Predicting State Uncertainty for GNSS-based UAVPath Planning using Stochastic Reachability

Akshay Shetty, University of Illinois Urbana-ChampaignGrace Xingxin Gao, Stanford University

Akshay Shetty received the B.Tech. degree in Aerospace Engineering from Indian Institute of Technology, Bombay, Indiain 2014. He received the M.S. degree in Aerospace Engineering from University of Illinois at Urbana-Champaign in 2017. Heworked as an Intern at NASA Ames Research Center during the summers of 2016 and 2017. He is currently pursuing his Ph.D.degree in the Department of Aerospace Engineering at University of Illinois at Urbana-Champaign. His research interests includesafe navigation and control for autonomous vehicles.

Grace Xingxin Gao Grace Xingxin Gao is an assistant professor in the Department of Aeronautics and Astronautics at StanfordUniversity. Before joining Stanford University, she was an assistant professor at University of Illinois at Urbana-Champaign. Sheobtained her Ph.D. degree at Stanford University. Her research is on robust and secure positioning, navigation and timing withapplications to manned and unmanned aerial vehicles, robotics and power systems.

Abstract

An important component of path planning algorithms is predicting state uncertainty in order to ensure probabilistic collision-freepaths. While predicting state uncertainty, path planning algorithms generally assume zero mean Gaussian distributions for motionand measurement error models. However, a zero mean Gaussian assumption is not applicable for Global Navigation Satellite System(GNSS) positioning measurements, which typically contain uncertain biases in urban areas due to multipath and non-line-of-sight(NLOS) effects. In this paper, we propose a method to predict the state uncertainty of a UAV in the presence of uncertain GNSSpositioning biases using stochastic reachability analysis. We enclose all possible GNSS positioning error distributions arising dueto uncertain biases with a probabilistic zonotope. For the on-board state estimation filter, we choose a Kalman filter (KF) and fix ahypothesis for the GNSS positioning error distribution. We first compute the stochastic set of estimation errors resulting from theabove hypothesis. We then compute the stochastic reachable sets for a linear motion model of the UAV with linear state feedbackcontrol along the candidate path. Finally, we validate the stochastic reachability analysis by evaluating the predicted state uncertaintyalong candidate paths in a simulated 3D urban environment.

Keywords

Unmanned Aerial Vehicle (UAV), Global Navigation Satellite System (GNSS), Path Planning, Stochastic Reachability,Probabilistic Zonotope

I. INTRODUCTION

Path planning under uncertainty in robotics [1]–[4] consists of two major components: predicting the state uncertaintyalong candidate paths, and building the planner tree [5] from start to goal state. Predicting the state uncertainty typicallyinvolves predicting the uncertainty in the robots position, which is used to ensure probabilistic collision-free paths. In [1], the

Fig. 1: (a) An urban scenario where multipath effects might exist along a candidate path. (b) These multipath effects result inuncertain GNSS positioning biases.

Fig. 2: Illustrations of (a) reachability analysis and (b) stochastic reachability analysis.

authors present linear-quadratic Gaussian motion planning (LQG-MP) that predicts state uncertainty taking into account a-priorimeasurement model and controller knowledge, and combines it with a rapidly-exploring random tree (RRT) planner. The authorsof [2] present rapidly-exploring random belief trees (RRBT), which implements a similar method to predict state uncertainty andfinds the optimal path to the goal state. In [3] and [4], the authors predict the state uncertainty of a vision-controlled unmannedaerial vehicle (UAV) and implement RRBT and RRT* planners respectively. This work focuses on predicting the state uncertaintyfor GNSS-based path planning.

Recently there have been some advances in GNSS-based path planning [6], [7]. In [6], the authors generate a GNSS positioningerror map, followed by implementing an A* for path planning based on path length and the error map. Similarly in [7], theauthors create a signal reliability map for positioning derived from GNSS and cellular signals, followed by implementing aDjikstra’s algorithm for path planning using the reliability map. While these methods focus on planning paths with lowerGNSS positioning errors, they do not explicitly check for collisions as done by path planning under uncertainty algorithmsdiscussed earlier. Additionally, these methods do not consider stochastic error models for the UAV motion and GNSS positioningmeasurements.

Path planning under uncertainty algorithms generally assume zero mean Gaussian distributions for motion and measurementerror models while predicting state uncertainty [1]–[4]. However, such an assumption is not always applicable for GNSSpositioning measurements which usually contain biases in urban areas due to multipath and non-line-of-sight (NLOS) effects.Numerous previous works exist for modeling the amount of GNSS positioning biases in urban areas. In [8]–[10], the authorsuse ray-tracing algorithms in order to predict the amount of positioning errors at a given location due to multipath and NLOSeffects. The authors of [11] take a machine learning approach and train a random forest model to predict GNSS positioningerrors due to multipath effects given details of the nearby environment. In [7], the authors derive an analytical upper bound forthe positioning bias due to multipath effects for a weighted nonlinear least-squares estimator. However, the GNSS positioningbiases observed during path execution are challenging to predict since they depend on various factors such as the details ofnearby urban structures [12] and the GNSS receiver-specific characteristics [13]. Fig. 1 shows an urban scenario where multipatheffects might exist along a candidate path, resulting in uncertain biases in GNSS positioning. Thus, in order to predict the state

Fig. 3: Overview of our method to predict state uncertainty.

uncertainty for GNSS-based path planning, it is important to consider the effects of these uncertain biases along the path.One approach to predict the state uncertainty in the presence of bounded uncertain GNSS positioning biases is to use reachability

analysis [14]. Classical reachability consists of computing a set of reachable states for a robot given an initial set of states, a setof system inputs and a set of model and measurement errors. The analysis was extended to stochastic reachability in [15], inorder to account for stochastic error models. Fig. 2 illustrates the concepts of reachability and stochastic reachability analysis. In[16], [17] the authors use reachability analysis to predict the state uncertainty. However, the authors do not consider the effect oferrors in the state estimation filters running on-board the robot. State estimation errors during path execution affect the feedbackcontrol inputs to the system, which consequently affect the reachable sets for the robot.

In this paper, our contribution is a method to predict the state uncertainty in the presence of uncertain GNSS positioning biasesusing stochastic reachability analysis. Fig 3 shows an overview of our method. We incorporate the effects of uncertain GNSSpositioning biases by enclosing all possible error distributions at a given location with a probabilistic zonotope (introduced in[15]). Since it is challenging to predict the biases observed during path execution, we fix a GNSS positioning error distributionhypothesis for the state estimation filter on-board the UAV. We choose a Kalman filter (KF) as the state estimation filter. In ourstochastic reachability analysis, we first compute the stochastic set of state estimation errors resulting from the above hypothesis.We then compute the stochastic reachable sets for the UAV along the candidate path, for a linear motion model with linearstate feedback control. In order to validate the stochastic reachability analysis, we evaluate the predicted state uncertainty alongcandidate paths in a simulated 3D urban environment. Additionally, we show that the computed stochastic reachable sets can beused to detect whether a candidate path is probabilistic collision-unsafe.

The rest of the paper is organized as follows. In section II we formulate our problem by describing the GNSS positioning errormodel and the UAV system. In section III we provide the details for our stochastic reachability analysis. We discuss simulationresults in section IV and finally conclude and discuss future directions for our work in section V.

II. PROBLEM FORMULATION

A. GNSS Positioning Error ModelWe first introduce the probabilistic zonotope set representation introduced in [15]. A probabilistic zonotope is an enclosing

probabilistic hull with a non-normalized probability density function. It is defined by a center, a width and Gaussian tails:

V = (cV ,gV ,ΣV)Z , (1)

where cV is the center of the probabilistic zonotope, gV is a set of vectors representing its width wV , and ΣV is the covariance ofits Gaussian tails. Probabilistic zonotopes are suitable to enclose multiple distributions. Fig. 4 shows two different examples wherea probabilistic zonotope can be used to enclose two mixtures of Gaussians or to enclose a Gaussian distribution with an uncertainmean. As discussed in section I, GNSS positioning measurements contain uncertain biases in urban areas. In order to considerthe effect of such uncertain positioning biases while predicting state uncertainty, we enclose the possible error distributions (asshown in Fig. 1(b)) with a probabilistic zonotope.

The problem of approximating bounds for the uncertain biases in the position domain has received a significant amount ofattention in recent works [7]–[11], and is beyond the scope of this paper. We assume the bounds for the uncertain biases areprovided, which we use to obtain the center cV and the width wV of the probabilistic zonotope V for a given location. Thevariance of GNSS pseudoranges have been shown to change with different multipath biases [18]. This would result in changes inthe GNSS positioning variance at different biases for a given location. For simplicity, we assume the same dilution of precision

Fig. 4: Examples of a probabilistic zonotope V , defined in equation (1), used to enclose: (a) two mixtures of Gaussians, or (b)a Gaussian distribution with an uncertain mean.

(DOP) [19] based covariance matrix at all possible biases, as shown in Fig. 1(b). More complicated non-Gaussian models forGNSS positioning errors can also be considered, as long as the distributions are enclosed by a suitable probabilistic zonotope.

In order to calculate the DOP matrix, we first compute the geometry matrix G:

G =

−11

x 1−12

x 1...

...−1M

x 1

, (2)

where 1mx is a line-of-sight unit vector from the UAV position x to the mth satellite position. Only satellites in a direct line-

of-sight are used for constructing the geometry matrix. The DOP matrix H is calculated as the pseudo-inverse of the geometrymatrix:

H = (GTG)−1, (3)

and the DOP-based covariance for GNSS positioning is calculated as:

RG = σG ·Hpos, (4)

where σG is a scale factor with units m2 and Hpos is the position sub-matrix of the DOP matrix. Thus, in order to enclose allpossible GNSS positioning error distributions, we set the covariance of the Gaussian tails of V as ΣV = RG.

B. UAV System DescriptionFor our UAV motion model, we assume a discrete linear model with velocity inputs:

xk = Axk−1 +Buk−1 + wk, (5)

where A is the state transition model, B is the control-input model, u is the velocity input to the system and wk is a zero meanGaussian noise with covariance Qk. We assume the availability of the nominal information for the candidate path as done inprevious works [2]. The nominal information includes a series of way-points xk, nominal inputs uk and the feedback controlgain K. The way-points and the nominal inputs along the candidate path are related as:

xk = Axk−1 +Buk−1. (6)

We assume that the UAV implements a linear state feedback control during path execution. Thus, the total control input duringpath execution is a sum of the nominal input and the feedback input:

uk = uk − K (xk − xk) , (7)

where xk is the state estimated by a filter running on-board the UAV. Probabilistic state estimation filters require normalizedmeasurement error models and cannot simultaneously consider all possible error distributions for GNSS positioning shown inFig. 1(b). Thus, as shown in Fig. 3, we fix a hypothesis for the GNSS positioning error model on the on-board state estimationfilter, and calculate the stochastic reachability of the UAV resulting from the hypothesis.

In this paper we choose a KF as the state estimation filter on-board the UAV, since we assume a linear motion model anduse GNSS positioning as measurements. For the GNSS positioning error model on the KF, we fix a hypothesis of a zero-meanGaussian distribution N (0, RG

k ), where RGk is the DOP-based covariance matrix from equation (3). The prediction step of the

KF is given by:

xk = Axk−1 +Buk−1, (8)Pk = APk−1A

T +Qk, (9)

where Pk is the state estimation covariance matrix. The update step of the KF uses GNSS positioning measurements zk and isgiven by:

zk = Cxk + νk, (10)

Lk = PkCT(CPkC

T +RGk

)−1, (11)

xk = xk + Lk(zk − Cxk), (12)Pk = (I − LkC)P , (13)

where C is the system measurement matrix, Lk is the Kalman gain, and νk is modeled according to our hypothesis as a zeromean Gaussian noise N (0, RG

k ). However, in the presence of GNSS positioning biases, our hypothesis would result in additional

state estimation errors. We consider the effect of these state estimation errors while computing the stochastic reachable sets inthe next section.

III. STOCHASTIC REACHABILITY COMPUTATION

Given the GNSS positioning error model and the UAV system description, we move on to our stochastic reachability analysis.For the computation of our stochastic reachable sets, we require the Minkowski sum and linear transform operations forprobabilistic zonotopes [15]:

V1 ⊕ V2 = (cV1 + cV2 ,gV1 ,gV2 ,ΣV1 + ΣV2)Z ,

T · V =(TcV , TgV , TΣVT

T)Z ,

(14)

where T is the matrix representing a linear transform. Thus, the stochastic reachable sets for the UAV along a candidate pathare computed as follows:

1) We begin by predicting the Kalman gain Lk and the state estimation covariance Pk of the KF on-board the UAV. We useequations (9), (11) and (13) to obtain Lk and Pk.

2) Since the KF uses our hypothesis of νk ∼ N (0, RGk ) for the GNSS positioning error model, we need to consider the

effect of GNSS positioning biases. Thus, in this step we compute the stochastic set of the state estimation errors of theKF on-board the UAV. We begin by defining the state estimation error and by using equations (12) and (5) to expand:

xk = xk − xk = xk + Lk (zk − Cxk)−Axk−1 −Buk − wk. (15)

On replacing xk by equation (8) and rearranging terms, we obtain:

xk = Axk−1 + Lk (zk − C (Axk−1 +Buk))− wk. (16)

On replacing zk by equation (10) and further rearranging of terms, the state estimation error can be written as:

xk = (I − LkC)Axk−1 − (I − LkC)wk + Lkνk. (17)

The stochastic set of the state estimation errors is obtained by considering the stochastic sets of the quantities on the righthand side of the above equation:

Xk = (I − LkC)AXk−1 ⊕ (I − LkC)Wk ⊕ LkVk, (18)

where Xk−1 is the corresponding stochastic set from the previous iteration, Wk is the probabilistic zonotope (0,0, Qk)Zthat encloses the motion model error distribution, and Vk is the probabilistic zonotope from equation (1) that enclosespossible error distributions due to uncertain GNSS positioning biases. It is important to note here that the term LkVk inequation (18) incorporates the effect of uncertain GNSS positioning biases in our stochastic reachability analysis.

3) Finally, we compute the stochastic reachable set for the UAV. We begin with the UAV motion model from equation (5):

xk+1 = Axk +Buk + wk+1. (19)

On replacing the control input from equation (7) and rearranging terms, we obtain:

xk+1 = Axk −BKxk +Buk +BKxk + wk+1. (20)

On replacing xk with xk + xk, and using equation (6), we obtain:

xk+1 = (A−BK)xk −BKxk + xk+1 −Axk +BKxk + wk+1. (21)

On further rearranging, the state at t = k + 1 can be written as:

xk+1 = xk+1 +(A−BK

)(xk − xk)−BKxk + wk+1. (22)

Thus, the stochastic reachable set for the UAV is obtained by again considering the stochastic sets of the quantities onthe right hand side of the above equation:

Xk+1 = xk+1 +(A−BK

)(Xk − xk)⊕BK

(−Xk

)⊕Wk+1. (23)

Thus, given an initial stochastic set X0 for the UAV along with initial estimation errors X0 and P0, the above steps can besolved iteratively to compute the stochastic reachable sets of the UAV along the candidate path.

Fig. 5: (a) Our 3D urban environment setup in the AirSim simulator [20]. (b) Simulated uncertain GNSS positioning biases inthe denser urban regions.

Fig. 6: (a) Candidate path from start to goal state. (b) Predicted state uncertainty along the candidate path using our stochasticreachability analysis. (c) 100 sample trajectories generated along the path. (d) Our computed stochastic reachable sets enclosethe sample trajectories.

IV. SIMULATIONS

In this section, we perform simulations in order to validate our computed stochastic reachable sets. We use the AirSim simulator[20] to setup a 3D urban environment with dimensions of 500m × 500m × 120m. To simulate the satellite positions, we usereal GNSS ephemeris data available online [21]. We simulate the presence of uncertain GNSS positioning biases in the denserurban regions of the environment. Fig. 5 shows our simulation environment and the regions we simulate to have uncertain GNSSpositioning biases. We choose zero-centered biases with a width of 20m in the horizontal directions and 10m in the verticaldirection: cVG = [0, 0, 0]m and wVG = [20, 20, 10]m. The DOP-based covariance matrix scale factor σG can be tuned for filterperformance. We set it as σ = 1m2. For the UAV motion model covariance we set Q = (0.5m2)I .

We introduce a metric for the size of the computed stochastic reachable sets, which are represented by probabilistic zonotopes.We define a 3σ bound size of a probabilistic zonotope V as:

|V| = ‖gV‖Z + 3√trace(ΣV), (24)

where ‖gV‖Z refers to the maximum norm of all points in the probabilistic zonotope width wV represented by the set of vectorsgV . Thus, |V| in equation (24) consists of a measure of the width size of V and a measure of the 3σ bound of the Gaussian tailcovariance of V .

For our simulations, we consider two candidate paths from start to goal: a longer path as shown in Fig. 6(a) and a shorter

Fig. 7: (a) Candidate path from start to goal state. (b) Predicted state uncertainty along the candidate path using our stochasticreachability analysis. The computed stochastic reachable sets intersect with nearby buildings. (c) 100 sample trajectories generatedalong the path, 40 of which result in collisions. Thus, our computed stochastic reachable sets can be used to detect a probabilisticcollision-unsafe path. (d) Our computed stochastic reachable sets enclose the sample trajectories.

path as shown in Fig. 7(a). For the longer path, we begin by computing the stochastic reachable sets for the UAV as shown inFig. 6(b). In the uncertain GNSS positioning bias regions, the stochastic set for the state estimation error (equation (18)) growswhich consequently leads to larger stochastic reachable sets (equation (23)). In order to validate the stochastic reachable sets, wegenerate 100 sample trajectories along the path as shown in Fig. 6(c). Fig. 6(d) compares the distance of the sample trajectoriesfrom the nominal waypoints x of the candidate path, and the size of the computed stochastic reachable sets along the path.The plot indicates that the predicted state uncertainty using our stochastic reachability analysis encloses trajectories along thecandidate path.

For the shorter path, we perform a similar analysis as above. We first compute the stochastic reachable sets for the UAV asshown in Fig. 7(b). Here the stochastic reachable sets intersect with nearby buildings, thus indicating a probabilistic collision-unsafe path. An upper bound for the collision probability can be calculated using the method proposed in [15]. We again generate100 sample trajectories along the path as shown in Fig. 7(c), 40 of which result in collisions. Thus, our computed stochasticreachable sets can be used to detect a probabilistic collision-unsafe path. Similar to Fig. 6(d), Fig. 7(d) again indicates that thepredicted state uncertainty using our stochastic reachability analysis encloses trajectories along the candidate path.

V. CONCLUSION AND FUTURE WORK

We proposed a method to predict the state uncertainty of a UAV using stochastic reachability analysis. We incorporateduncertain biases in GNSS positioning by enclosing possible error distributions using a probabilistic zonotope. We formulatedour problem for a linear motion model of the UAV and chose a KF on-board along with a linear state feedback control. Forthe GNSS positioning error model on the KF, we fixed a hypothesis of a zero-mean Gaussian distribution with a DOP-basedcovariance matrix. Next, we computed a stochastic set of state estimation errors for the KF and used it to compute the stochasticreachable sets along the candidate path. Finally, we performed simulations and validated our stochastic reachability analysis inthe presence of uncertain GNSS positioning biases. We showed that the computed stochastic reachable sets were able to enclosetrajectories generated along the path. We also observed that our stochastic reachable sets can be used to detect probabilisticcollision-unsafe paths.

For future work we plan to analyze using different hypotheses for the GNSS positioning error model on the KF, such as anover-bounding Gaussian which is commonly used for GNSS integrity monitoring purposes [22]. We also plan to explore thecomputational load of our method and the possibility of integrating our method in a path planning framework.

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