Nonlinear Complementary Filters on the Special Orthogonal Group

This paper considers the problem of obtaining good attitude estimates from measurements obtained from typical low cost inertial measurement units. The outputs of such systems are characterized by high noise levels and time varying additive biases. We formulate the filtering problem as deterministic observer kinematics posed directly on the special orthogonal group SO (3) driven by reconstructed attitude and angular velocity measurements. Lyapunov analysis results for the proposed observers are derived that ensure almost global stability of the observer error. The approach taken leads to an observer that we term the direct complementary filter. By exploiting the geometry of the special orthogonal group a related observer, termed the passive complementary filter, is derived that decouples the gyro measurements from the reconstructed attitude in the observer inputs. Both the direct and passive filters can be extended to estimate gyro bias online. The passive filter is further developed to provide a formulation in terms of the measurement error that avoids any algebraic reconstruction of the attitude. This leads to an observer on SO(3), termed the explicit complementary filter, that requires only accelerometer and gyro outputs; is suitable for implementation on embedded hardware; and provides good attitude estimates as well as estimating the gyro biases online. The performance of the observers are demonstrated with a set of experiments performed on a robotic test-bed and a radio controlled unmanned aerial vehicle.

I. Introduction

The recent proliferation of microelectromechanical systems (MEMS) components has lead to the development of a range of low cost and light weight inertial measurement units. The low power, light weight and potential for low cost manufacture of these units opens up a wide range of applications in areas such as virtual reality and gaming systems, robotic toys, and low cost mini-aerial-vehicles (MAVs) such as the Hovereye (Fig. 1). The signal output of low cost IMU systems, however, is characterized by low-resolution signals subject to high noise levels as well as general time-varying bias terms. The raw signals must be processed to reconstruct smoothed attitude estimates and bias-corrected angular velocity measurements. For many of the low cost applications considered the algorithms need to run on embedded processors with low memory and processing resources.