Visual Feedback Attitude Control of a Bias Momentum Micro Satellite using Two Wheels Fuyuto Terui a, Nobutada Sako b, Keisuke Yoshihara c, Toru Yamamoto c, Shinichi Nakasuka b a National Aerospace Laboratory of Japan, 7-44-1 Jindaiji-Higashimachi, Chofu-shi, Tokyo 182-8522, Japan b Department of Aeronautics and Astronautics, University of Tokyo, Hongou 7-3-1, Bunkyou-ku, Tokyo 113, Japan c National Space Development Agency, Sengen 2-1-1, Tsukuba-shi, Ibaraki-ken 35-47, Japan Abstract A 55 kg micro-piggyback satellite, µ-lab Sat, is currently under development. One of its missions will be to release a truncated cone-shaped target and obtain camera images of it. Since there are uncertainties regarding the release direction of the target, it would be advantageous if the satellite were able to adjust its attitude to track it, performing three-axis attitude maneuvers. µ-lab Sat is a bias momentum microsatellite with two wheels, and although such maneuvers are not easy for this hardware configuration, two algorithms have been developed to enable the satellite to track the target; one utilizes an optimal control algorithm and the other applies the sliding mode control algorithm. This paper describes these algorithms and presents the results of their evaluation by numerical simulation. Nomenclature p Rodriguez parameter. J Satellite moment of inertia matrix (principal axes are the satellite s body-fixed reference frame, J = diag[j 1,J 2,J 3 ]). ω Satellite attitude angular velocity expressed in the body-fixed reference frame, [ω 1,ω 2,ω 3 ] T. J ωi Moment of inertia matrix of the i-th wheel (i=1, 3). θ ωi Rotational angle of the i-th wheel (i=1, 3). z i Unit vector representing the direction of each axis of the body-fixed reference frame (i=1, 2, 3). n ωi Control torque applied to the i-th wheel (i=1, 3). n max H i l Control torque limit for each wheel (.1 Nm). Satellite angular momentum vector expressed in the inertial reference frame. Angular momentum vector of wheels expressed in the body-fixed reference frame, [l 1,l 2,l 3 ] T.
Φ b i l θ t cmos x cmos y Direction cosine matrix from inertial reference frame to body-fixed reference frame. Initial angular momentum of WHL-M. Total maneuver angle of the 3rd axis of the body frame. x axis position of the target in the CMOS camera image. y axis position of the target in the CMOS camera image. Introduction Figure 1: µ-lab Sat external configuration Figure 2: The target release mission (left) and µ-lab Sat attitude control devices (right) A group of Japanese space-related organizations are jointly developing a micro piggyback satellite µ-lab Sat which is planned to be launched in near future. The satellite has a maximum diameter of 688 mm, is 515 mm in height, and has a mass of approximately 55 kg. One of µ-lab Sat s missions is to evaluate the use of machine vision to estimate the relative position and attitude between a satellite and a target. To this end, µ-lab Sat will release a truncated cone-shaped target with representative dimensions of approximately 1 cm and then obtain images of it using a camera with a CMOS image sensor. The target will be released from the satellite at a velocity of approximately 1 cm/s by means of springs. Since there is a degree of uncertainty regarding the precise direction of release, it would be advantageous if the satellite were able to alter its attitude to track the target to maintain it near the center of the image. Fig. 2 (right) shows the configuration of µ-lab Sat s attitude control devices. The satellite is equipped with two wheels (WHL-R and WHL-M) which rotate about mutually orthogonal axes. During the mission, WHL-M has an angular momentum of
approximately.97 Nms. The signals from three Fiber Optic Gyros (FOG-1, 2, 3), one for each axis, and information from the images of the target are used as feedback signals for tracking. µ-lab Sat is a bias momentum satellite with two wheels, and 3-axis attitude maneuvers are not easy for this configuration. This paper presents two algorithms developed for 3-axis attitude maneuvering of this satellite: a Switching Time Search Controller algorithm, and a controller that applies the Sliding Mode control algorithm. The performance of these algorithms are evaluated and compared through numerical simulation of target tracking maneuvers considering the constraints of the actual on-orbit experiments to be conducted. Attitude Controller Design This section describes the two algorithms developed for three-axis attitude maneuvering of a two-wheel bias momentum satellite. Switching Time Search Controller The Switching Time Search Controller (SWSC) is a quasi-time-optimal controller which approximates the optimal solution obtained by off-line algorithms such as the Sequential Conjugate Gradient Restoration Algorithm (SCGRA)[2]. Characteristics of the Optimal Solution obtained by SCGRA The optimal control input which minimizes the time to reach a final condition can be obtained using optimal control algorithms such as SCGRA. Such off-line optimization algorithms tend to be computation-intensive and so are unsuitable for realtime applications, particularly in the case of small autonomous satellites where the amount of on-board computing power is limited. The optimal control input trajectory obtained always possesses certain characteristics, and it may be possible to use approximation to obtain quasi-optimal solution for real-time applications. Fig. 3 (left) shows an example of a control input solution given by SCGRA for a target tracking maneuver. Through many trials using various initial and final conditions, it was found that the control input profiles had the following common characteristics: WHL-R maintains maximum torque ±n max throughout the maneuver. WHL-M maintains maximum torque ±n max and switches polarity several times. SWSC (t < t low ) It is assumed that initially WHL-M has angular momentum l and WHL-R has zero angular momentum. If the total maneuver angle of the 3rd axis of the satellite s bodyfixed reference frame is θ t, then the angular momenta of WHL-R and WHL-M at the
Figure 3: SCGRA torque profile (left) and SWSC torque profile (right) final condition should be l sinθ t and l cosθ t respectively. Since the torque of WHL- R is constant, the assumed minimum time to complete the maneuver t low is given by Using this t low, the final state is defined as t low = l sin θ t n max (1) ω 1 (t low ) = ω 2 (t low ) = ω 3 (t low ) = cmos x (t low ) = cmos y (t low ) = (2) l 1 (t low ) = l sinθ t, l 3 (t low ) = l cosθ t (3) and the SWSC torque profile can be defined as follows. WHL-R maintains constant maximum torque until the angular momentum of WHL-R reaches l sinθ t, as shown in the left of Fig. 3 (right). WHL-M maintains maximum torque output and changes its polarity twice such that its angular momentum becomes l cosθ t, as shown in the right of the Fig. 3 (right). Two parameters, t 1,t 2 can be regarded as design parameters. Since the total amount of the angular momentum deviation of WHL-M is fixed, t 2 t 1 is fixed. To determine the value of t 1, the attitude motion of the satellite is simulated using the torque input profile mentioned above with various values of t 1 and the sub-optimal torque profile that moves the target closest to the final states at t = t low is found. Feedback Control (t t low ) After SWSC is applied, a feedback control is used to compensate the error between the final state achieved by SWSC and the desired state. n w1 = k 11 ω 1 + k 12 ω 2 + k 13 ω 3 + k 14 (l 1 l sinθ t ) + k 15 cmos x + k 16 cmos y (4)
n w3 = k 31 ω 1 + k 32 ω 2 + k 33 ω 3 + k 34 (l 3 l cosθ t ) + k 35 cmos x + k 36 cmos y (5) The feedback gains k i j (i = 1,...,3, j = 1,...,6) are tuned so that stability and convergence are achieved. Sliding Mode Controller Satellite Angular Momentum Using the actual data of the moment of inertia, J + J ω1 + J ω3 can be approximated as J + J ω1 + J ω3 J = diag [ 2.93 2.94 3.9 ] [Nms] (6) Since the external torque is zero, the angular momentum of the satellite is constant and the angular velocity ω is expressed as follows ω = z 1 J(1,1) 1 J ω1 (1,1) θ ω1 z 3 J(3,3) 1 J ω3 (3,3) θ ω3 + J 1 Φ b i H i (7) Attitude Kinematics The Rodriguez parameter p = [p 1, p 2, p 3 ] T, which describes the attitude of a satellite, is defined as p = atan φ 2 where a is the Euler axis (the principal vector) and φ is the rotation angle around it (the principal angle). The time derivative of Rodriguez parameter is (8) ṗ = 1 2 (1 + p + ppt )ω (9) Substituting eq. (7) into eq. (9), the equation giving the relation between the time derivative of the Rodriguez parameters is obtained in a state-space representation.[1] where ṗ1 h 1 ṗ 2 h 2 ṗ 3 h 3 = 1 p 3 p 1 1 [ ṗ1 h 1 ṗ 3 h 3 ] = B(p)U (1) h = 1 2 (1 + p + ppt )J 1 Φ b i H i (11) It is interesting to note that ṗ 2, which can be regarded as attitude rate along the 2-axis, can be changed using attitude rates along the 1- and 3-axes, ṗ 1 and ṗ 3, assuming that p 3 and p 1 are non-zero. Therefore, U in eq. (1) is used as the control input for the Sliding Mode Controller as described in the following subsection.
Design of the Sliding Surface The sliding surface is defined as a linear combination of the Rodriguez parameter and its time derivative σ = Sp + ṗ = (12) where S is a 3 3 matrix designed so that the dynamics on the sliding surface are stable and p represents relative attitude error for the tracking and is calculated using error Euler angles obtained from the target position in the CMOS camera image. ṗ is calculated using p and attitude angular velocity ω, which is measured by fiber optic gyros using eq. (9). Control Input To force the state [p ṗ] T to go to the sliding surface from any initial condition and then constrain it to the sliding surface, the wheel angular velocity command θ ω1, θ ω3 is given as follows [ ] [ ˆθ ω1 2 J1 J 1 ][ ] ω = 1 (1 + p 2 3 ) p 1 p 3 + p 2 ˆθ ω3 1 + p 2 1 + p2 2 + p2 J 3 3 Jω 1 3 p 1 p 3 p 2 (1 + p 2 1 ) {(SB) T (SB)} 1 (SB) T {K sgn(σ) + Sh} (13) For the calculation of h in eq. (13), Φ b i Hi is required as shown in eq. (11), and this requires that attitude angle and the initial angular momentum vector in the inertial reference frame be known. Although in theory these may be obtained from the output of magnetic sensor MS(X,Y,Z) or by integration of the outputs of the FOGs, neither of these sensors has sufficient accuracy. However, since the value of Sh is negligibly small compared to K sgn(σ), Sh can be ignored in this implementation. Since the wheels are driven by commanded torques, the control torques n ω1, n ω3 are approximated from ˆθ ω1, ˆθ ω3 as follows n ω1 = J ω1 k w ( ˆθ ω1 θ ω1 ) (14) n ω3 = J ω3 k w ( θ ω3i + ˆθ ω3 θ ω3 ) (15) where k w is a design parameter and θ ω3i is the initial angular velocity of the momentum wheel. Numerical Simulation The performance of the two algorithms were evaluated using numerical simulation, and the results compared. Fig. 4 shows two examples of the closed-loop position of the target in the camera image during a maneuver. Relative attitude is calculated using the target position information extracted from the image and used by the attitude controller. The figures along the paths show time elapsed (in seconds) since the start
of the maneuver. In both cases it seems that responses by SWSC is faster than ones by Sliding Mode Controller. Further evaluation of these attitude controllers are necessary for implementation of the planned on-orbit experiment, with consideration of electrical power consumption, communication link requirements, robustness against parameter uncertainty, and performance degradation due to noise. 8 6 4 2 2 4 6 1-1 -2 Sliding Mode Controller Quasi Time Optimal Controller 8 1 4 8 6 6 cmosy[deg] 2-2 Sliding Mode Controller Quasi Time Optimal Controller 4 8 1 8 6 cmosy[deg] -3-4 -5-6 2 2 4-4 -8-7 -6-5 -4-3 -2-1 1 cmosx[deg] -7-6 -5-4 -3-2 -1 1 cmosx[deg] Figure 4: Target position in the camera image Concluding Remarks Two types of controllers that enable a bias momentum micro-satellite with 2 wheels to perform 3-axis attitude maneuvers were designed and compared by numerical simulation. References [1] F.Terui, et.al., Target Tracking Attitude Maneuver of a Bias Momentum Micro Satellite Using Two Wheels, AIAA-2-4144, AIAA Guidance, Navigation, and Control Conference, 2 [2] A.K.Wu and A.Miele, Sequential Conjugate Gradient-Restoration Algorithm for Optimal Control Problems with Non-Differential Constraints and General Boundary Conditions, Part1, Optimal Control Applications & Methods, Vol.1, pp.69-88, 198.