Unobservable Directions of VINS Under Special Motions
In this paper, we study the unobservable directions of vision aided inertial navigation systems (VINS) under special motion profiles. It is well known that the VINS has four unobservable directions, corresponding to the global translation and the rotation around gravity, under the assumption of generic 3D motions. Here we study the case when the motion is constrained to be of either constant local acceleration or no rotation, as commonly found with ground vehicles. We show and prove the additional unobservable directions due to these special motions analytically, as well as the corresponding physical meanings of these new directions. 1 Vision aided Inertial Navigation System (VINS) First, we provide a brief review of the VINS. The VINS estimates the following state vector: x IqTG b T g v I b T a GpTI f 1 f N ]T (1) where IqG is the unit quaternion that represents the orientation of the global frame {G} in the IMU frame {I} at time t. GvI and GpI are the the velocity and position of {I} in {G} respectively, and the gyroscope and accelerometer biases are denoted by bg and ba, respectively. Finally, the positions of point features in {G} are denoted by Gfj j 1, N The IMU provides measurements of the rotational velocity, om, and the linear acceleration, am, as: om(t) o(t) bg(t) ng(t) (2) am(t) C( IqG(t) a(t) g) ba(t) na(t) where the noise terms, ng(t) and na(t) are modelled as zero mean, white Gaussian noise processes, C( IqG(t) denotes the rotational matrix associated with the quaternion IqG at time t, while the gravitational acceleration, g, is considered a known constant. The IMU's rotational velocity o(t) and linear acceleration a(t) in (2) can be used to derive the continuous time system equations: IqG(t) 1 2 O(om(t) bg(t) ng(t))IqG(t) bg(t) nwg(t) GvI(t) C( IqG(t) T (am(t) ba(t) na(t) g ba(t) nwa(t) GpI(t) GvI(t) Gfj(t) 0, j 1, N (3)