
||||
|
||||
:param d: Known direction
|
||||
:param idx: index of the related calibration state
|
||||
:return: numpy array representing the measurement matrix C0
|
||||
"""
|
||||
|
||||
Cc = np.zeros((3, 3 * self.__n_cal))
|
||||
|
||||
# If the measurement is related to a sensor that has a calibration state
|
||||
if idx >= 0:
|
||||
Cc[(3 * idx) : (3 + 3 * idx), :] = Rot3.Hat(d.d.unitVector())
|
||||
|
||||
return Rot3.Hat(d.d.unitVector()) @ np.hstack(
|
||||
(Rot3.Hat(d.d.unitVector()), np.zeros((3, 3)), Cc)
|
||||
)
|
||||
|
||||
def __outputMatrixDt(self, idx: int) -> np.ndarray:
|
||||
"""Return the measurement output matrix Dt
|
||||
|
||||
:param idx: index of the related calibration state
|
||||
:return: numpy array representing the output matrix Dt
|
||||
"""
|
||||
|
||||
# If the measurement is related to a sensor that has a calibration state
|
||||
if idx >= 0:
|
||||
return self.__X_hat.B[idx].matrix()
|
||||
else:
|
||||
return self.__X_hat.A.matrix()
|
||||
Loading…
Reference in New Issue