1995 IEEE Int. Conf. on Robotics and Automation ( Osaka, Japan )
1
Attitude Control without Angular Velocity Measurement: A Passivity Approach Fernando Lizarralde
G.S.C.A.R. Department of Electrical Eng. - COPPE/UFRJ 21945/970 - Rio de Janeiro - BRAZIL e-mail:
[email protected]
Center for Advanced Technology in Automation and Robotics Dept. of Electrical, Computer and Systems Eng. Rensselaer Polytechnic Institute - Troy, NY 12180 e-mail:
[email protected]
Abstract
It is well known that the linear feedback of the quarternion of the attitude error and angular velocity globally stabilizes the attitude of a rigid body. In this note, we show that the angular velocity feedback can be replaced by a nonlinear lter of the quaternion, thus removing the need of direct angular velocity measurement. In contrast to other approaches, this design exploits the inherent passivity of the system; a dynamic observer reconstructing the velocity is not needed. An application of the proposed scheme is illustrated for the robot control problem. Simulation results are included to illustrate the theoretical results.
1 Introduction
Stable control of a rigid body orientation is required in the pointing and slewing for spacecrafts [17, 18] and the orientation control of a rigid payload held by a single (or multiple) robot [11, 12, 16]. In the case of robot arm control, the arms or hand ngers can be viewed as actuators maneuvering the attitude of the held object. There are many possible parameterizations of the manifold SO(3) on which the orientation evolves. Unit quarternion, which consists of four parameters subject to the unity length constraint, is a popular choice in the spacecraft control literature [17, 18]. This is due in part to the fact that the unit quarternion is a globally non-singular parameterization. An extensive treatment of the attitude control problem using quaternion was presented in [15]; where a wide range of stabilizing control laws were obtained: from model independent proportional-derivative (PD) control to adaptive control. These controllers are motivated by the passivity property inherent in this problem: the rigid body dynamics is passive with torque as input and angular velocity as output. Supported by
Brazilian Research Council (CNPq).
John T. Wen
In most of the attitude control strategies presented in the literature, the angular velocity measurement is required. However, this assumption may not always be satis ed (e.g., not all the manipulators are equipped with tachometer). In such cases, the angular velocity is typically obtained through the approximate di erentiation of some orientation representation. Alternatively, an angular velocity observer was proposed in [13]. However, a separation principle-like property was conjectured but not proved; consequently, there was no closed-loop stability analysis. In this paper, we show that the passivity based attitude control strategies can be extended to the case that the angular velocity is not available. In contrast to [13], this approach does not explicitly construct an observer (i.e., it does not use input/output and/or model information), instead, the angular velocity is replaced by the vector quarternion ltered by a lead lters. Global asymptotic stability is shown by extending the passivity analysis in [15]. Our approach is very similar in spirit to the recently proposed velocity-less PD control for robotic manipulators [1, 3, 8]. The main complication comes from the fact that the angular velocity cannot be integrated to an equivalent position variable. We show that in the case of the unit quarternion representation, this diculty can be overcome by noting that the columns at the representation Jacobian are of unit length and mutually orthogonal. This paper is organized as follows. In Section 2, the attitude control problem is formulated in term of the unit quaternion. Section 3 presents the main result of the paper. The global asymptotic stability under only the error quarternion feedback is shown. An extension to a robot control in task space is presented in Section 4. Section 5 presents some simulation results and the paper is closed with some conclusions.
2
Rigid Body Kinematics, Dynamic, and Control with Unit Quaternion
where is the rotation about equivalent axis k, subject to the constraint qT q = 1. The rotation matrix R can be related to q through the Rodrigues Formula:
This section summarizes the kinematics and dynamics of the rigid body orientation. Consider two orthonormal right-handed coordinate frames: the inertial (world) coordinate, E O = [eO1 ; eO2 ; eO3 ], and body coordinate (attached to the rigid body), E B = [eB1 ; eB2 ; eB3 ]. De ne the 3 3 attitude matrix as R = EB EO where () denotes the adjoint operation. Then the rigid body kinematic and dynamic equations can be written in the inertial coordinate in the following form:
R = I + 2q0(qv ) + 2(qv )(qv ) (7) Algorithm for computing q from R can be found in [10]. In general q both represent R and this sign ambiguity
can be resolved by using the kinematic equation below: (8) q_ = 21 E(q)! where
M !_ + ! M! = ; M = RM 0RT (1) d R = ! R = R(!B ) (2) dt
0 ?q ?q ?q 1 1 2 3 Tv B ? q q q ? q 0 3 E(q)= q0I ? (qv ) = B@ ?q3 q0 q21 CCA (9)
where !; !B 2
