Efficient Representation in 3D Environment Modeling for Planetary ...

Efficient Representation in 3D Environment Modeling for Planetary Robotic Exploration Narunas Vaskevicius,

Andreas Birk, S¨oren Schwertfeger

Kaustubh Pathak,

Jacobs University, Campus Ring 1, 28759 Bremen, Germany http://robotics.jacobs-university.de, {n.vaskevicius, a.birk}@jacobs-university.de

Abstract Good situation awareness is an absolute must when operating mobile robots for planetary exploration. 3D sensing and modeling data gathered by the robot are hence crucial for the operator. But standard methods based on stereo vision have their limitations, especially in scenarios where there is no or only very limited visibility, e.g., due to extreme light conditions. 3D Laser Range Finders (3D-LRF) provide an interesting alternative, especially as they can provide very accurate, high resolution data at very high sampling rates. But the more 3D range data is acquired, the harder it becomes to transmit the data to the operator station. Here, a fast and robust method to fit planar surface patches into the data is presented. The usefulness of the approach is demonstrated in two different sets of experiments. The first set is based on data from our participation at the ESA Lunar Robotics Challenge 2008. The second one is based on data from a Velodyne 3D-LRF in a high fidelity simulation with ground truth data from Mars.

keywords: space robotics, planetary exploration, 3D mapping, surface representation, plane fitting published in: Advanced Robotics, Vol. 24, Iss. 8-9, Brill, 2010

1

Introduction

The Mars Exploration Rover (MER) missions [1][2][3][4] are an impressive demonstration of the possibilities of using mobile robots for planetary exploration. Though intelligent autonomous functionalities onboard of the robots play an important role, especially for locomotion control [5][6][7][8][9], there is still a significant amount of human control in form of advanced teleoperation needed for operating the systems [10]. In doing so, it is of high importance that the operators have an excellent situation awareness through 3D representations of the environment, typically based on stereo vision [11] [12] [13] [14] [15] or other vision based techniques [16][17], possibly in combination with structured laser light [18]. Stereo vision has its limitations for planetary exploration, especially in planetary settings with few features [19]. Furthermore, vision techniques are in general challenged by darkness or other extreme lighting conditions. There is hence an interest in exploring in addition alternative technologies for robotic planetary exploration like 3D Laser Range Finders (3D-LRF) [20], which as further benefits provide a large field of view and quite precise, high resolution data. But what in principle is a significant advantage, namely the provision of large amounts of high resolution 3D data, is also a drawback when it comes to the processing of the data, respectively the transmission of the data to the operator’s station on earth. Here, a way out of this dilemma is presented. Based on previous own work [21], it is shown how large planar surface patches can be fitted in a fast and robust manner into 3D range point clouds, hence significantly reducing the amount of data. In doing so, potential noise in the sensor is compensated [22] without the need for advanced sensor models of the laser range finder [23][24].

1

There is a tremendous amount of different map representations including for example occupancy grids [25], correspondence graphs [26], elevation maps [12][27] or behavior based representations [28]. In this article, we focus on full 3D representation, for which point clouds are the predominant form of representation [29, 30, 31, 32, 33, 34] for two main reasons. First, points clouds, i.e., sets of coordinates of surface points, are the prototypical raw data delivered by 3D range sensors. Second, point clouds are the main basis for state of the art 3D perception and mapping algorithms, which - as also pointed out in [35] - are usually based on the iterative closest point (ICP) algorithm [36] operating on point clouds.

Figure 1: The two Jacobs robots during the ESA Lunar Robotics Challenge 2008. On the left, the relay robot with a 3D Laser Range Finder to generate models of the environment. In the middle, the probe robot to take soil samples in the crater. On the right, the robots heading for the crater rim, during the actual ESA Lunar Robotics Challenge taking place in the night with simulated light conditions as found in the polar regions of the moon.

Figure 2: The setup of the ESA Lunar Robotics Challenges features a typical scenario where 3D sensing and modeling is highly desirable but vision based approaches are unfeasible. The crater that is to be explored is in complete darkness and RF-based communication from the lander is not possible. A communication relay robot can hence serve in addition model the environment to ease the operation of the robot exploring the crater. Our approach is validated in two different planetary exploration experiments. The first set of experiments is based on data collected at the Lunar Robotics Challenge (LRC) [37] of the European Space Agency (ESA) in October 2008 in the volcanic landscape of the Teide National Park on the island of Tenerife. Two robots of Jacobs university (figure 1) participated in this field test based on a scenario where vision based approaches to environment modeling are infeasible due to the light conditions (figure 2

1, on the right). The motivation for the ESA LRC is scientific interest to search for the presence of water on moon, especially at the bottom of craters in the polar regions. For this purpose, a probe robot has to get from a landing site to the crater, to descend into it, to take probes of interesting soil spots at the bottom, to climb out of the crater, and to return back to the lander where the soil probe has to be delivered for an automated analysis. This leads to several particular challenges (figure 2). First, the crater is steep and loose soil makes locomotion challenging. Second, there is no RF-communication from the lander - that provides the communication link to the operator station on earth - into the crater, hence direct teleoperation is not possible. Third, there is very bright, horizontal illumination by sunlight on the top and the rim of the crater and absolute darkness inside of it, making teleoperation via video-streams extremely hard and vision based environment modeling impossible. The Jacobs team hence used a second robot in addition to a probe robot. This second robot serves as communication relay between the lander and the probe robot and furthermore as 3D environment modeler. For this purpose, the robot is equipped with an actuated laser range finder (LRF). The sensor is based on a commercial 2D LRF of the type SICK S300 combined with a simple servo for a pitching motion for the additional degree of freedom. The SICK S300 has a horizontal field of view of 270o of 541 beams. The servo allows a maximum motion from −90o to +90o at a maximum spacing of 0.5o . This gives a 3D point-cloud of a total size of 541 × 361 = 195, 301 per sample in the highest resolution. The maximum range of the sensor is about 20 meters. The time to take one full scan in maximum resolution is about Tscan ≈ 32 seconds. This scan time is the main bottleneck for the environment modeling. But as shown in the second experiment, there are - more costly - off the shelf 3D-LRF alternatives that provide even higher resolution 3D data at significantly faster rates.

Figure 3: Planetary exploration in the Unified System for Automation and Robotics Simulator - USARsim. The simulator features artificially generated planetary surface types as well as the option to important in ground truth models from space missions, which is used for the experiments here. In a second set of experiments, the Unified System for Automation and Robot Simulation (USARSim) [38] is used where a high fidelity simulation of a high end 3D LRF is available for testing purposes. Furthermore, we have imported ground truth data from Mars to model the experimentation environment (figure 3). USARsim is a high fidelity simulator built on top of the Unreal Tournament [39] game engine. Its feature include a commercial physics engine (Karma [40]) and a real-time, three-dimensional visualisation engine. It is important that the components and models in USARsim have been tested for their physical fidelity [41, 42, 43, 44]. USARsim has already proven to be useful for the investigation of planetary exploration scenarios, including the usage of models based on ground truth data from space missions. One example is the terrain of the Eagle crater on Mars. The underlying environment model is based on ground truth data from the Mars Exploration Rover (MER) mission data archives [45]. The according models were used in USARsim for example to investigate terrain classification and autonomous robot behaviors for planetary exploration [46][47]. In this article, we are interested in high resolution 3D range data that can ideally be acquired as fast as possible. Hence the simulation of a Velodyne HDL-64E is used in the second set of experiments. This sensor is particularly interesting as it can be considered to be the top of the state of the art of 3D sensing as for example shown by its usage by different teams in the DARPA Grand Challenge [48, 49]. The Velodyne HDL-64E has 64 lasers, 360◦ horizontal field of view (0.09◦ angular resolution), 26.8◦ vertical field of view (from ±2◦ up to −24.8◦ down with 64 equally spaced angular subdivisions) and 3

produces more than 1.3M points per second with maximum range of 50 − 120 meters depending on the reflectivity of the surface [50]. The rest of the article is organized as follows: in sec. 2, it is shown how approximate estimates of plane parameters and their uncertainty can be calculated from 3D measurements with the radial Gaussian noise. In sec. 3 a real-time algorithm for a range image segmentation is presented, which is followed by a discussion in sec. 4 about memory efficient representation of the extracted planar surfaces. The experimental results are provided in sec. 5 where we mainly concentrate on runtimes, compression rates and visual quality of created 3D models. Finally, conclusions are discussed in sec. 6.

2

Estimation of Plane Parameters and their Uncertainty

A general idea of an analytical solution to the Approximate Least Squares Problem (ALSP) formulated for the optimal plane fitting under the assumption of the radial Gaussian noise is presented here. It is based on previous own work [22], [21]. We concentrate in this article on the efficient representation of single 3D scans by our plane fitting technique and its application in planetary exploration scenarios. Please note that this plane based representation has been shown to be a suited basis for 6 Degree of Freedom registration of scans [51], which is suited for full 3D Simultaneous Localization and Mapping (SLAM) in unstructured domains as already shown in the case of disaster scenarios [52]. ˆ · r = d, where n ˆ is the plane’s unit normal and d the distance to The equation of a plane is n ˆ j , j = 1 . . . N , where, m ˆ j are the the origin. Assume that the sensor returned a point-cloud rj = ρj m measurement directions for the sensor, usually accurately known, and ρ are the respective ranges which j
ˆ · m) ˆ , where, ρ¯j = nˆ ·dm are noisy. We make an assumption that ρj ∼ N ρ¯j , σ 2 (¯ ρj , n ˆ j is the true range of j -th measurement. Therefore the likelihood of plane-parameters (ˆ n, d) given range sample ρi along ˆ j is measurement direction m   ˆ j )) 1 1 (ρi − d/(ˆ n·m ˆ j) = √ p(ρi |ˆ n, d, m exp − (1) ˆ ·m ˆ j) 2 σ 2 (¯ ρj , n ˆ ·m ˆ j) 2πσ(¯ ρj , n For commonly available 3D sensors like the Swiss-Ranger and Laser-Range-Finder (LRF) the standard deviation is modeled as ([53], [54], [23]) σ ˆ (¯ ρj ) , ˆ j| |ˆ n·m

ˆ ·m ˆ j) = σ(¯ ρj , n

σ ˆ (¯ ρj ) , κ¯ ρ2j

(2)

ˆ is the local normal to the surface the point rj lies on. The coefficient κ > 0 can be estimated where n by doing initial calibration experiments with the sensor. From equations (1) and (2) we get the following log-likelihood function L=K−

N X

N

log

j=1

ˆ · m) ˆ ˆ j )ρj − d]2 1 X [(ˆ n·m σ ˆ (¯ ρj , n − ˆ j| |ˆ n·m 2 j=1 σ ˆ 2 (¯ ρj )

(3)

ˆ , d. This can not be handled analytically, especially as σ is a function of L has to be maximized w.r.t. n d , therefore we make the assumption σ ˆ (ρ¯j ) ≈ σ ˆ (ρj ). We define σ ˆj , σ ˆ (ρj ) and note that σ ˆj is now ˆ ·m ˆj n ˆ and d. Using Eq. (3) and ignoring constant terms together no longer a function of ρ¯j and hence of n with sum of logarithms we get ALSP formulation N

max LALSP = − ˆ ,d n

2

n · rj − d) 1 X (ˆ 2 j=1 σ ˆj2

(4)

Now we have a constrained optimization problem which can be solved using Lagrange multipliers. Defining Lagrangian N

L1

= −

n · rj − d)2 1 X (ˆ ˆ − 1) + λ(ˆ nT n 2 j=1 σj2 4

(5)

and maximizing it gives us the following estimates PN ∗

∗T

d

ˆ = n

λ∗

=

1 j=1 σj2 rj

!

PN

1 j=1 σj2

ˆ ∗T rG , =n

(6)

N

ˆ ∗T (rj − rG )(ˆ 1Xn n∗T rj ) 2 j=1 σj2

(7)

ˆ ∗ is the eigenvector corresponding to the smallest eigenvalue of By substituting d∗ in (4) we get that n PN the positive semi-definite weighted scatter matrix M = j=1 σ12 (rj − rG )(rj − rG )T . j Now the Hessian of log-likelihood function at d∗ , n∗ and λ∗ can be evaluated " #   ∂2L ∂2L Hnn Hnd ˆ2 ˆ ∂n ∂d∂ n H= = (8) 2 2 T Hnd Hdd ( ∂ L )T ∂ L2 ˆ ∂d∂ n

∂d

ˆ ∗ , d∗ , λ∗ ≡ H ∗ H evaluated at n ∗ Hnn

= −

N X rj rT j j=1

∗ Hnd

=

σj2

"N # Xn ˆ ∗T (rj − rG )(ˆ n∗T rj ) + I3 σj2 j=1

(9)

N N X X rj 1 ∗ , H = − dd 2 2 σ σ j=1 j j=1 j

Then the covariance matrix is [55] C(ˆ n∗ , d∗ ) = −(H ∗ )+

(10)

H ∗ has a zero eigenvalue in the direction of (ˆ n∗ , d∗ )T therefore Moore-Penrose generalized inverse has to be used. This property of the Hessian is discussed in [22], [56].

3

Range Image Segmentation

In this section, the principle of our algorithm for identifying regions of points that lie on one plane is presented. This extends the work presented in [57] by modifying the algorithm by reformulating the underlying mathematics to an incremental version, which allows a highly efficient implementation. The algorithm proceeds as follows. We select a point r1 and its nearest neighbor r2 from point cloud data P C. For selection we can use random sampling or most planar place, where the flatness at the point is evaluated by the mean square error of the local optimal plane. This is our initial set of points - region Π (Algorithm 1, Line 4 - 5). Then we try to extend this region by considering points in increasing distance from set Π. Now suppose point r0 is such that the distance between it and the region is less than the distance δ. Then if the mean square error (MSE) to the optimal plane Ω of the region Π ∪ r0 is less than  and if the distance between the new point and the optimal plane Ω is less than γ, then r0 is added to the current region Π. We grow this region until no points can be added (Algorithm 1, Line 6 - 11). Afterwards if the region size is more than θ we add it to the set of regions R, else we treat those points as unexplained and add them to the set R0 (Algorithm 1, Line 12 - 16). This is repeated until each point from P C is either in R or in R0 . The approach for regions identification can deal with local noise and therefore it is suitable for the data produced by fast but error-prone 3D range sensors. Actually, the complexity of the model and sensitivity to noise can be controlled by using aforementioned parameters (δ, , γ, θ). The meaning and impact of parameters is quite intuitive, however selecting the right parameters can play a crucial role in the production of a good model. Fortunately, a single calibration step is sufficient, i.e., the parameters can be tuned for one sensor on a single point cloud and then they can be used from then on without adaptation. 5

The high level formulation of Algorithm 1 is almost the same as in [57]. However, a naive implementation of it would lead to a high computational complexity and unreasonable work time even for small point clouds of a size around 104 . Here, an incremental approach is presented, which is discussed later in the implementation section 3.1. From the theory in section 2 we know that calculating the optimal fit for a set of 3D points rj = ˆ ∗ corresponding to the smallest eigenvalue of the (xj , yj , zj )T , j = 1 . . . k requires finding eigenvector n matrix k X 1 T (11) Mk = 2 (rj − rG )(rj − rG ) σ j j=1 where rG is the weighted gravity center of the data Eq. (6). This operation is crucial as it has to be performed whenever a new point is investigated, therefore it should be efficient. The numerically optimized methods of the GNU Scientific Library (GSL) are used for this purpose. Algorithm 1 Range Image Segmentation 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17:

R←∅ R0 ← ∅ while ( P C \ (R ∪ R0 ) 6= ∅ ) do select points r1 and r2 in P C \ (R ∪ R0 ) Π ← {r1 , r2 } while ( new point can be found ) do select nearest neighbor r0 with distance d(Π, r0 ) < δ if (MSE(Π ∪ {r0 }) <  && d(Ω, r0 ) < γ ) then Π ← Π ∪ r0 end if end while if ( size(Π) > θ ) then R←R∪Π else R0 ← R0 ∪ Π end if end while

3.1

Efficient Implementation

As mentioned before, it would be very time consuming to use a naive implementation for region growing Algorithm 1. Here, we introduce a minimization of the computational complexity for this. 3.1.1

Incremental optimal plane and its error update

Suppose the matrix Mk would be calculated from the start every time a new point is added to the region. This would mean that one would need to traverse every point in the current region leading to a huge overhead. Here a way to an incremental update of the matrix Mk and its mean square error is presented, which takes previous calculations into account. Lets define three variables which will describe a state of k points rj Pk • The sum of all weights - wk = j σ1j • The weighted sum of all points - sk =

rj j σj2

Pk

• The weighted sum of all product matrices - Pk =

Pk j

rj rT j σj2

We can express the matrix Mk and mean square error M SEk using these state variables in the following

6

way Mk

=

=

k k k k k X X X X X rj rT rT 1 rj T 1 j j T T (r − r )(r − r ) = − r − ( )r + r r G j G G G G 2 j 2 2 2 G 2 σ σ σ σ σ j j=1 j j=1 j=1 j j=1 j j=1 j k X rj rT j j=1

M SEk

σj2

− rG

k X rT j

σ2 j=1 j

= Pk −

sk sTk , wk

(12)

=

  k k T X r r 1 2d∗ ∗ 1 X 1 ∗T j j n  ˆ ∗k − k n ˆ ∗T ˆ k · sk + d∗2 (ˆ nk · rj − d∗k )2 = n k k 2 2 wk j=1 σj wk σ w k j j=1

=

1 ∗T 2d∗ ∗ ˆ k Pk n ˆ ∗k − k n ˆ · sk + d∗2 n k . wk wk k

(13)

Now suppose we want to add a new point rk+1 . We can easily update state variables wk+1 = wk +

1 2 σk+1

, sk+1 = sk +

rk+1 rT rk+1 k+1 , P = P + k+1 k 2 2 σk+1 σk+1

(14)

From equations (12), (13) and (14) we get Mk+1 and M SEk+1 . In other words we have updated Mk+1 and M SEk+1 indirectly by updating variables sk , wk and Pk . Please note that for M SEk+1 new normal ˆ ∗k+1 and d∗k+1 has to be calculated based on Mk+1 matrix. vector n Note that Hessian matrix Eq. (9) can be efficiently evaluated during the optimal fit calculation in the following way. Suppose we have a set of k points {rj } with the state {Pk , sk , wk } and the estimate
T ˆ ∗T , d∗ , where n ˆ ∗ is the eigenvector corresponding to the smallest eigenvalue λ of matrix Mk Eq. n (11), then it can be shown that  P Pk rj  rj rT k  ∗  j ∗ + λI − 2 3 j=1 σj2 j=1 σj Hnn Hnd  H∗ = = T ∗T ∗ P P r k k Hnd Hdd j 1 − j=1 σj2

 =

−Pk + λI3 sT k

sk −wk

j=1 σj2

 (15)

As we always keep track of the state {Pk , sk , wk } and the eigenvalue λ is calculated during the optimal fitting, we need only a few extra operations to find the Hessian matrix, which can be used to obtain the covariance matrix Eq. (10). 3.1.2

Finding nearest neighbors

The important part of the region growing algorithm is the nearest neighbor selection for a current region (Algorithm 1, line 7), which is performed every time the inner loop starts. For that either a priority or a FIFO (First In, First Out) queue can be employed. In the former case a priority queue Q with the minimum distance on the top should be used. Whenever a new point is added to the region, its k unvisited nearest neighbors nbt , t = 1 . . . k (from its Moore neighborhood in the range image) are investigated and if the distance d(r, nbt ) < δ then nbt is added to the priority queue. This allows to extract the nearest neighbor for the region just by calling Q.TOP(). As the distances in the priority queue are bounded by the parameters δ and γ they are in a relatively small range. With the assumption that variation of the distances in this range is insignificant we can justify the usage of a FIFO queue. We observed that priority queue is more advantageous when random selection of a region seed is used. In case where initial point is selected based on the flatness criteria1 a simple FIFO queue showed better performance. The latter combination was used in experiments of this paper. 1 The local flatness of a point is evaluated by fitting a plane in a small window around the point and calculating the mean square error of the optimal fit. This is done in a preprocessing step.

7

3.1.3

Algorithm complexity

All optimizations described in the previous sections lead to a really fast algorithm. Suppose we have a point cloud data of size n. All operations inside the loops are performed in constant time except the nearest neighbor search for the current region, which has logarithmic complexity when the priority queue is used. Note that in case of a FIFO queue the approximate nearest neighbor is obtained in constant time. Now assume that there are a few points lying in the intersection of the planes, which means that most of the points will be visited only once. Then the complexity of the algorithm can be considered to be O(n · log(n)) in case of priority queue and O(n) in case of FIFO queue. The memory complexity is clearly linear - O(n).

4

Finding Polygonal Representations of Planar Surfaces

This is the part where most of the compression of the original point cloud can be made. Once the range image is segmented the points are projected to their optimal planes, thus reducing representation dimension to 2, i.e., each segment can be represented by parameters (ˆ n∗ , d∗ )T , H ∗ 2 of the optimal plane and a set of 2D points defined in the planes’ frame. Furthermore, we note that the most important part per segment is the boundary of its 2D set of points. As the neighborhood information of points is sustained from the pixel space of the range image to the Cartesian space the boundary of the segment is the same in both spaces. Finding border pixels of a segment in a range image is straightforward. One just need to check Moore neighborhood of range 1 for each pixel in the segment. If all neighbors of the pixel are from the same segment then it is an inner pixel otherwise it is a boundary pixel. Taking corresponding Cartesian coordinates of border pixels gives us the outline of the planar patch. The inner points are redundant and can be removed. Note that the outline usually consists of several components - one outer boundary and several boundaries of inner holes. The boundary representation already reduces the initial size quite significantly, however it is usually very dense and can be approximated, thus allowing to increase memory efficiency even more. One option is to use a Minimum Description Length (MDL) encoding of the chain code, which is computationally expensive. A simple, but sufficient alternative is a convex hull algorithm in form of Graham’s scan, which achieves very high compression rates in a short runtime of O(n · log(n)). Convex hull has certain limitations such as holes closing and loosing concave features, which sometimes ends in very rough approximations. Therefore one should always consider which detail level is needed before applying it. Usually in datasets with large openings (which is true in our case), the large scale features are more interesting, therefore convex hull is a suitable candidate.

5

Experiments

The experiments section consists of three parts. The fidelity of models created by our algorithm in nonplanar environments is investigated in the first part in section 5.1. The performance of the presented approach is then evaluated in section 5.2 based on data from the two typical planetary exploration scenarios introduced in section 1. As discussed before in the introduction, there is a wide range of different representations for mapping data. We concentrate in section 5.2 on point clouds as comparison basis for our approach as this is the predominant representation for 3D perception and mapping. In addition, a discussion and experimental comparison to octrees and elevation maps is presented in section 5.3.

5.1

An Analysis of the Approximation Error

Natural landscapes typically feature curved surface instead of planes. The polygonal representation hence always leads to a certain loss of information as it is an approximation of the terrain. This can be seen as a structural error, i.e., an error due to the false representation of the true structure. As shown 2 Actually,

it is enough to store only Hessian H ∗ , as (ˆ n∗ , d∗ )T is in its nullspace

8

in the following experiments, this approximation error can be controlled by the choice of the parameters in the plane fitting process. Concretely, the fidelity of our planar approach is evaluated in the following framework. Three quadratic surfaces with different curvature complexities are sampled by a simulated ALRF (figure 4). No noise is added to have only approximation errors in the planar models. The plane extraction is run with different sets of parameters thus allowing to investigate their impact on the model precision.

(a) Surface with low curvature

(b) Surface with medium curvature

(c) Surface with high curvature

Figure 4: The quadric surfaces used for structural error evaluation. The ground truth surfaces are shown in yellow, the sampled points in blue and the sensor location in red. The surfaces used as ground truth curved terrain (figure 4) are hyperbolic paraboloids. Their canonical form is described by the following implicit equation y2 x2 − 2 −z =0 2 a b

(16)

where (a, b) are shape controlling parameters. We used the values (10, 10), (5, 10), (5, 5) respectively for the surfaces plotted in figures 4(a), 4(b) and 4(c). The three different surfaces represent three amounts of curvature in the environment, which covers 40 m by 40 m. All surfaces are moved by 3 meters down the z-axis. Three different sets of parameters are used for the plane extraction. Concretely, we investigate the  and γ thresholds from the algorithm 1. The exact values are given in the table 1. , mm2 γ, mm

p1 300 250

p2 100 75

p3 50 30

Table 1: The different parameters for region growing used in the fidelity experiments. The qualitative results are shown in figure 5. The models consist of planar segments and the points from the raw data, which could not be approximated by region growing. The boundaries of the polygons are not simplified to preserve more details. An example with the results of convex hull polygonalization is given in figure 6. The quantitative results are shown in table 2. First, the size of each point cloud, i.e., the number of points in it, is shown for each of the three curved surfaces that represent ground truth. Second, the number of planes generated by our approach is given for the different parameter sets and surface types. As one can expect, both the surface types as well as the parameter settings influence the number of planes generated to approximate the curved surfaces. The planes never perfectly cover all points from the input point clouds. These remaining points are denoted as “unassigned”. The unassigned points are visualized in figures 5 and 6 as little black dots. They occusionally occur at the boundaries of planes and mainly at regions on the far end of the laser scanners’ range where the density of samples is not high enough anymore for a reasonable approximation. They can be included in the model, but in practice they tend to be in far away regions, which are not sufficiently sampled by the ALRF anyway.

9

(a) p1 parameters, low curvature

(b) p1 parameters, medium curvature.

(c) p1 parameters, high curvature

(d) p2 parameters, low curvature

(e) p2 parameters, medium curvature

(f) p2 parameters, high curvature

(g) p3 parameters, low curvature

(h) p3 parameters, medium curvature

(i) p3 parameters, high curvature

Figure 5: Planar models created by region growing with the different parameters. Different planar segments have different colors.

(a) Surface with low curvature

(b) Surface with medium curvature

(c) Surface with high curvature

Figure 6: Models created using convex hull polygonalization with p2 parameters

10

7000

4000

6000

3500 3000 occurances

occurances

5000 4000 3000 2000

2500 2000 1500 1000

1000 0 −100

500 −50

0 error, mm

50

0 −100

100

−50

0 error, mm

50

100

(a) The example of the typical error histogram with its (b) The example of the typical error histogram with its scaled Gaussian approximation for low curvature surface scaled Gaussian approximation for high curvature surface and p1 parameters and p3 parameters 0.08 0.07

0.07 p1

σ = 5.5917 mm

0.06

p3

σ = 9.4127 mm

0.03

0.03 σ = 17.0328 mm

0.02

σ = 17.0029 mm

0.01

0.01 0 −100

p3

0.04

σ = 9.2538 mm

0.02

p2

0.05

0.05 0.04

p1

σ = 5.9669 mm

0.06

p2

−50

0 error, mm

50

0 −100

100

(c) surface with low curvature

−50

0 error, mm

50

100

(d) surface with medium curvature

0.07 0.06

p1

σ = 6.2545 mm

p2 p3

0.05 0.04 σ = 9.4013 mm

0.03 0.02

σ = 16.7478 mm

0.01 0 −100

−50

0 error, mm

50

100

(e) surface with high curvature

Figure 7: The distributions of the signed approximation error for different region growing parameters and different types of curved surfaces.

11

parameter set p1

p2

p3

surface curvature low medium high low medium high low medium high

point cloud size 37661 41141 28707 37661 41141 28707 37661 41141 28707

number of planes 95 137 140 174 248 234 287 375 298

unassigned points number percent 560 1.5 566 1.4 812 2.8 878 2.3 1066 2.6 1371 4.8 1572 4.2 1577 3.8 2150 7.5

Table 2: The statistics of the plane fitting per ground truth point cloud and parameter set.

Surface with low curvature Surface with medium curvature Surface with high curvature

p1 13.3216 13.5333 13.2706

p2 7.2835 7.5116 7.4665

p3 4.3120 4.6779 4.9383

Table 3: The average absolute errors of the plane approximations in millimeters. In the following, a quantitative evaluation of the approximation error is conducted. Concretely, the fidelity of the models in form of their numerical deviation from the ground truth curved surfaces is investigated. Consider a point pi in one of the point clouds, i.e., it lies on one of the curved surfaces as the point clouds represent ground truth due to the absence of sensor noise. Suppose pi ∈ Rj where Rj is one of the segments returned by the plane extraction with its optimal plane (ˆ nj , dj ). Then a signed distance - from here forth denoted as error - between pi and its optimal plane is ˆ j · pi − dj ei = n

(17)

This error captures the amount of approximation of the curved surface by the plane in this point. Note that the absolute error |ei | is the Euclidean distance between a point pi and a plane (ˆ nj , dj ). The average absolute error per point cloud is given in table 3 and the histograms/distributions of the signed error are shown in figure 7. First of all, the important observation can be made that the approximation error is mostly dependent on the thresholds in plane extraction, but not on the surface complexity. This is also reflected in table 2. The more curved the environment, the more planes are automatically generated to represent it. Therefore the model precision can be controlled using the parameters of the segmentation algorithm. It can in addition be observed that the approximation is very precise. Each of the curved ground truth terrains spans 40 meters by 40 meters. But the average structural error is in the order of millimeters. Even the highest amount of average approximation error of about 14 mm with parameter setting p1 is so small that it roughly corresponds to the noise level found in the ALRF used on the robot. This means that with these parameter settings, the effects of the approximation are roughly comparable to the noise introduced by the sensor itself.

5.2

Planetary Exploration Scenarios

The first set of 3D measurements were made in the volcanic landscape of the Teide National Park on Tenerife where the ESA Lunar Robotics Challenge (LRC) took place. For the LRC, one of the Jacobs robots was equipped with an actuated Laser Range Finder (LRF). As described in section 1, this is a rather low end solution for acquiring 3D LRF data. The second dataset is from a virtual environment, where the Eagle crater on Mars is modeled in USARsim based on ground truth data from the Mars Exploration Rover (MER) mission data archives. There, a high end 3D LRF sensor in form of a Velodyne HDL-64E is used. Examples of the point clouds used in the experiments are shown in figures 8 and 9.

12

Figure 8: Several views of a 3D-LRF point cloud from the ESA Lunar Robotics Challenge.

Figure 9: Three point clouds from the Velodyne 3D-LRF on Mars. On the left, two views of a scan taken on relative flat terrain in the center of the crater. In the middle, two views of a scan taken near the rim with more complex terrain. On the right, two views of scanned terrain in a mountainous part.

13

Figure 10: Results of the plane fitting for the point cloud shown in figure 8. On top, two views of a representation where the boundaries of the plane patches are computed using the convex hull; on the bottom, the boundaries are computed by MDL polygonalization.

Figure 11: Results of the plane patch fitting for the three point clouds from the Velodyne 3D-LRF on Mars shown in figure 9. Again, two views are shown for each case. Despite the compression by three orders of magnitude, the general overview is still very useful; easy as well as hard to negotiable areas can be well recognized.

14

Examples of the results of the plane fitting and polygonalization are shown in figures 10 and 11. Multiple views of the 3D models are shown in these two figures; nevertheless, a full 3D visualization like for example the one provided in the operator GUI of the Jacobs robots - provides a much better overview. For the convenience of the reader, several movies with 3D visualizations of the raw point clouds as well as of the results of the plane fitting can be downloaded from • http://robotics.jacobs-university.de/datasets/ESA-LRC-08 and • http://robotics.jacobs-university.de/datasets/USARsim-Velodyne-Mars. Tables 5.2 and 5.2 show the compression rates and run-times. All experiments were carried out on a 64-bit mobile platform with Intel Core 2 Duo T7100 (1.8GHz) CPU. As can be seen, the plane fitting with convex hull polygonalization leads to compressions up to a factor 25. As discussed before, the polygonal representation does not cover all points, some remain unassigned. We consider them to be not necessary for a good environment representation but one may be interested in keeping them in the model. Hence, all values are also given for the case where unassigned points are included. Then, the compression is still up to about a factor 15. The computations are very fast, the total runtimes are in the order of a few hundred milli-seconds; the approach is hence well suited for online computations on the robot. Polygonal Representation Boundary Convex Hull

point cloud size (KB) 326 326

size (KB) 58 21

incl. unassig. compression (#PC/#PMincl ) 5.62 15.18

percent unassigned (%) 2.54 2.54

Size, (KB) 52 13

excl. unassig. compression (#PC/#PMexcl ) 6.30 24.92

Table 4: Average performance statistics on the dataset from the ESA LRC (11 scans, ∼ 4 · 104 points per scan). The values are given for point clouds (PC) as well as polygonal maps including (PMincl ), respectively excluding unassigned points (PMexcl ).

Polygonal Representation Boundary Convex Hull

Runtimes (sec) Segmentation Polygonalization 0.66 0.05 0.66 0.14

Total 0.71 0.80

Table 5: Average runtimes on the dataset from the ESA LRC (11 scans, ∼ 4 · 104 points per scan)

An important aspect is that this compression is achieved with a high level of fidelity in the representation of the surface. Figure 12 shows the distribution of the “error”, i.e., the deviation of the points in the input point cloud from the fitted planes. Please note that this “error” captures both the effects of the real structural error, i.e., the deviation from the - unknown - ground truth, plus the effects of sensor noise. The real structural error is hence smaller. It can be seen that the representation has a very high fidelity. Concretely, the average absolute “error” is 19.3 millimeter, which is not far from the actual noise level of the ALRF. Tables 5.2 and 5.2 show the compression rates and the run-times for the Velodyne Mars data. Again, the plane fitting with convex hull polygonalization leads on average to high compression rates. The computations take - due to the much higher resolution and range of the sensor - longer, namely in the order of several seconds. Nevertheless, it is suited for online computations as the robot only needs to occasionally take a scan for getting an environment overview. For the virtual Velodyne data it is interesting to have a more detailed look at the individual scans as they differ in the complexity of the terrain where they are taken. As can be seen in table 5.2, the compression rates range from a minimum of 29.01 for the scan near the rim of the crater to a maximum of 452.6 for the scan in the center of the crater in the case when unassigned points are excluded. The 15

12000

occurances

10000

σ = 27.7005 mm

8000 6000 4000 2000 0 −300

−200

−100 0 error, mm

100

200

Figure 12: Distribution of the approximation “error” for the real point cloud shown in figure 8. The absolute value of the “error’ is on average 19.3 millimeter, which is not far from the noise level of the sensor.

point cloud size (KB) Fig.9 left Fig.9 middle Fig.9 right average

4711 3806 4726 4311

Fig.9 left Fig.9 middle Fig.9 right average

4711 3806 4726 4311

incl. unassig. percent size compression unassigned (KB) (#PC/#PMincl ) (%) Polygonal Representation: Boundary 533 8.83 2.51 818 4.65 0.78 806 5.87 3.03 702 6.14 2.11 Polygonal Representation: Convex Hull 129 36.61 2.51 161 23.65 0.78 211 22.43 3.03 163 26.47 2.11

Size, (KB)

excl. unassig. compression (#PC/#PMincl )

415 788 663 607

11.35 4.83 7.13 7.10

10 131 68 68

452.60 29.01 69.82 63.28

Table 6: Performance statistics on the virtual Mars dataset(3 scans, ∼ 4·105 points per scan). The values are given for point clouds (PC) as well as polygonal maps including (PMincl ), respectively excluding unassigned points (PMexcl ).

Polygonal Representation Boundary Convex Hull

Runtimes (sec) Segmentation Polygonalization 4.98 1.15 4.98 1.19

Total 6.13 6.17

Table 7: Average runtime statistics on the virtual Mars dataset (3 scans, ∼ 4 · 105 points per scan)

16

same effect can also be observed when unassigned points are included though in a smaller extent. This again shows that the segmentation of the planar patches works as expected and that it adapts to the complexity of the terrain. 4

4

x 10

12 σ = 21.9 mm

5

2

4 3 2 1

x 10

σ = 26.6948 mm

1.8

σ = 26.3634 mm

10

occurances

occurances

5

x 10

1.6

1.4

8

occurances

6

6 4

1.2

1

0.8

0.6

0.4

2

0.2

0 −100

−50

0

50 error, mm

100

150

(a) top PC

200

0 −300

−200

−100

0 error, mm

100

200

0 −400

300

−300

−200

−100

0

100

200

300

400

500

600

error, mm

(b) middle PC

(c) bottom PC

Figure 13: The “error” distributions for the virtual point clouds (PC) shown in figure 9 An other very important point is that the representation have a high fidelity. Figure 13 shows the “error” distributions for the plane approximations of the three different point clouds. As discussed before, this “error” captures both the effects of the real structural error plus the effects of sensor noise, i.e., the real structural error is even smaller. The average of the absolute values for the three virtual Mars scans are all in the order of 2 cm, namely 21.9 mm in the crater center (top), 19.6 mm at the crater side (middle), and 18.5 mm in the mountains (bottom). Again, this is within the noise level of the sensor, which is included in the simulation. Polygonal Representation

point cloud (KB)

Boundary Convex Hull

290 290

Boundary Convex Hull

3342 3342

incl. unass. compression (KB) (#PC’/#PM’incl ) ESA LRC data 55 5.31 19 15.35 virtual Mars data 607 5.51 135 24.81

excl. unass. (KB)

compression (#PC’/#PM’excl )

47 11

6.18 25.90

530 58

6.31 57.52

Table 8: Additional file compression on the data using gzip almost exactly preserves the compression rates in form of relative sizes (see tables 5.2 and 5.2 for the values without additional zip-compression). The values given below are for average sizes of zipped point clouds (PC’) as well as zipped polygonal maps including (PM’incl ), respectively excluding unassigned points (PM’excl ). Finally, the effects of file compression are studied. Table 5.2 shows the sizes of the different formats after applying the gzip file compression to them. When comparing to tables 5.2 and 5.2, it is interesting to notice that a bit of additional compression can be achieved though both the raw point clouds and the polygonal maps are represented in a binary format. But the most important aspect is that the compression ratio due to the different representations hardly changes at all. Concretely, the ratio of unzipped point cloud over the corresponding unzipped polygon map is always almost exactly the same as the one for zipped point cloud over the corresponding zipped polygon map.

5.3

Comparison with Octrees and Elevation Maps

Occupancy grids are the predominant form of representation for 2D mapping [58]. Unfortunately, their canonical extension to 3D, i.e., 3D occupancy grids are highly inefficient with respect to memory usage. An alternative option are octrees [59, 60], which provide the same access operations as 3D occupancy grids and which are more memory-efficient, at least when compared to full 3D grids. We have included a comparison here for the sake of completeness, though octrees have several disadvantages as illustrated in this section. 17

resolution (cm × cm× cm) 1×1 5×5 10 × 10 20 × 20 30 × 30 40 × 40 50 × 50

size (KB) 723 288 138 63 34 24 18

relative size point cloud polygon map (#OCT/#PC) (#OCT/#PMch ) 1.87 55.19 0.75 21.98 0.36 10.53 0.16 4.77 0.09 2.56 0.06 1.83 0.05 1.34

Table 9: Map sizes for octree representations with different resolutions of the point cloud shown in figure 8. The absolute size in kilobytes (KB) and the relative factor of the size of the octrees (#OCT) to the raw point cloud (#PC), respectively to the polygon map with convex hull boundaries (#PMch ) are shown.

For the following comparison experiments, a highly efficient implementation of an octree is used [61]. This Fast Octree (FOT) is highly efficient both with respect to computation time as well as memory usage. The FOT is optimized for binary (occupancy) information per cell. The nodes and leaves of a FOT consume just 4 bytes of memory. In addition to small memory usage, this allows computationally efficient integer and logical operations for accessing the FOT. But even a highly optimized FOT requires significant amounts of memory compared to a polygonal map. The octree representation may be an advantage over full 3D grids, but it is already less so in comparison to raw point clouds. The exact memory requirement depends of course on the spatial resolution of the octree. Table 5.3 shows the sizes of a FOT representation of the ESA LRC scan shown in figure 8 for different spatial resolutions. Visualizations of the different octrees are shown in figure 14. The octree with the finest spatial resolution of a cubic centimeter per cell captures approximately all points in the original point cloud. This octree is about 1.9 times larger than the raw point cloud due to the overhead of the tree structure. The efficiency of the octree increases with decreasing spatial resolution as increasingly more points can be collapsed in a single octree cell. A cell size of 5×5×5 cm3 already leads to a reduction in size to approximately 75% of the raw point cloud data. But the octree is not really efficient compared to the polygonal representation. An extremely coarse octree cell size of 50×50×50 cm3 is required to get roughly similar memory requirements as table 5.3 shows. But the main problem with the octree representation are the side effects that result from the discretization of the cells. A halfway smooth representation of the environment’s surface requires a small cell size. Otherwise, there are step artifacts as illustrated in figures 15 and 16. If the cell size is too big, the discretization will always lead to illusionary obstacles, which are not present in the polygonal representation. But a suited cell size of 5×5×5 cm3 leads to memory requirements, which are about 22 times higher than for the polygonal representation. A further possible representation are elevation maps. As mentioned before, we are mainly interested in full 3D data. Please note that our planar surface patches can represent tunnels, overhanging rocks, and other full 3D structures. But in many natural landscapes, a 2.5D representation is sufficient as long as two points with different vertical z-coordinates never have the same horizontal x,y-coordinates, i.e., if the environment can be described by an elevation surface. According 2.5D elevation maps - also known as digital elevation models (DEM) or digital terrain models (DTM) - have been used in the context of planetary exploration [12][62]. Hence, a comparison is included here. Elevation maps come in two different flavors, namely as irregular triangle mesh - short trimesh - or as raster data based on regular grids. In case an irregular trimesh representation is used, there is no size difference between the elevation map and a raw point cloud. Concretely, the points in the cloud are simply identical with the corner points of the Delaunay triangulation of the elevation surface and hence their most efficient representation. The situation is different if a regular grid is used as basis for the elevation map. The size of the map is then dependent on the resolution of the grid.

18

(a) 5 cm × 5 cm × 5 cm

(b) 10 cm × 10 cm × 10 cm

(c) 20 cm × 20 cm × 20 cm

(d) 30 cm × 30 cm × 30 cm

(e) 40 cm × 40 cm × 40 cm

(f) 50 cm × 50 cm × 50 cm

Figure 14: Topviews of octree representations of the ESA LRC point cloud shown in figure 8. The octrees are computed with different spatial resolutions, which directly influence their sizes but also their usability.

Figure 15: The discretization used in octrees can lead to the illusion that there are steps in the terrain, which can not be negotiated by the robot. The effect obviously depends on the resolution with which the octree is created and it is hence directly related to the overall size of the octree. The problem does not occur with the polygon representation based on plane fitting.

19

Figure 16: An example of the step illusion (see also figure 15) shown here in the 20×20×20 cm3 octree representations of the ESA LRC point cloud from figure 8.

elevation maps (EM) resolution ESA LRC Mars (cm × cm) (KB) (KB) 10 × 10 977 5625 20 × 20 244 1406 30 × 30 109 625 40 × 40 61 352 50 × 50 39 225 60 × 60 27 156 70 × 70 20 115 80 × 80 15 88

relative size ESA LRC Mars (#EM/#PMch ) (#EM/#PMch ) 74.55 82.57 18.64 20.64 8.28 9.17 4.66 5.16 2.98 3.30 2.07 2.29 1.52 1.69 1.16 1.29

Table 10: Map sizes for elevation maps with different grid resolutions. In addition to the absolute size in kilobytes (KB), the relative factor of the size of the elevation maps (#EM) to the polygon maps with convex hull boundaries (#PMch ) is shown.

20

Table 5.3 shows the sizes of elevation maps in the two test scenarios with different grid resolutions. The raster varies from a tile size of 10 cm by 10 cm up to 80 cm by 80 cm with steps of 10 cm. It can be seen that the elevation maps with reasonable grid resolutions require more space than the plane fitting representation with convex hull boundaries. Only with very coarse grid resolutions, the sizes are comparable. In addition, elevation maps are as mentioned only 2.5D and they can hence not handle full 3D structures. But they at least do not suffer from the step illusion effect like octrees.

6

Conclusions

3D environment information is crucial for operating a robot in planetary exploration scenarios. It is hence desirable to get long range 3D data with high resolution, large field of view, and very fast update rates. 3D Laser Range Finders (3D-LRF) have a high potential in this respect, especially devices at the top of the state of the art like the Velodyne HDL-64E. In addition, 3D-LRF can operate under conditions where standard vision based methods fail, e.g., under extreme light conditions. But it is non-trivial to transmit the huge amounts of data delivered by a 3D-LRF from a planetary exploration robot to an operator station. Based on previous own work on plane fitting, it is shown in this article how the huge amounts of 3D point cloud data from 3D-LRF can be tremendously reduced. In doing so, large sets of points are replaced by planar surface patches that are fitted into the data in an optimal way. The underlying computations are very efficient and hence suited for online computations onboard of the robot. The approach is tested with two sets of experiments. The first is based on data gathered during our participation at the Lunar Robotics Challenge (LRC) of the European Space Agency (ESA) that took place in October 2008 in the volcanic landscape of the Teide National Park on Tenerife. The data is collected by a robot with a SICK S300 2D-LRF combined with a simple servo for a pitching motion to get the additional degree of freedom. A high end solution is used for the second set of experiments. There, a Velodyne HDL-64E is used in a high fidelity simulation with ground truth data from Mars. The generated models are suited to give an overview of the environment while compression rates of up to two orders of magnitude are achieved in both sets of experiments.

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