Having finalized the LVLH attitude control in Chapter 8, we now proceed with the syn-thesis and analysis of the final approach position control. The trajectory profile for the different phases of the mission is recalled from Figure 2.4 and Figure 2.5. The final approach is from points3tos4and further on until docking. The hand over from GPS to RVS sensors takes place ats3, as well as a change from COM to COM control to port to port control.
The research here will focus on the latter part after change over to the final approach as this is the more critical and demanding part.
The position control is3 DOF and the attitude control of the chaser remains with respect to the LVLH frame. This is the second situation illustrated in Figure 7.3. The lateral control objective along the y and z-axis is to keep the center of the docking ports aligned, but at this large distance it is not desirable to track the target port oscillatory motion. Instead we will design a controller which will track the mean motion of the target docking port and thereby conserve fuel without any loss of performance. This means controlling the relative differences to zero along the y and z-axes. The control objective along the x-axis of the LVLH frame is to follow the guidance profile developed in Section 7.7 and detailed in Figure 7.10 and tables 7.5 and 7.6. Further to this servo type problem the control system shall also handle disturbance rejection, which is mostly the presence of differential drag, which is described in Section 3.4.2.
9.1 Control Requirement Detailing
The location ofs3 ands4is recalled from Table 2.1 to bes3 = [−500,0,0]Tm COM to COM ands4= [−20,0,0]Tm port to port respectively. It shall be recalled thats4is along the target docking port x-axis in theFdt0frame.
From Table 2.5 it is evident that ats3the chaser docking port shall stay inside a cube of20m with corresponding rates smaller than0.1m/s. This box is parallel to LVLH and will move around as the target docking port is moving with respect to LVLH due to the
ISS attitude motion.
Ats4 there are two sets of requirements namely the arrival and departure ones as listed in Tables 2.6 and 2.7 respectively. This location is used for changing control structure from3DOF to6DOF and some transients are anticipated before reaching the departure requirements. The arrival box is1m and rates less than 0.1m/s for y and z-axes and0.02m/s for the x-axis. The departure requirements are not applicable for3 DOF controller. The approach velocity shall be in the range of[0.05; 0.35]m/s.
From Figure 2.5 we see that the approach corridor in the AE and into the KOZ is conical, but there are no formal requirements betweens3ands4except the definition of the KOZ.
For practical reasons we will consider the requirement linearly connected between s3ands4and we will try to aim directly fors4 departure requirements, as defined in Table 2.7, with the3DOF controller.
9.2 Target and Sensor Characteristic
The target docking port motion is described in detail in Section 3.2.1 and from that we will derive the needs for propulsion and control bandwidth in order to be able to track the motion.
The worst case of fastest reverse timed= 8s and largest amplitude0.7 degis used as a basis. From the data in Section 3.1.2 forrbtandrgdtwe can compute the lever arm as the first component ofrdtto be about31m. From the angular ratevtin Table 3.2 we can find the largest acceleration needed to bea∼= 2vt|drdt| = 2.7·10−3m/s2. For the largest chaser mass from Equation (C.5) we then need a force of at leastFmin= 55N.
This is well covered by theFmaxspecified in Equation (8.1) of150N.
A good estimate of the minimum bandwidth required can be obtained by considering the acceleration of a sinusoidalAω2=agivingω= 0.014Hz.
The only sensor used is the RVS which measures only distance and angles. There are no measurements of rates or accelerations or other sensors to measure such. The RVS is described in Section 3.3.4 and Table 3.3 which reveals a complex internal functioning with many modes. We will not use all those details for controller synthesis, but extract the essential worst case characteristics.
We will use the range valuesRdirectly for the x-axis direction and the lateral values we approximate withRsin(LOS). The simplified data is presented in Table 9.1
x-axis y,z-axis Bias[m] [0.01; 5] [0; 2.6]
Noise3σ[m] [0.005; 25] [0; 1.75]
Table 9.1: Simplified sensor data extracted from Table 3.3. The lower and upper values corre-spond to a range of[0; 500]m.
9.3 Plant Description and Variation 177
The data in Table 9.1 is considered to vary linearly between the two boundary values, though it is not exactly so.
This can also be expressed such that the range has5% noise of the range and the lateral noise is8.75%. This formulation will be convenient in the following control design.
If we compare the requirements ats4, but use the docking requirements departings4, as is often done, the sensor bias does not fit departings4initially. The bias is0.2m and the requirement is half of that. One could estimate the bias with a filter likeH(s) =s−ps and subtract it from the measurements. The problem with that approach is the filter time constant, as in (Ignagni 1990), needs to be comparable to the travel time froms3tos4. The filter then needs to be initialized fairly close to the real values, which might then be used directly.
The latter approach will be used and the sensor calibration values for the bias will be subtracted before used for control.
9.3 Plant Description and Variation
The model for the relative dynamics was developed in Section 4.1 and we will use the LTV models in Equations (4.16) and (4.18) for the controller designs.
We see that the spacecraft mass does not affect the plant dynamics, but only the orbital parameter variations.
Out of plane: The pole variations for the rigid plant are all on the jω axis and are symmetric over half an orbit. The variation is listed in Table 9.2.
ε= 0.1 θ= 0deg θ= 180deg %∆
0±1.12·10−3 0±8.29·10−4 −26%
Table 9.2:Pole variation for the out of plane plant for the true anomalyθ∈[0; 180]deg.
In plane: The pole variation is more complex than for the out of plane and is better illustrated by the graphs in Figure 9.1.
The root locus in Figure 9.1 is for half an orbit only. The locus for the complete orbit is symmetric around thejωaxis. This is caused by the fact that the derivative of the orbital rateω˙ in Equation (4.16) changes sign on each side of the semi major axis.
The locus for the full orbit is available in Figure 7.5.
We observe in Figure 9.1 that2poles remain with an imaginary component in the LHP. The2other poles move in the RHP, then become real and one of them relocate to the LHP. This transition from RHP to LHP happens when the true anomaly is of such value that the spacecraft in the orbit intersects the minor axis of the elliptical orbit. This is exactly one quarter and three quarters around the orbit measured from perigee and where one can say the orbit is the widest.
Figure 9.1 illustrates the locus for3different orbital eccentricitiesεand one observes that they are not all that different, mostly a faster dynamics for higherεaround perigee.
−2 0 2 x 10−4
−1.5
−1
−0.5 0 0.5 1
1.5x 10−3In Plane ε = 0.01
jω
−5 0 5 10
x 10−4
−1.5
−1
−0.5 0 0.5 1
1.5x 10−3In Plane ε = 0.1
jω
−1 0 1
x 10−3
−2
−1.5
−1
−0.5 0 0.5 1 1.5
2x 10−3In Plane ε = 0.7
jω
θ = 0 deg θ = 180 deg
A B
Figure 9.1: Pole variation for the in plane plant for the true anomalyθ ∈ [0; 180]deg and for 3different eccentricitiesε ∈ [0.01,0.1,0.7]. The point marked as A are where two complex conjugate poles become real and point B is the smallest value for the other complex poles.
In Figure 9.1 the point A indicates where one set of complex poles becomes real. The point B indicates the smallest real value for the other set of poles. Both these phenomena happen exactly at the points of maximum acceleration of the true anomaly.
For ε = 0.1, the eccentricity for this research work, the poles are bounded by [−4.3; 5.5]·10−4for the real parts and±1.2·10−3for the imaginary parts.
The poles originating from the flexible modes and sloshing are as listed in Table 8.2, though the couplings to the rigid dynamics are different than they were for the attitude dynamics. Those modes will not be considered for the control design, though loop shaping might be applied if deemed necessary by stability and performance analysis.
9.4 Position H
∞Control Design
For the complete range of the relative position control we will perform the synthesis by means ofH∞ control. This is a worst case design method and suits well for the critical uncertainties and variations present in the system. Further this is a novel ap-proach applied to all the phases of the relative motion in a multi variable manner, where a single axis design for final mode is presented in (Bourdon, Delpy, Ganet, Quinquis &
Ankersen 2003).
For completeness and insight the theory and assumptions for theH∞ design will be explained followed by general scaling of the plants. Then the simpler out of plane control will be performed followed by the coupled in plane control.
The control problem can be formulated as a2×2system illustrated in Figure 9.2 The system in Figure 9.2 is formulated by
z v
=P(s) w
u
=
P11(s) P12(s) P21(s) P22(s)
w u
(9.1)