On the Design of an Optimal GNC Sensor Architecture

On the Design of an Optimal GNC Sensor Architecture For Autonomous Planetary Landers V.M.G. Silva and S.M. Parkes University of Dundee, Applied Computing, Dundee, DD1 4HN, Scotland, UK, Email: [email protected]

Abstract The definition of guidelines for the design of an optimal GNC sensor architecture for spacecraft is a recurrent problem. Since the beginning of Space Age several generations of GNC sensors have been developed, flight tested and successfully used in space missions. With the advent of new technologies and materials, GNC sensors performances have significantly improved on previous decades and new sensors have emerged. Concepts are evolving from ground-based to autonomous GNC systems. Hazard avoidance systems for planetary landers are replacing hazard tolerance systems. Both Lidar-based and Visionbased GNC systems are currently being developed to support autonomous landings. The optimum mix of sensors and the best way to fuse sensor data is an important area of research for future planetary lander space missions. Kalman filters play an important role in fusing data from several sensors to give an optimal estimate of the dynamic state of the spacecraft system. This paper briefly introduces Kalman filters and then gives a series of examples to demonstrate their use. It then goes on to describe current research at the University of Dundee on GNC sensor architectures for autonomous planetary landers. 1. - Introduction Previous lander missions to the surface of the moon and other planets, from Luna 9 in 1966 to the present day, have relied on hazard tolerance designs which use physical means to protect the payload from the shock of impact on the surface. Relatively flat, hazard free areas have been chosen for landing where accurate positioning was not necessary. As the space exploration proceeds to targets further away from the Earth, the communications delay make ground-control based on Earth too risky if not impracticable. An effective exploration of celestial bodies requires the capability of safe and precise landing on hazardous areas, often the most relevant for scientific purposes. The future exploration of the solar system and beyond will require autonomous spacecraft capable of safe and precise landing. New concepts for autonomous GNC sensor architectures have emerged in recent years and are still being researched [1], [2], [3]. Hazard avoidance is a promising landing design approach. Hazard avoidance systems can be of two main types: vision-based or lidar-based [4]. A planetary lander will carry navigation sensors like a camera, a radar altimeter, a scanning Lidar, star trackers, sun sensors, and horizon sensors. A camera may be used for visionbased navigation. A radar altimeter may be used to determine the height of the spacecraft above the surface. A scanning Lidar may be used to generate a 3D surface model for hazard avoidance. A star tracker may be used to determine its attitude or orientation.

Accelerometers and/or gyroscopes may be used to measure changes in the velocity of the spacecraft and thus track its movement from a known position. These sensors can be used to support the primary system making the estimate of position much more robust than that provided by a single sensor on its own. The optimum mix of sensors and the best way to fuse sensor data is an important area of research for future planetary lander missions [5]. Two essential elements of this research work are the effective simulation of the different types of navigation sensor and the design of an optimal navigation system for planetary landers. The sensor simulations must be a close reflection of the way in which real instruments operate and they must be synchronised in both time and space so that all sensors are observing a corresponding spot on the planet’s surface. This synchronisation is determined by the way in which the sensors are mounted on the spacecraft, so that if two sensors are co-axial their observations will be centred on the same point on the surface. The achievable accuracy of a navigation system can be described as a function of three main factors: the accuracies of the individual components, the geometry of the measurements, and the dynamics of the system. The performance of a given GNC sensor architecture can be accessed at the beginning of the design phase using the covariance equations [6]. The Extended Kalman Filter can then be used to integrate measurements from the previously determined optimum mix of sensors. This paper proceeds as follows. In Section 2 the Kalman filter is briefly introduced. Section 3 gives three examples of Kalman filter implementation. This is followed by an account on GNC sensor architecture for planetary landers, GNC sensor integration, and navigation system performance in Section 4. Finally, section 5 gives a summary of the work done, conclusions and future work plans.

2. – The Kalman Filter In this section the linear Kalman filter and the Extended Kalman filter are briefly described. 2.1- Methods for attitude and position determination

Methods for autonomous position and attitude determination can be broadly classified into two main types: deterministic and state estimation techniques. Deterministic methods use the minimum number of necessary measurements to compute the position/attitude of the spacecraft. These methods cannot take advantage of redundant measurements. State estimation techniques use redundant measurements to determine the position/attitude of the spacecraft. Estimation techniques can be batch or recursive. In batch processing the state estimate is based on processing a batch of data spread over a time interval. Recursive techniques use sequential measurements to update the estimate. The most commonly used estimation technique in space navigation is the Kalman filter.

2.2- The Linear Kalman Filter

In 1960, R.E. Kalman introduced a new formulation for the least squares filtering problem using state-space methods [7]. The Kalman filter is an estimator for the linear-quadratic problem. This problem consists of estimating the instantaneous state of a linear dynamic system perturbed by white noise. The Kalman filter uses measurements linearly related to the state but corrupted by white noise. Its recursive approach for determining the successive minimum variance state estimates makes it ideal for real-time on-board applications. The Kalman filter requires a dynamic model for the propagation of the system state over time and a model of the sensor measurements to achieve the most accurate estimate of the system state. Figure 1 illustrates the linear Kalman filter. Measurement Update (“Correct”) Input ( Xˆ k −1 , Pˆk −1 , Q, Rk , z k ) Time Update (“Predict”) (1) Project the state ahead xˆ k = Axˆ k −1 + Bu k (2) Project the error covariance ahead Pk− = APk −1 AT + Q

(1) Compute the Kalman gain K k = Pk− H T ( HPk− H T + R ) −1 (2) Update estimate with measurement z k xˆ k = xˆ k− + K k ( z k − Hxˆ k− ) (3) Update the error covariance Pk = ( I − K k H ) Pk−

Output ( Xˆ k , Pˆk ) Figure 1- Diagram illustrating the operation of the linear Kalman filter. Time update (“Predict”) and Measurement update (“Correct”) phases. Initial values comprise the best estimates for the initial state vector, the state covariance matrix, P , the process-noise covariance matrix, Q , and the measurement-noise covariance matrix,

R.

2.3- The Extended Kalman Filter

Numerous orbit and attitude estimation problems involve nonlinear models. The Kalman filter can also be applied to nonlinear systems and nonlinear measurements, provided that a linearization is performed. The linearization process assumes that the true state is sufficiently close to the estimated state, so that the error dynamics can be described by a linearized first-order Taylor series expansion. The linearization may lead to divergence problems in the error covariance matrix. Several solutions have been proposed to mitigate this problem [8]. The process and measurement noise are assumed to be modelled by a zero-mean Gaussian stochastic process with known covariance. Figure 2 gives a complete picture of the operation of the extended Kalman filter.

Measurement Update (“Correct”) Input ( Xˆ k −1 , Pˆk −1 , Q, Rk , z k )

(1) Compute the Kalman gain K k = Pk− H kT ( H k Pk− H kT + Vk Rk VkT ) −1

Time Update (“Predict”)

(1) Project the state ahead xˆ k = f ( xˆ k −1 , u k ,0) (2) Project the error covariance ahead Pk− = Ak Pk −1 AkT + Wk Qk −1WkT

(2) Update estimate with measurement z k xˆ k = xˆ k− + K k ( z k − h( xˆ k− ,0)) (3) Update the error covariance Pk = ( I − K k H k ) Pk−

Output ( Xˆ k , Pˆk ) Figure 2- Diagram illustrating the operation of the Extended Kalman filter

3. – Kalman Filter Examples In this section several examples of Kalman filter implementations are presented. 3.1- Estimating a scalar constant

Perhaps the simplest example is the estimation of a scalar constant using a set of randomly generated measurements [9]. Let’s suppose we want to estimate the value of z=-0.5 (the “truth”) based on 50 noise corrupted measurements of z. We start simulating the 50 measurements using a random function, which generates values with a Normal distribution with zero mean and 0.01 variance. The variance accounts for white measurement noise (RMS=0.1). The process can be described by the linear difference equation

x k = Ax k −1 + Bu k + wk where A = 1 (the system is time-invariant) and u = 0 because there is no control input. The measurements model is given by

z k = Hx k + v k where H = 1 . We then apply the discrete Kalman filter algorithm. Figure 3 illustrates the results of this simulation. It can be seen that the smaller is R , the closer the estimates approach the measurements. Increasing R , the filter is slower to respond to the measurements. The plot of the error covariance, P , versus the iteration shows the filter convergence. The initial value of P ( P0 = 1 ) for the error covariance was appropriate. The process of changing the measurement noise covariance, R , and the process noise covariance, Q , to obtain better performance is called “tuning” the filter.

Figure 3- Linear Kalman filter to estimate a scalar constant.

3.2- Planetary Lander falling in a constant gravity field

To illustrate how the Kalman Filter works applied to space navigation we start with a simple example of a planetary lander falling in a constant gravity field after MECO (Main Engine CutOff). The planetary lander carries a laser range finder to measure the altitude of the lander relative to the surface of the planet. The position, velocity and acceleration of the planetary lander falling in a constant gravity field after MECO are estimated using a discrete linear Kalman filter. Neglecting unknown perturbation forces, one can assume that the only force acting on the planetary lander is the gravity of the planet. In this circumstances the dynamic model consists of a simple acceleration, g. The dynamic model equations are a (t ) = a 0 ,

v(t ) = v0 + a 0 ∆t ,

r (t ) = r0 + v0 ∆t +

1 a 0 ∆t 2 2

and the state vector is given by x = [x 1 x 2 x 3 ] = [r v a] = [r r! !r!] . The system state at discrete times can be obtained propagating the state vector using the formula T

T

T

x(t k ) = Φ (t k , t k −1 ) x(t k −1 ) + Γ(t k −1 )u (t k −1 ) where the integral tk

Γ(t k −1 ) = ! Φ (t k , σ)C (σ)dσ t k −1

accounts for the control input (in the present example is 0) and the state transition matrix is given by

∆t 2 $ " 2 " ∆t 1 " " #"

' %1 ∆t Φ = %0 1 %0 0 % %&

The state space representation of the system can then be written as

' x1 $ xk = % x2 " %x " & 3 #k

' %1 ∆t % = %0 1 %0 0 %&

∆t 2 $ " 2 " ∆t " 1 " "#

' x1 $ % x2 " %x " & 3 # k −1

100m

g

range

Figure 4- Planetary Lander falling in a constant gravity field after MECO

We consider the planetary lander falling from an altitude of 100m in the gravity field of the planet Mars (Figure 3). The initial velocity is set to zero. The gravity of Mars is about 38% of the gravity of the Earth ( 9.8 × 0.38 ≅ 3.724 m / s 2 ). Assuming the initial conditions with 1σ uncertainty we have x10 = r0 = 100m ± 5m,

x 2 0 = v 0 = 0 m / s ± 5m / s ,

x30 = a0 = −3.724m / s 2 ± 0.01m / s 2

T which gives the best estimate of the initial state vector Xˆ 0 = [500 0 − 3.724] . An initial best estimate for the state covariance matrix P is also required. This can be obtained by squaring the standard deviations of the initial state vector components

0 $ '25 0 Pˆ0 = % 0 25 0" %& 0 0 0.0001"#

The initial estimate for the process-noise matrix, Q(t ) , can be defined by analysing the system at hands. In the present example we neglect all external forces that potentially could

cause disturbance torques, such as solar radiation pressure, atmospheric drag, perturbations due to non-spherical planet, etc. Bearing in mind these simplifications, we can assume that our model is very accurate and, therefore, the noise statistics should be very small. We assume the following statistics 'range noise 2 0 % 0 Q(t ) = velocity noise 2 % 0 0 &%

$ '1 0 " = %0 0 " % acceleration noise 2 #" &0

0 0.01 0

0 $ 0 " 0.01"#

Finally, we need to define an initial value for the measurement-noise covariance matrix, R(t ) . A reasonable assumption for R(t ) is R(t ) = σ 2 = 10m 2 . This standard deviation should not be confused with the initial standard deviations for the initial conditions. Since the only measurement is the range obtained using the laser range finder, the measurement model is given by z k = Hx k + v k , where H is the sensitivity matrix ( H = [1 0 0] ) and vk is a white Gaussian noise with zero mean. Figure 5 illustrates the

Figure 5- Planetary lander trajectory falling in a constant gravity field.

quick convergence of the Kalman filter. It also shows different performances of the Kalman filter for different values of R and Q . The update estimates at each time depend on the initial conditions, and the number and frequency of the measurements. 3.3. - Planetary Lander thruster-supported descent followed by free-fall in a constant gravity field after MECO

A more realistic example of a planetary lander system is introduced next. In the previous example the lander was only subjected to a constant gravity field. We will now consider a thruster-supported descent to reduce the velocity of the lander before MECO. In this case we have a linear time-varying system in contrast with the previous example where we had a linear time-invariant system. Assuming the acceleration provided by the thruster ( at ) as constant and the mass of the thruster negligible when compared with the total mass of the lander, we can write the state-space representation as

' x1 $ xk = % x2 " %& x3 "# k

' %1 ∆t = %0 1 %0 0 % %&

∆t 2 $ " 2 " ∆t 1 " " #"

$ '1 at ∆t 2 " % ' x1 $ 2 % x 2 " + % at ∆t "u k %& x3 "# k −1 % at " " % # &

We consider the planetary lander falling from an altitude of 500m in the gravity field of the planet Mars. The initial velocity is set to zero. The gravity of Mars is about 38% of the gravity of the Earth ( 9.8 × 0.38 ≅ 3.724 ). Assuming the initial conditions with 1σ uncertainty we have x10 = r0 = 500m ± 25m x 20 = v0 = −40m / s ± 5m / s x30 = a 0 = a t + g = 2m / s 2 with

at = 5.724m / s 2 ± 0.001m / s 2 g = −3.724m / s 2 ± 0.001m / s 2

which gives the best estimate of the initial state vector ' 500 $ Xˆ 0 = %− 40" %& 2 "#

Figure 6 illustrates the thruster-supported descent.

500m

t

range 100m

Figure 6- Thruster-supported descent. The thrusters are fired from 500m to MECO (at 100m) to compensate for the acceleration of gravity. After MECO the lander is on free-fall.

The initial best estimates for the state covariance matrix, P , and the process noise covariance matrix, Q(t ) , were '625 0 Pˆ0 = % 0 25 %& 0 0

0$ 0" 1"#

and

'1 Q(t ) = %0 %&0

0 0.01 0

0 $ 0 " 0.01"#

Figure 7- Position estimation of a planetary lander during thruster-supported descent using a Kalman filter.

The measurement covariance was R(t ) = σ 2 = 25m 2 . Figure 7 illustrates the position estimation during thruster-supported descent. The time interval for state update was ∆t = 0.5 seconds. It can be seen that the Kalman filter estimates closely track the reference trajectory both during the thruster-supported and the free-fall phases.

4. – GNC Sensor Architecture for Planetary Landers

GNC sensor architecture design can be defined as the process of selecting the appropriate set of sensors for a given mission. This involves making decisions about the sensor types, as well as the interfaces with the other GNC subsystems and the spacecraft. The main questions driving the design process of a GNC sensor architecture are the required accuracy and the robustness of the navigation system. The required level of accuracy will determine the type of sensors to use. The required robustness will help to establish the best set of sensors for a given mission and the sensor redundancy if any has to be used. Whenever redundancy has to be used it is preferable to use analytical redundancy rather than hardware redundancy. Unlike hardware redundancy, analytical redundancy does not add extra mass to the system and is less prone to error. 4.1- GNC sensor integration

A GNC sensor architecture may comprise an inertial navigation system (INS) and sensors that provide external information. The INS provides very good high frequency information but poor low frequency data due to its slow rate drift with time. External sources of information can provide three main types of data: Position data (radar, laser ranging, position fixes (star sightings, landmarks), and radiometric area correlation, etc.), Velocity data (Doppler radar), and Altitude data (radar altimeter, lidar). External sources provide good low frequency information but are usually corrupted by high frequency noise. An optimal GNC sensor architecture should comprise INS complemented by external data. The integration of data from both sources can be achieved through a kalman filter. In space navigation the Kalman filter is usually implemented in conjunction with an INS. The Kalman filter formulation can either be total state space (direct filtering) or error state space (indirect filtering). Another important aspect to take into account is the configuration of the aided INS. This can be feedforward (using an ordinary linearized Kalman filter) or feedback (using an extended Kalman filter). Both configurations have been successfully used in several applications. The best configuration depends on the particular situation at hand.

4.2- Possible GNC sensor architectures for Planetary Landers

GNC sensor architectures for planetary landers are evolving from traditional hazard tolerance ground-based control to hazard avoidance autonomous systems. Unlike hazard tolerance, hazard avoidance systems use onboard sensors to detect hazards and can autonomously select a safe landing site and manoeuvre the spacecraft to that site. Hazard

avoidance systems can be of two main types: vision-based or Lidar-based. Both designs present advantages and disadvantages. Vision-based hazard avoidance systems use vision to detect potential hazards like boulders and small craters in the vicinity of the target landing area. The main disadvantages are that they require very good lit conditions on the landing site and cannot be used for landing in the dark. Often, the landing site is not well lit, for several reasons, such as shadows due to local topography, or the relative position of the Sun, e.g. low Sun elevation angles [10]. Their main advantages include low mass and cost ratios to the amount of navigation information provided, and the potential to implement analytical redundancy, e.g. the camera can provide roll and pitch information, typically obtained by horizon sensors. Lidar-based hazard avoidance uses a 3D surface model for hazard detection, which is generated by a scanning Lidar. As an active sensor, a scanning Lidar provides its own illumination and can be used virtually in any lit conditions, including in the dark. This advantage makes it a suitable technology to be used on planetary landers, especially for missions requiring landing in the dark or poor lit conditions. It can also be used as redundant/diverse system in conjunction with a camera. A disadvantage of Lidar-based systems is the reliability of the scanning mechanism. 4.3- Navigation System Performance

The achievable accuracy of a navigation system can be described as a function of three main factors: the accuracies of the individual sensors, the geometry of the measurements, and the dynamics of the system. We can think of system-level accuracy as a function P comprising three factors P=P(R, H, F). P is the system-level accuracy, R is the componentlevel accuracies, H is the geometry of the problem, and F is the system dynamics. P corresponds to the state error covariance matrix in the Kalman filter. The performance of a given GNC sensor architecture can be assessed computing the covariance matrix P. This can be accomplished by recursively calculating xˆ k +1 and Pk +1 for k ≥ 0 . Convergence to a steady-state covariance can be achieved after a sufficient number of iterations, i.e., if k → ∞ , then Pk +1 → P∞ . A more straightforward approach consists in solving for the steady-state covariance directly. This involves closed-form solutions to the steady-state covariance of the Algebraic Riccati Equation. The solution to the Riccati equation can be used to predict the performance of a Kalman filter. All that is required in design analyzes for characterizing the performance of a proposed GNC sensor architecture before it is actually implemented are the covariance equations [6].

5. – Summary and Conclusions

The Kalman filter as a GNC sensor integration tool for space navigation has been introduced through a series of simple examples of relevance to planetary landers. Different landing design approaches have been considered and possible GNC sensor architectures for planetary landers briefly discussed. Possible GNC sensor architectures for autonomous

planetary landers have been briefly analysed. Methods for assessing the performance of a given GNC sensor architecture have been presented. Future research at the University if Dundee aims to answer the following questions: •

What GNC sensors are most appropriate to use on a planetary lander?

•

Can we simulate GNC sensors fast enough and realistically enough to enable integration in a close-loop simulation of a lander?

•

How can we test which are the best set of sensors for a particular mission?

•

How can sensor data best be fused to give required navigation information?

6. - References

[1] Jean-Marius, T, and Strandmoe, SE, “Integrated Vision and Navigation for a Planetary Lander”, IAF, 1998. [2] Johnson, A., Klumpp, A., Collier, J., Wolf, A.: “LIDAR-based Hazard Avoidance for Safe Landing on Mars”, AAS 01-120 in the AAS/AIAA Space Flight Mechanics Meeting, Santa Barbara, CA, February 2001. [3] ESTEC, “Statement of Work for the study Navigation for Planetary Approach and Landing on Mercury”, ESA doc. Ref. ESA-ESC-SES-SOW-00.01, February 2001. [4] Steve Parkes, Victor Silva, Martin Dunstan, Iain Martin, David Matthews:”LIDAR-based GN&C for Safe Landing Using PANGU”, DASIA-2003 [5] S. M. Parkes and V. Silva, “GNC sensors for planetary landers – a review”, in: DASIA 2002 Data Systems in Aerospace (Dublin, Ireland 13-16 May 2002) SP-509, pp.1-9. [6] Grewal, Mohinder S. and Andrews, Angus P.: “Kalman Filtering – Theory and Practice using Matlab”, Second Edition, John Wiley & Sons, 2001 [7] Kalman, R. E.:”A New Approach to Linear Filtering and Prediction Problems”, Transaction of the ASME--Journal of Basic Engineering, pp. 35-45 (March 1960). [8] Hwang, P.Y.C., Robert, G.B.: "Introduction to Random Signals and Applied Kalman Filtering - with Matlab Exercises and Solutions", Third-Edition, John Wiley & Sons, 1997. [9] Welch, G. and Bishop, G.:”An Introduction to the Kalman Filter”, Department of Computer Science, University of North Carolina at Chapel Hill, 2002 [10]Martin, I.: “Virtual Planetary Surfaces for Testing Vision Guided Planetary Landers”, PhD Thesis, University of Dundee 2001