Jie Shi
email: jie.shi@asu.edu

Yuting Wang

Paul M Thompson

Yalin Wang

Hippocampal Morphometry Study by Automated Surface Fluid Registration and its Application to Alzheimer's Disease

Alzheimer's disease (AD) is a severe and growing public health crisis. Efforts are underway to look for AD early detection in an efficient manner. Among all the AD biomarkers, hippocampal atrophy assessed on high-resolution T1-weighted MRI is the best established and validated. Hippocampal morphometry is increasingly used in the AD research, with modeling the hippocampus as a 3D parametric surface mesh. However, a major question in the analysis is how to align corresponding surface regions across subjects. Here we develop a system for detecting AD symptoms on hippocampal surfaces with an automated surface fluid registration method, which is based on conformal surface representation and mutual information regularized image fluid registration. Since conformal mappings are diffeomorphic and the mutual information method is able to drive a diffeomorphic flow that is adjusted to enforce appropriate surface correspondences in the surface parameter domain, combining conformal and fluid mappings will generate 3D shape correspondences that are diffeomorphic. We also incorporate in the system a novel method to compute curvatures using surface conformal parameterization. Experimental results in three hippocampal datasets show that the new system outperformed an early similar method and the popular SPHARM tool. * ω = -g(x α , y α )dx α + f (x α , y α )dy α

(2)

Introduction

Alzheimer's disease (AD) doubles in frequency of onset every 5 years after age 60, afflicting 1% of those aged 60 to 64, and 30-40% of those over 85. Many MRIbased measures of atrophy in several structural measures, including whole-brain, hippocampus [START_REF] Morra | Automated mapping of hippocampal atrophy in 1-year repeat MRI data from 490 subjects with Alzheimer's disease, mild cognitive impairment, and elderly controls[END_REF][START_REF] Qiu | Regional shape abnormalities in mild cognitive impairment and Alzheimer's disease[END_REF][START_REF] Thompson | Mapping hippocampal and ventricular change in Alzheimer disease[END_REF], and ventricular enlargement [START_REF] Thompson | Mapping hippocampal and ventricular change in Alzheimer disease[END_REF], etc., correlate closely with changes in cognitive performance, supporting their validity as markers of AD progression [START_REF] Frisoni | The clinical use of structural MRI in Alzheimer disease[END_REF]. Among all the biomarkers, hippocampal atrophy assessed on high-resolution T1-weighted MRI is the best established and validated. As a result, detection of valid and efficient morphometry changes in hippocampus and their correlation with other cognitive functions and biomarkers becomes a key research topic for clinical diagnosis and monitoring of patients with suspected Alzheimer's disease. Although most hippocampus analysis used volume as the atrophy measurement [START_REF] Holland | Subregional neuroanatomical change as a biomarker for Alzheimer's disease[END_REF], recent researches [START_REF] Apostolova | Subregional hippocampal atrophy predicts Alzheimer's dementia in the cognitively normal[END_REF][START_REF] Qiu | Regional shape abnormalities in mild cognitive impairment and Alzheimer's disease[END_REF][START_REF] Thompson | Mapping hippocampal and ventricular change in Alzheimer disease[END_REF][START_REF] Yushkevich | Continuous medial representation of brain structures using the biharmonic PDE[END_REF] demonstrated that surface-based subcortical structure analysis may offer more advantages because these methods studied patterns of hippocampal subfield atrophy and produced detailed point-wise correlation between atrophy and cognitive functions/biological markers [START_REF] Thompson | Mapping hippocampal and ventricular change in Alzheimer disease[END_REF].

Brain surface deformation studies typically require the computation of dense correspondence vector fields that match one surface with another. Many brain surface registration methods have been proposed [START_REF] Durrleman | Sparse approximation of currents for statistics on curves and surfaces[END_REF][START_REF] Vaillant | Surface matching via currents[END_REF]27].Often, higher order correspondences must be enforced between specific anatomical points and curved landmarks lying within the two surfaces. This can be achieved by first mapping each of the 3D surfaces to canonical parameter spaces such as a sphere [START_REF] Fischl | High-resolution inter-subject averaging and a coordinate system for the cortical surface[END_REF] or a planar domain [START_REF] Wang | Automated surface matching using mutual information applied to Riemann surface structures[END_REF]. A flow, computed in the parameter space of the two surfaces [START_REF] Davatzikos | Spatial normalization of 3D brain images using deformable models[END_REF], induces a correspondence field in 3D. Artificial neural networks can rule out or favor certain types of feature matches [START_REF] Pitiot | Learning object correspondences with the observed transport shape measure[END_REF]. Correspondences may be determined by using a minimum description length (MDL) principle, based on the compactness of the covariance of the resulting shape model [START_REF] Davies | A minimum description length approach to statistical shape modeling[END_REF]. A key direction in surface registration research has been the computation of a diffeomorphic surface map that also matches automatically identified surface features.

Using holomorphic 1-forms, a global conformal parameterization can be developed to conformally map a surface with complex topology (e.g., a surface with branching topology) to a set of rectangular domains in the Euclidean plane. The resulting parameterization helps in discretizing partial differential equations (PDEs) for smoothing, denoising, or matching signals defined on the surface. The mutual information (MI) method has been widely used to drive a diffeomorphic flow in image registration. By adjusting the mutual information method to enforce appropriate surface correspondences in the parameter domain, any scalarvalued signals defined on the surfaces can also be aligned using the same flow field. Conformal maps and fluid registration techniques can be combined to avoid having to define a large set of manually-defined landmarks to constrain brain surface correspondences. Since they generate diffeomorphic mappings, conformal and fluid mappings together could generate 3D shape correspondences that are diffeomorphic (i.e., smooth one-to-one correspondences). In [START_REF] Wang | Automated surface matching using mutual information applied to Riemann surface structures[END_REF][START_REF] Wang | Mutual information-based 3D surface matching with applications to face recognition and brain mapping[END_REF], Wang et al. proposed an automated surface fluid registration method based on conformal mapping and mutual information regularized image fluid registration and applied it to register human faces and hippocampus. Here we develop a system based on this technique for studying hippocampus in AD and incorporate a novel method to compute surface curvatures as proposed in [START_REF] Lui | Computation of curvatures using conformal parameterization[END_REF]. Our major contributions can be summarized as: [START_REF] Apostolova | Subregional hippocampal atrophy predicts Alzheimer's dementia in the cognitively normal[END_REF]. Introduction of a new stable method to compute surface curvatures. ( 2). An automated hippocampal surface registration system validated in three AD datasets with better performance than a previous similar method [START_REF] Wang | Multivariate tensor-based brain anatomical surface morphometry via holomorphic one-forms[END_REF] and SPHARM [START_REF] Styner | Framework for the statistical shape analysis of brain structures using SPHARM-PDM[END_REF]. (3). The system will be publicly available [START_REF] Wang | Multivariate tensor-based subcortical morphometry system[END_REF]. Last, although the current system finds applications in AD detection, it is a general method which may be applied to many other applications.

Method

Surface Conformal Parameterization

Let S be a surface in R 3 with an atlas {(U α , z α )}, where (U α , z α ) is a coordinate chart defined on S. The atlas thus is a set of consistent charts with smooth transition functions between overlapping charts. Here z α : U α → C maps an open set U α ⊂ S to a complex plane C. If on any chart (U α , z α ) in the atlas, the Riemannian metric or the first fundamental form can be formulated as ds 2 = λ(z α ) 2 dz α d zα , and the transition maps

z β • z -1 α : z α (U α U β ) → z β (U α U β )
are holomorphic, the atlas could be called conformal. Given a conformal atlas, a chart is compatible with the atlas if adding this chart still generates a conformal atlas. A conformal structure is obtained by adding all possible compatible charts to a conformal atlas. A Riemann surface is a surface with a conformal structure. One coordinate chart in the conformal structure introduces a conf ormal parameterization between a surface patch and the image plane. The conformal parameterization is angle-preserving and intrinsic to the surface geometry.

Let (x α , y α ) be the local parameter on the chart (U α , z α ) on S, a differential 1-form in (x α , y α ) can be defined as

ω = f (x α , y α )dx α + g(x α , y α )dy α (1)
where f, g are smooth functions. ω is a closed 1-form, if in each local parameter (x α , y α ), ∂f ∂yα -∂g ∂xα = 0. If ω is the gradient of another function defined on S, it can be called an exact 1-form. An exact 1-form is also a closed 1-form. If a closed 1-form ω satisfies ∂f ∂xα + ∂g ∂yα = 0, then it is a harmonic 1-form. The gradient of a harmonic 1-form is an exact harmonic 1-form. The Hodge star operator acting on a differential 1-form gives the conjugate differential 1-form Intuitively, the conjugate 1-form * ω is obtained by rotating ω by a right angle everywhere. If ω is harmonic, so is its conjugate * ω. The holomorphic 1-form consists of a pair of conjugate harmonic 1-forms:

τ = ω + √ -1 * ω (3)
For a Riemann surface S with genus g > 0, its conformal structure can always be represented in terms of a holomorphic 1-form basis, which is a set of 2g functions τ i : [START_REF] Wang | Mutual information-based 3D surface matching with applications to face recognition and brain mapping[END_REF]. Any holomorphic 1-form τ is a linear combination of these functions. This finite-dimensional linear space generates all possible conformal parameterizations of surface S and the quality of a global conformal parameterization is fundamentally determined by the choice of the holomorphic 1-form [START_REF] Wang | Mutual information-based 3D surface matching with applications to face recognition and brain mapping[END_REF]. Then the conformal parameterization φ at point p can be computed by integrating the holomorphic 1-form: φ(p) = γ τ , where γ is any path joining p to a fixed point c on the surface. Figure 1 (a) illustrates a brain cortical surface and its conformal parameterization to a square.

K i → R 2 , i = 1, 2 • • • , 2g

Surface Conformal Representation

It has been known that surface registration requires defining a lot of landmarks in order to align corresponding functional regions. Labeling features could be accurate but time-consuming. Here we show that surface conformal parameterization could represent surface geometric features, thus avoiding the manual definition of landmarks. For a general surface and its conformal parameterization φ : S → R 2 , the conformal factor at a point p can be determined by the formulation:

λ(p) = Area(B (p)) Area(φ(B (p))) (4)
where B (p) is an open ball around p with a radius . The conformal factor λ encodes a lot of geometric information about the surface and can be used to compute curvatures and geodesic. In our system, we compute the surface mean curvatures only from the derivatives of the conformal parameterization as proposed in [START_REF] Lui | Computation of curvatures using conformal parameterization[END_REF], instead of the three coordinate functions and the normal, which are generally more sensitive to digitization errors. Mathematically, the mean curvature is defined as:

H = 1 2λ sign(φ)|∆φ|, where sign(φ) = < ∆φ, - → N > |∆φ| . (5)
Using this formulation of H, we need to use the surface normal -→ N only when computing sign(φ), which takes the value 1 or -1. Thus, the surface normal does not need to be accurately estimated and still we can get more accurate mean curvatures. Using the Gauss and Codazzi equations, one can prove that the conformal factor and mean curvature uniquely determine a closed surface in R 3 , up to a rigid motion. We call them the conf ormal representation of the surface. Figure 1 (b) shows the computed conformal factor (left) and mean curvature (right) on a hippocampal surface with color indices according to the values. Since conformal factor and mean curvature could represent important surface features and they are intrinsic to the surface, they may be used for surface registration.

Surface Fluid Registration Regularized by Mutual Information

After computing intrinsic geometric features, we align surfaces in the parameter domain with a fluid registration technique.Using conformal mapping, we essentially convert the surface registration problem to an image registration problem. The mutual information (MI) method has been successfully used to drive a diffeomorphic flow in rigid [START_REF] West | Comparison and evaluation of retrospective intermodality brain image registration techniques[END_REF] and non-rigid [START_REF] Meyer | Demonstration of accuracy and clinical versatility of mutual information for automatic multimodality image fusion using affine and thinplate spline warped geometric deformations[END_REF][START_REF] Rueckert | Nonrigid registration using free-form deformations: application to breast MR images[END_REF] image registration. Image registration will be optimized when MI between two images is maximized. For MI to work, a monotonic mapping in grayscales between images is not required, so images from different modalities can be registered [START_REF] Kim | Mutual information for automated unwarping of rat brain autoradiographs[END_REF]. Hermosillo et al. [9] adopted linear elasticity theory to regularize the variational maximization of MI. D'Agostino et al. [3] extended this approach to a viscous fluid scheme allowing large local deformations, while maintaining smooth, one-to-one topology [START_REF] Christensen | Deformable templates using large deformation kinematics[END_REF]. We call this approach MI regularized fluid registration.

In [START_REF] Wang | Automated surface matching using mutual information applied to Riemann surface structures[END_REF][START_REF] Wang | Mutual information-based 3D surface matching with applications to face recognition and brain mapping[END_REF], Wang et al. proposed an automated surface fluid registration method combining conformal mapping and image fluid registration [3]. Let I 1 , I 2 be the conformal representations of the target and the deforming template surfaces, respectively, the MI between two surfaces was defined as [START_REF] Wang | Automated surface matching using mutual information applied to Riemann surface structures[END_REF][START_REF] Wang | Mutual information-based 3D surface matching with applications to face recognition and brain mapping[END_REF]

I(u) = R 2 p u (i 1 , i 2 )log p u (i 1 , i 2 ) p(i 1 )p u (i 2 ) di 1 di 2 (6)
where

p(i 1 ) = P (I 1 (x) = i 1 ), p u (i 2 ) = P (I 2 (x -u) = i 2 ), and p u (i 1 , i 2 ) = P (I 1 (x) = i 1 & I 2 (x -u) = i 2 )
. Since conformal mapping and MI regularized fluid registration generate diffeomorphic mappings, a diffeomorphic surface-tosurface mapping is then recovered that matches surfaces in 3D. In our system, we adopt their methods of conformal mapping and fluid registration. However, our system differs from theirs in the computation of surface features as introduced in Sec. 2.2. The new way to compute mean curvature is more stable and less sensitive to normal computation, thus gives better representation of the surface features for registration.

Results

Synthetic Surface Registration Result

First, we illustrate the algorithm on synthetic surfaces. Figure 1 (c) illustrates a synthetic surface example. A pair of simple S-shape surfaces were generated. Corresponding 2D images were generated based on the sum of the local conformal factor and the mean curvature, expressed in the conformal parameterization domain. Some black horizontal lines were drawn on the surfaces to show equal distances on the surfaces and represent the differences in their shapes.The locations of the highest and lowest intensities are different (as shown by the positions of the horizontal stripes in the 2D images below). Using surface-based fluid registration, in the last image, the obtained horizontal line positions demonstrated an improved matching between features lying in the two surfaces.

In this experiment, we aim to visually explain the work flow of the algorithm. First, we conformally mapped each surface onto a planar domain and computed their conformal representation by combining conformal factor and mean curvature. Second, we scaled the conformal representation to form the 2D feature images as shown by the first two 2D images in Fig. 1 (c). As we pointed out in Sec. 2.2, the 2D images, i.e. the conformal representations of the surfaces clearly show the characteristics of the surfaces. Third, by registering the two feature images using fluid registration method, we induced a change in both the template feature image and the template surface. The last image in Fig. 1 (c) demonstrates that without changing the shape of the surface, the features on the template surface are well aligned with the target surface.

Hippocampal Surface Morphometry in Alzheimer's Disease

In this experiment, we test the robustness of our system by applying it to two clinical studies of hippocampal changes in Alzheimer's disease (AD). In our system, we leave two holes at the front and back of the hippocampal surface, representing its anterior junction with the amygdala, and its posterior limit as it turns into the white matter of the fornix. The resulting structure can then be logically represented as an open boundary genus-one surface, i.e., a cylinder. To better visualize the matching of surface features, we chose to encode surface features using a compound scalar function based on the local conformal factor and the mean curvature. A similar technique for AD study in hippocampus was proposed in [START_REF] Wang | Multivariate tensor-based brain anatomical surface morphometry via holomorphic one-forms[END_REF], but with surface registration using constrained harmonic map. We take their method as a comparison. After the cross-subject registration is computed with one target surface selected, we examine shape differences using the multivariate tensor-based morphometry (MTBM) [START_REF] Wang | Multivariate tensor-based brain anatomical surface morphometry via holomorphic one-forms[END_REF]. MTBM computes statistics from the Riemannian metric tensors that retain the full information in the deformation tensor fields, thus is more powerful in detecting surface differences than many other statistics [START_REF] Wang | Multivariate tensor-based brain anatomical surface morphometry via holomorphic one-forms[END_REF].

Figure 2 illustrates the experimental results on a group of hippocampal surface models extracted from 3D brain MRI scans of 12 individuals with AD and Fig. 2. The comparison of our method and a previous similar method on map of local shape differences (p-values), based on the multivariate TBM method with hippocampal surfaces in 12 AD versus 14 control subjects [START_REF] Thompson | Mapping hippocampal and ventricular change in Alzheimer disease[END_REF].

14 matched healthy control subjects, the same dataset as in [START_REF] Thompson | Mapping hippocampal and ventricular change in Alzheimer disease[END_REF], where the data was manually traced by experienced neurologists. With MTBM, we can see that the significant areas detected with two methods are consistent.But with surface fluid registration, we detected larger areas with more significant group differences in the surface parameterization tensor, which is related to the relative area of regions in disease versus normality.The overall statistical significance of these group difference maps, based on permutation testing of the suprathreshold area of statistics (and therefore corrected for multiple comparisons) were p=0.0063 for the left hippocampal surface and 0.0298 for the right. This outperformed the previous method [START_REF] Wang | Multivariate tensor-based brain anatomical surface morphometry via holomorphic one-forms[END_REF] (0.0205 for the left hippocampal surface and 0.1026 for the right). The cumulative distribution function (CDF) plot shows the comparison of resulting p-values for each method. While the line y = x represents null hypothesis, which is according to no difference, steeper curve shows more differences detected.

Figure 3 illustrates the experimental results on another hippocampal dataset from the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset (http:// www.loni.ucla.edu/ADNI). Mild cognitive impairment (MCI) is an intermediate stage between the expected cognitive decline of normal aging and the more pronounced decline of dementia. If MCI could be found and treated, the risk of AD will be significantly reduced. However, at MCI stage, changes in brain surface are not significant thus impose more difficulty on the detection. Here we randomly chose 40 AD, 40 MCI, and 40 control subjects to test the effectiveness of our system. The hippocampus data was automatically segmented by a prior work [START_REF] Morra | Automated mapping of hippocampal atrophy in 1-year repeat MRI data from 490 subjects with Alzheimer's disease, mild cognitive impairment, and elderly controls[END_REF]. With MTBM, we can see that, in the three experiments, our system demonstrated better results than the previous method. Particularly, our system gave better MCI detection results when comparing with both AD and control subjects. In the experiment, all group difference p-maps were corrected using false discovery rate (FDR). The FDR method decides whether a threshold can be assigned to the statistical map that keeps the expected FDR below 5% (i.e., no more than 5% of the voxels are false positive findings). The CDF plots show the uncorrected p-values (as in a conventional FDR analysis). The x value at which the CDF plot intersects the y = 20x line represents the FDR corrected p-value or q-value. It is the highest statistical threshold that can be applied to the data, for which at most 5% false positives are expected in the map. In general, a larger q-value indicates a more significant difference in the sense that there is a broader range of statistic threshold that can be used to limit the rate of false positives to at most 5%. The use of the y = 20x line is related to the fact that significance is declared when the volume of suprathreshold statistics is more than 20 times that expected under the null hypothesis. Table 1 gives the FDR corrected p-values comparison.

Comparison with SPHARM

In this study, we tested our system on the whole ADNI baseline dataset (http:// www.loni.ucla.edu/ADNI). The dataset consists of 233 healthy controls, 410 subjects with MCI, and 200 patients with AD. We excluded 1 subject from the control group and 2 subjects from the MCI group due to name duplication. For subjects with duplicated names, we retained the one which is the repeated scan. The hippocampal surfaces were automatically segmented using FIRST (http://www.fmrib.ox.ac.uk/fsl/first/index.html). FIRST is an integrated surface analysis tool developed as part of the FSL library, which is written mainly by members of the Analysis Group, FMRIB, Oxford, UK. We compared our system's results with the popular SPHARM tool [START_REF] Styner | Framework for the statistical shape analysis of brain structures using SPHARM-PDM[END_REF] in surface registration on the same dataset. We adopted the suggested parameters for hippocampus in the manual of SPHARM [START_REF] Styner | Framework for the statistical shape analysis of brain structures using SPHARM-PDM[END_REF]. In this experiment, 1 subject from each group (AD, MCI, control) failed the FIRST segmentation step probably due to the original images' resolution or contrast. After FIRST segmentation, we extracted the left and right hippocampi and saved each of them into a binary image. We fed the binary images as the input of both systems. As a result, 231 control, 199 AD, and 407 MCI subjects were successfully registered by our system, which shows the robustness of the system compared with SPHARM, in which 5 control, 17 AD, 14 MCI subjects failed either due to segmentation failure or parameterization failure. Fig. 4 shows the experimental results. From the p-map and the CDF plots, we can see that, with MTBM, our system outperformed the SPHARM 1. FDR corrected p-values on hippocampal surfaces in all three group difference studies. Table 2 gives the FDR corrected p-values comparison. Considering fairness, we also made comparisons by excluding those subjects that failed in SPHARM from our system in the statistical study. Table 3 gives the FDR corrected p-values comparison with the failed subjects excluded, i.e., 226 control, 182 AD, and 393 MCI subjects were studied in this experiment. From the table, we can see that our system still outperformed SPHARM on this dataset. Furthermore, because AD and MCI failed more than control subjects, the AD-MCI group has been more affected as expected.

Conclusion and Future Work

We develop an automated surface fluid registration system for hippocampal surface registration. Experiments on different types of AD hippocampal datasets all demonstrate our system's stronger statistical power. Ongoing work is to apply this system to automatically map lateral ventricle enlargements in Alzheimer's disease and those at risk.

Fig. 1 .

 1 Fig. 1. Illustration of conformal parameterization (a) and geometric features (b). (c) shows the matching of geometric features in the 2D parameter domains using fluid registration with synthetic surfaces. Geometric features on 3D surfaces were computed and mapped to 2D conformal parameter domains.

Fig. 3 .

 3 Fig. 3. Comparison of surface fluid registration and the previous method on map of local shape differences (p-values), based on the mutivariate TBM method with hippocampal surfaces in 40 AD, 40 MCI, and 40 control subjects which were automatically segmented [14]. (a), (c), (e) are our results on group difference between AD and control, MCI and AD, MCI and control, respectively. Similarly, (b), (d), (f) are results of a prior work [23] on AD and control, MCI and AD, MCI and control, respectively. The p-map color scale is the same as Figure 2.

Fig. 4 .

 4 Fig. 4. Comparison of surface fluid registration and SPHARM on map of local shape differences (p-values), based on the mutivariate TBM method with hippocampal surfaces from ADNI baseline data, which were automatically segmented by FIRST. (a), (c), (e) are our results on group difference between AD and control, MCI and AD, MCI and control, respectively, in 231 control, 199 AD, and 407 MCI subjects. Similarly, (b), (d), (f) are results of SPHARM on AD and control, MCI and AD, MCI and control, respectively, in 226 control, 182 AD, and 393 MCI subjects. The p-map color scale is the same as Figure 2.

Table 2 .

 2 FDR corrected p-values on hippocampal surfaces.

		Our Method SPHARM [18]
	AD-CTL	0.0485	0.0459
	MCI-AD	0.0215	0.0047
	MCI-CTL	0.0483	0.0288


Paul Backes

Daniel Helmick

Max Bajracharya

Results of Coring from a Low Mass Rover

Technology for coring from a low-mass rover has been developed to enable core sample acquisition where a planetary rover experiences moderate slip during the coring operation. A new stereo vision technique, Absolute Motion Visual Odometry, is used to measure rover slip during coring and the slip is accommodated through corresponding arm pose updating. Coring rate is controlled by feedback of the measured force of the coring tool against the environment. Test results in the JPL Marsyard showthat coring from a low-mass rover with slip is feasible.

INTRODUCTION

Technology to enable core sample acquisition from a low- mass rover with modest slip during coring has been developed. The current state-of-the-art in coring and drilling from a planetary rover arm-mounted tool, represented by the upcoming 2009 Mars Science Laboratory (MSL) mission, assumes that the arm and rover act as a rigid platform for the coring tool or drill. The MSL mission rover will have a drill mounted on a manipulator. The manipulator will anchor the drill against a rock by pressing tines against the rock to provide a rigid base for the drill. The drill will then provide its own translational degree-offreedom (DOF) and control of interaction with a rock. It is reasonable to assume that the rover will not move during the coring operation if the rover is big enough, as is the case for the MSL mission, but this assumption is not reasonable when the rover's mass is low relative to the interaction forces with the environment or when the rover is on a slope where slippage of the rover is possible.

Future missions could benefit by enabling arm-mounted drilling or coring from a low-mass rover where the rover 1 1 1-4244-1488-1/08/$25.00 C 2008 IEEE. 2 IEEEAC paper # 1122, Version 2, Updated December 21, 2007 Oussama Khatib, S kha Figure 1: Cori may experience m coring operation. of 100-200kg. A fu might utilize a low that are stored and to eliminate the n environment via pu and instead allow t the coring tool rel allow for a lower tool-environment i arm such as motor coring from a low Program sponsored Figure 1 illustrates A low-mass rover arm and we need t a 26 degree slope. rover slip can be e to the vibration fro be small relative t the slippage can be reconfigure the arm hole during the cor

The overall approa to accommodate sl is to measure the r the tool remains al slips far enough s workspace, then the rover is driven so that the arm is in the middle of its workspace with the coring bit in the hole. During this process the coring bit is not removed from the coring hole. There might be scenarios where removing the bit and reinserting it might be feasible, but it is preferred to find a solution where the bit remains in the hole until the sample has been acquired. In our approach a stereo vision technique, Absolution Motion Visual Odometry (AMVO), described here for the first time, is used to measure the rover motion, from slip or driving, relative to its initial pose. This motion is accommodated by reconfiguration of the arm to keep the tool aligned with the coring hole. Single axis force control is used for motion of the tool along the tool axis in the hole during coring. Tool rotational velocity is used to measure binding as a safety feature. It was decided to not use six-axis force sensing and control of the tool in the hole because there is no flight qualified six axis force-torque sensor available for use. The approach described here enables coring with modest slip without the use of tines and using existing planetary rover manipulation and sensing capabilities.

ABSOLUTE MOTION VISUAL ODOMETRY

We developed Absolute Motion Visual Odometry (AMVO) to measure the six degrees of freedom (DOF) of motion of a rover for small motions of the vehicle (less than about 50 cm). We use this technique to accurately detect and measure the slippage that a rover undergoes while coring a sample from a rock in its environment. AMVO uses the same algorithm as visual odometry (VO) [START_REF] Matthies | Dynamic stereo vision[END_REF][4] [START_REF] Hirschmuller | Fast, unconstrained camera motion estimation from stereo without tracking and robust statistics[END_REF][6][7] [START_REF] Maimone | Two Years of Visual Odometry on the Mars Exploration Rovers[END_REF] but it uses the concept of a key pair to avoid the measurement drift associated with VO.

Stereo camera motion estimation, or visual odometry, is used to sense the 6 DOF rover motion due to slip during the course of coring activity. This 6 DOF of motion is represented by a transform that is used in a transform equation resulting in an updated arm configuration to keep the tool at the original pose (and then offset along the coring axis for the coring operation). Because only small motions need to be estimated, standard camera motion estimation methods are modified to use a key frame (image pair) taken before coring. All subsequent frames are then compared to this key frame and thus avoid the accumulation of error between successive frames. Because all motion is measured with respect to a single fixed frame, we term this method Absolute Motion Visual Odometry.

In order to use AMVO to compensate for rover motion during coring, it must be able to detect very small motions and run fast enough to be used in the real-time closed-loop system. Two implementations of AMVO were evaluated. The first was the more common technique of detecting, tracking, and robustly matching feature points [3][4][6] [START_REF] Maimone | Two Years of Visual Odometry on the Mars Exploration Rovers[END_REF].

The second implementation makes use of Euclidean constraints to select inliers and avoid tracking features [START_REF] Hirschmuller | Fast, unconstrained camera motion estimation from stereo without tracking and robust statistics[END_REF].

Both techniques begin by autonomously selecting features in the first frame, matching them between left and right camera images, and triangulating their 3D positions. The first implementation then uses a correlation method to track the features to the next frame and triangulate the new positions, and then uses a rigidity test to reject outliers. Finally, this approach uses a least median squares with a weighted least squares matching method to robustly choose an initial motion estimation, and then refines this estimate with a non-linear maximum likelihood procedure that incorporates the stereo error covariances directly into the estimation. The second method does not track features, but rather detects features in the next frame, uses the SAD (sum of absolute difference) score from stereo matching to generate initial feature matches, and then refines the matches using geometric (point rigidity and angular motion) constraints. The motion of the matched points is then calculated using a non-linear optimization of image-space error. As a result, this algorithm can be implemented faster, but could suffer from some reduction in accuracy.

We found that both VO methods performed similarly for computing absolute motion, although the second (feature matching rather than tracking) could be implemented to be slightly faster (as expected). Both methods were able to detect 0.5mm motions, and could achieve better than 2mm accuracy for 4mm to 20cm motions in realistic outdoor Mars-like environments (more precise measurements could not be made due to limitations of our outdoor groundtruthing equipment). The full 6 DOF motion of the vehicle was ground-truthed by measuring three rover-mounted fiducials with an outdoor laser surveying instrument. Motion magnitudes ranging from 4mm to 20cm were measured, and visual odometry was performed on the vehicle's 1024x786 (but makes use of image pyramiding) grayscale body-mounted "hazcams", which have an approximately 110 degree diagonal field-of-view. On the onboard 1.2GHz Pentium M CPU, our implementation of the first algorithm can be run at approximately 3Hz, whereas the second can be run at approximately 5Hz. Since the two algorithms produced similar results but the second algorithm was slightly faster, we used the second algorithm in our implementation of AMVO.

Results of AMVO in a small motion test are shown in Figure 2. The rover was commanded with the smallest possible motion and AMVO was used to measure the motion. This resulted in a measured motion of 0.54 mm.

COORDINATED ROVER-ARM CONTROL

The arm motion to push the coring tool into the hole and accommodate rover slippage was controlled at 20 Hz with the arm command computed using Equations ( 1) and (2).

RCT(i)=RciT(i)* RTTTC T(i) (1) Ti Ti T, c TCT(i)=TCT(i -1)*T TS (i) (2)
RiT(i) is the 6 DOF transform from the Rover frame initial pose where the coring operation started to the Rover frame current pose. This transform is computed by AMVO at each sample interval. TiT is the transform from the Tool frame to the Rover frame at the start of the coring operation and is a constant. fiT(i) is the transform from the Tool frame current commanded pose to the Tool frame initial pose when coring was started. TcTS(i) is the transform that contains the sensor based motion command generated in the current sample interval. For coring motion, TCTS(i) has only a tool translation axis (Z axis) motion, Az, generated from force control. Az is computed with Equation ( 3)

Az = Kf * (fzd-fza) (3)
Where (fd -fza) is the difference between the desired average weight on bit of the coring tool and the measured average weight on bit, and Kf is the control gain. RcT(i) is computed each sample interval and then used in inverse kinematics to compute arm joint angles which are then commanded and controlled.

When the rover moves so the arm reaches the edge of its workspace, the rover is driven back to its pose at the start of coring to put the arm back in the center of its workspace.

The rover motion is controlled in a separate process from the arm control so the arm control based on AMVO is used to autonomously accommodate the rover motion without During a Mars mission a rover might approach a coring target from below, as is done in the tests shown here, or from other orientations such as transverse of the face of the slope. For any approach orientation, the rover's pose relative to the target needs to be computed. One approach assumes an approach direction defined by the geometry of the target and computes the rover position along that approach direction that maximizes arm manipulability [START_REF] Backes | Automated Rover Positioning and Instrument Placement[END_REF]. An alternative approach has been developed that considers the rover mobility characteristics, wheel-soil force distribution, and the rover arm's ability to apply forces. This has been implemented and tested within the Stanford Simulation and Active Interface (SAI) system [10]. SAI is unique in that it is a real-time interactive environment that allows the user to apply and sense forces within the virtual world via haptic devices. SAI features multi-contact resolution for multi-body systems, efficient algorithms for articulated body dynamics, and simulated friction and ground reaction forces. SAI is based upon a general framework [START_REF] Ruspini | Collision / Contact Models for Dynamic Simulations and Haptic Interaction[END_REF] for the resolution of multi-contact between articulated multi-body systems in the context of operational space control for robots [12]. Using this framework, the dynamic relationships between all existing contact points can be described. These relationships are characterized by the masses as perceived at the contact points. A force exerted at a contact point, whether from a collision with another object or from interaction with a user, can be translated into forces at all related contact points. The necessary computations are performed with an efficient recursive algorithm. The contact space representation allows interaction between groups of dynamic systems to be described easily without having to examine the complex equations of motion of each individual system. A collision model can be developed with the same ease as if one were considering interaction only between simple bodies. Impact and contact forces between interacting bodies can then be solved efficiently. The rover pose algorithm computes the optimal coring configuration given the coring conditions: terrain slope, coring point location, and ground type. This requires the definition of the criterion used to characterize the optimality of a given configuration. We used different manipulability measures [START_REF] Walker | Impact configurations and measures for kinematically redundant and multiple armed robot systems[END_REF] so that the force capabilities of the robot along the coring tool axis are maximized and the overall manipulability is maximized in all operational directions.

IMPLEMENTATION ENVIRONMENT

A coring tool was attached to the arm on a 65 kg six wheel rocker-bogie rover called Pluto, shown in Figure 1, and coring tests were performed. The Pluto rover has the same mechanical mobility system as the FIDO rover [START_REF] Schenker | FIDO: a Field Integrated Design & Operations rover for Mars surface exploration[END_REF] but has a different mast and avionics system and has a five DOF manipulator. The avionics system has a 1.2GHz Pentium M CPU running Linux O/S. Arm and wheel motor control is done using ION commercial motor controllers from Performance Motion Devices, Inc. The coring tool is the LSAS coring tool from Alliance Spacesystems [START_REF] Stanley | The Lowforce Sample Acquisition System[END_REF]. The LSAS tool is a rotary percussive coring tool weighing 400 grams and requires about 15 N average weight on bit. It rotates at 800 RPM with 3 hits per revolution, and requiring about 20 Watts.

RESULTS

The objective of this work was to develop the technology needed to demonstrate that coring from a low-mass rover is feasible for future Mars rover missions. The first step was to find a coring tool that represents the functionality of the coring tool that would be on such a mission. It was concluded that a rotary percussive coring tool would be used because a percussive coring tool requires significantly lower weight on bit than a rotary friction coring tool. Since a representative percussive coring tool had never been used to acquire a core from a rover previously, an LSAS coring tool was mounted to the end of the Pluto rover five DOF arm and used to acquire a core from limestone, as shown in Figure 3.

Figure 4 shows the bit used in the coring test, the hole, and the resulting core. The results were significant in demonstrating for the first time the feasibility of core sample acquisition with such a device. The first significant result was that the core started without tines or a centering bit. The core started without a centering bit or tines because with a percussive tool the bit rebounds from the surface after each impact and returns to a nominal position. Disturbance forces that would cause the tool to skitter (move away from the coring position) are accommodated by the stiffness of the arm. The core was acquired at a rate of 2 cm/hour in limestone. Cuttings were artificially removed using a vacuum. Cuttings removal is a critical capability for coring tools and is not available in this first version of the tool. The test was stopped after one hour. The core was removed from the hole by inserting a thin metal rod down the side of the core and breaking it off. Core breakoff is another capability that needs to be added to such a device for planetary applications. The core was 2cm long with constant 12mm diameter along its length. A longer core could have been acquired in a longer coring operation. This rate of coring, about 2 cm/hour, is representative of the Figure 6: AMVO output and arm motion during whole coring operation expected rate of coring for Mars mission applications. A wrist force-torque sensor was used to measure the maximum impact forces felt by the arm during the coring operation, and were measured to be about 60 N. The core quality was a second significant result. Coring attempts with other tools had resulted in broken cores. The low average weight on bit, 15 N, was also significant because that is an achievable applied force for a low-mass rover.

Coring on a slope with slip is shown in Figure 1 and Figure 5. The rover was placed on a 26 degree slope in the JPL Marsyard and the tool was deployed to a limestone rock for coring. The AMVO control technique was used to keep the tool aligned with the coring target while the rover moved during the coring operation. Rover slip when starting the Figure 8: AMVO output and arm position during drives core and during the coring operation were autonomously accommodated by updating the arm pose based on AMVO feedback as described in section 3. Figures 6,7, and 8 show the AMVO output in the Rover frame X direction (forward) and the measured arm Tool frame position in the same direction (changed sign so that they would overlap; the actual motions are in opposite directions) during the coring operation.

The coring tool was turned on just before contacting the rock. The core was autonomously started and the coring operation continued for about 260 seconds. Then, in order to show robustness of the system to larger rover motions, the rover was purposely driven down the hill in steps of about 1 cm. The arm accommodation using AMVO kept the coring tool safely in the coring hole and continuing to core. Then to demonstrate driving the rover back up the hill to put the arm closer to the center of its workspace, it was driven up about 1.5 cm. Then it was driven down to the limit of about 13 cm from the starting point. The arm accommodated all motions, keeping the tool safely coring in the hole. Figure 6 shows the data over the whole motion where the rover moved 13 cm down the hill during the coring operation. Figure 7 shows the motion at the start of the coring operation. As expected, there is a small amount of slip by the rover (about 1.5 mm over 50 seconds), AMVO measures the slip motion, and the arm is commanded to change configuration to accommodate the slip motion. Figure 8 shows the AMVO output and arm motion when the rover was driven down and then up the hill at about 390 -410 seconds. There is a slight lag in the arm motion to accommodate the AMVO outputs as expected. Figure 9 shows a back hazcam image used by AMVO during the coring operation. The results show that use of AMVO to generate arm commands works over the 13 cm of motion.

CONCLUSION AND FUTURE WORK

Technologies to enable coring from a low-mass rover for planetary applications were developed and demonstrated including starting a core without a centering bit or stabilizing tines, coring with low weight on bit, and coring from a rover that is experiencing slip. Absolute motion visual odometry was developed and shown to provide sufficiently accurate measurement of rover motion during the coring operation to be used to compute the arm accommodation to keep the coring tool in the coring hole. The tool motor rotational speed was found to be an indicator of the binding of the tool in the hole. We plan on integrating motor current and speed for fine adjustment of tool position to minimize friction between the tool and hole. Since acquiring a core can take hours, the shadows of features will change during this time, potentially affecting the accuracy of AMVO relative to the initial key frames. We will investigate the affect of changing shadows and compensate for the shadow affects. Separately, coring tool technology is being advanced to include required capabilities for planetary applications including core breakoff, core retention, core ejection, and bit changeout.

X 5 Figure 2 :

 52 Figure 2: AMVO small motion test

Figure 3 :Figure 4 :

 34 Figure 3: Percussive coring test

Figure 5 :

 5 Figure 5: Coring on a slope in the Marsyard

Figure 7 :Figure 9 :

 79 Figure 7: AMVO output and arm position at start

ACKNOWLEDGEMENTS

The research described in this publication was carried out at the Jet Propulsion Laboratory of California Institute of Technology and Stanford University under contract from the National Aeronautics and Space Administration (NASA), with funding from the Mars Technology Program, NASA Science Mission Directorate.

June 1999 he has been working at the Jet Propulsion Laboratory on robotics research projects involving vision/sensor based control of robots, state estimation, and navigation and mobility algorithms. He has worked on robotic vehicles covering a wide range of functionality, including: Mars research rovers for rough terrain mobility; small, tracked robots for urban mobility; a cryobot for ice penetration; and reconfigurable wheeled robots for Mars exploration. His research interests include: sensor-based control of robots, sensor fusion and state estimation, and rover navigation and mobility. His research mainly focuses on the modeling and control of redundant and complex systems such as wheeled mobile manipulators and humanoid robots. He is also involved in the field of human motion synthesis using model learning techniques and in development of dynamic simulation and collision detection software.

Max Bajracharya received M. Eng., B.S.,EEICS degreesfrom MIT in 2001. He is a member of the Mobility and Manipulation Group at the Jet Propulsion Laboratory, Pasadena,


Florentin Smarandache

Arnaud Martin
email: arnaud.martin@univ-rennes1.fr

Christophe Osswald
email: christophe.osswald@ensta-bretagne.fr

Contradiction measures and specificity degrees of basic belief assignments

Keywords: Belief function, uncertainty measures, specificity, conflict

In the theory of belief functions, many measures of uncertainty have been introduced. However, it is not always easy to understand what these measures really try to represent. In this paper, we re-interpret some measures of uncertainty in the theory of belief functions. We present some interests and drawbacks of the existing measures. On these observations, we introduce a measure of contradiction. Therefore, we present some degrees of non-specificity and Bayesianity of a mass. We propose a degree of specificity based on the distance between a mass and its most specific associated mass. We also show how to use the degree of specificity to measure the specificity of a fusion rule. Illustrations on simple examples are given.

I. INTRODUCTION

The theory of belief functions was first introduced by [START_REF] Dempster | Upper and Lower probabilities induced by a multivalued mapping[END_REF] in order to represent some imprecise probabilities with upper and lower probabilities. Then [START_REF] Shafer | A mathematical theory of evidence[END_REF] proposed a mathematical theory of evidence.

Let Θ be a frame of discernment. A basic belief assignment (bba) m is the mapping from elements of the powerset 2 Θ onto [0, 1] such that:

X∈2 Θ m(X) = 1. (1)
The axiom m(∅) = 0 is often used, but not mandatory. A focal element X is an element of 2 Θ such that m(X) = 0.. The difference of a bba with a probability is the domain of definition. A bba is defined on the powerset 2 Θ and not only on Θ. In the powerset, each element is not equivalent in terms of precision. Indeed, θ 1 ∈ Θ is more precise than θ 1 ∪θ 2 ∈ 2 Θ . In the case of the DSmT introduced in [START_REF] Smarandache | Applications and Advances of DSmT for Information Fusion[END_REF], the bba are defined on an extension of the powerset: the hyper powerset noted D Θ , formed by the closure of Θ by union and intersection. The problem of signification of each focal element is the same as in 2 Θ . For instance, θ 1 ∈ Θ is less precise than θ 1 ∩ θ 2 ∈ D Θ . In the rest of the paper, we will note G Θ for either 2 Θ or D Θ .

In order to try to quantify the measure of uncertainty such as in the set theory [START_REF] Hartley | Transmission of information[END_REF] or in the theory of probabilities [START_REF] Shannon | A mathematical theory of communication[END_REF], some measures have been proposed and discussed in the theory of belief functions [START_REF] Dubois | A note on measures of specificity for fuzzy sets[END_REF], [START_REF] George | Quantification of conflict in Dempster-Shafer framework: a new approach[END_REF], [START_REF] Klir | Measures of uncertainty in the Dempster-Shafer theory of evidence[END_REF], [START_REF] Yager | Entropy and Specificity in a Mathematical Theory of Evidence[END_REF]. However, the domain of definition of the bba does not allow an ideal definition of measure of uncertainty. Moreover, behind the term of uncertainty, different notions are hidden.

In the section II, we present different kinds of measures of uncertainty given in the state of art, we discuss them and give our definitions of some terms concerning the uncertainty. In section III, we introduce a measure of contradiction and discuss it. We introduce simple degrees of uncertainty in the section IV, and propose a degree of specificity in the section V. We show how this degree of specificity can be used to measure the specificity of a combination rule.

II. MEASURES OF UNCERTAINTY ON BELIEF FUNCTIONS

In the framework of the belief functions, several functions (we call them belief functions) are in one to one correspondence with the bba: bel, pl and q. From these belief functions, we can define several measures of uncertainty. Klir in [START_REF] Klir | Measures of uncertainty in the Dempster-Shafer theory of evidence[END_REF] distinguishes two kinds of uncertainty: the non-specificity and the discord. Hence, we recall hereafter the main belief functions, and some non-specificity and discord measures.

A. Belief functions

Hence, the credibility and plausibility functions represent respectively a minimal and maximal belief. The credibility function is given from a bba for all X ∈ G Θ by: bel

(X) = Y ⊆X,Y ≡∅ m(Y ). ( 2
)
The plausibility is given from a bba for all X ∈ G Θ by:

pl(X) = Y ∈G Θ ,Y ∩X ≡∅ m(Y ). (3)
The commonality function is also another belief function given by:

q(X) = Y ∈G Θ ,Y ⊇X m(Y ). (4)
These functions allow an implicit model of imprecise and uncertain data. However, these functions are monotonic by inclusion: bel and pl are increasing, and q is decreasing. This is the reason why the most of time we use a probability to take a decision. The most used projection into probability subspace is the pignistic probability transformation introduced by [START_REF] Ph | Constructing the pignistic probability function in a context of uncertainty[END_REF] and given by: betP

(X) = Y ∈G Θ ,Y ≡∅ |X ∩ Y | |Y | m(Y ), (5)
where |X| is the cardinality of X, in the case of the DSmT that is the number of disjoint elements corresponding in the Venn diagram.

From this probability, we can use the measure of uncertainty given in the theory of probabilities such as the Shannon entropy [START_REF] Shannon | A mathematical theory of communication[END_REF], but we loose the interest of the belief functions and the information given on the subsets of the discernment space Θ.

B. Non-specificity

The non-specificity in the classical set theory is the imprecision of the sets. Such as in [START_REF] Ristic | The TBM global distance measure for association of uncertain combat ID declarations[END_REF], we define in the theory of belief functions, the non-specificity related to vagueness and non-specificity.

Definition The non-specificity in the theory of belief functions quantifies how a bba m is imprecise.

The non-specificity of a subset X is defined by Hartley [START_REF] Hartley | Transmission of information[END_REF] by log 2 (|X|). This measure was generalized by [START_REF] Dubois | A note on measures of specificity for fuzzy sets[END_REF] in the theory of belief functions by:

NS(m) = X∈G Θ , X ≡∅ m(X) log 2 (|X|). (6)
That is a weighted sum of the non-specificity, and the weights are given by the basic belief in X. Ramer in [START_REF] Ramer | Uniqueness of information measure in the theory of evidence[END_REF] has shown that it is the unique possible measure of non-specificity in the theory of belief functions under some assumptions such as symmetry, additivity, sub-additivity, continuity, branching and normalization.

If the measure of the non-specificity on a bba is low, we can consider the bba is specific. Yager in [START_REF] Yager | Entropy and Specificity in a Mathematical Theory of Evidence[END_REF] defined a specificity measure such as:

S(m) = X∈G Θ , X ≡∅ m(X) |X| . (7)
Both definitions corresponded to an accumulation of a function of the basic belief assignment on the focal elements. Unlike the classical set theory, we must take into account the bba in order to quantify (to weight) the belief of the imprecise focal elements. The imprecision of a focal element can of course be given by the cardinality of the element.

First of all, we must be able to compare the non-specificity (or specificity) between several bba's, event if these bba's are not defined on the same discernment space. That is not the case with the equations ( 6) and [START_REF] George | Quantification of conflict in Dempster-Shafer framework: a new approach[END_REF]. The non-specificity of the equation ( 6) takes its values in [0, log 2 (|Θ|)]. The specificity of the equation [START_REF] George | Quantification of conflict in Dempster-Shafer framework: a new approach[END_REF] can have values in [ 1 |Θ| , 1]. We will show how we can easily define a degree of non-specificity in [0, 1]. We could also define a degree of specificity from the equation [START_REF] George | Quantification of conflict in Dempster-Shafer framework: a new approach[END_REF], but that is more complicated and we will later show how we can define a specificity degree.

The most non-specific bba's for both equations ( 6) and ( 7) are the total ignorance bba given by the categorical bba m Θ : m(Θ) = 1. We have NS(m) = log 2 (|Θ|) and S(m) = 1 |Θ| . This categorical bba is clearly the most non-specific for us. However, the most specific bba's are the Bayesian bba's. The only focal elements of a Bayesian bba are the simple elements of Θ. On these kinds of bba m we have NS(m) = 0 and S(m) = 1. For example, we take the three Bayesian bba's defined on Θ = {θ 1 , θ 2 , θ 3 } by:

m 1 (θ 1 ) = m 1 (θ 2 ) = m 1 (θ 3 ) = 1/3, (8) m 2 (θ 1 ) = m 2 (θ 2 ) = 1/2, m 2 (θ 3 ) = 0, (9)
m 3 (θ 1 ) = 1, m 3 (θ 2 ) = m 3 (θ 3 ) = 0. ( 10
)
We obtain the same non-specificity and specificity for these three bba's. That hurts our intuition; indeed, we intuitively expect that the bba m 3 is the most specific and the m 1 is the less specific. We will define a degree of specificity according to a most specific bba that we will introduce.

C. Discord

Different kinds of discord have been defined as extensions for belief functions of the Shannon entropy, given for the probabilities. Some discord measures have been proposed from plausibility, credibility or pignistic probability:

E(m) = - X∈G Θ m(X) log 2 (pl(X)), (11)
C(m) = - X∈G Θ m(X) log 2 (bel(X)), (12)
D(m) = - X∈G Θ m(X) log 2 (betP(X)), (13)
with E(m) ≤ D(m) ≤ C(m). We can also give the Shanon entropy on the pignistic probability:

-

X∈G Θ betP(X) log 2 (betP(X)). (14)
Other measures have been proposed, [START_REF] Klir | Measures of uncertainty in the Dempster-Shafer theory of evidence[END_REF] has shown that these measures can be resumed by:

-

X∈G Θ m(X) log 2 (1 -Con m (X)), ( 15
)
where Con is called a conflict measure of the bba m on X. However, in our point of view that is not a conflict such presented in [START_REF] Wierman | Measuring Conflict in Evidence Theory[END_REF], but a contradiction. We give the both following definitions: Definition A contradiction in the theory of belief functions quantifies how a bba m contradicts itself.

Definition (C1) The conflict in the theory of belief functions can be defined by the contradiction between 2 or more bba's.

In order to measure the conflict in the theory of belief functions, it was usual to use the mass k given by the conjunctive combination rule on the empty set. This rule is given by two basic belief assignments m 1 and m 2 and for all X ∈ G Θ by: 16) k = m c (∅) can also be interpreted as a non-expected solution.

m c (X) = A∩B=X m 1 (A)m 2 (B) := (m 1 ⊕ m 2 )(X). (
In [START_REF] Yager | Entropy and Specificity in a Mathematical Theory of Evidence[END_REF], Yager proposed another conflict measure from the value of k given bylog 2 (1k).

However, as observed in [START_REF] Liu | Analyzing the degree of conflict among belief functions[END_REF], the weight of conflict given by k (and all the functions of k) is not a conflict measure between the basic belief assignments. Indeed this value is completely dependant of the conjunctive rule and this rule is non-idempotent -the combination of identical basic belief assignments leads generally to a positive value of k. To highlight this behavior, we defined in [START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF] the auto-conflict which quantifies the intrinsic conflict of a bba. The autoconflict of order n for one expert is given by:

a n = n ⊕ i=1 m (∅). (17)
The auto-conflict is a kind of measure of the contradiction, but depends on the order. We studied its behavior in [START_REF] Jousselme | Conflict measure for the discounting operation on belief functions[END_REF]. Therefore we need to define a measure of contradiction independent on the order. This measure is presented in the next section III.

III. A CONTRADICTION MEASURE

The definition of the conflict (C1) involves firstly to measure it on the bba's space and secondly that if the opinions of two experts are far from each other, we consider that they are in conflict. That suggests a notion of distance. That is the reason why in [START_REF] Jousselme | Conflict measure for the discounting operation on belief functions[END_REF], we give a definition of the measure of conflict between experts assertions through a distance between their respective bba's. The conflict measure between 2 experts is defined by:

Conf(1, 2) = d(m 1 , m 2 ). (18)
We defined the conflict measure between one expert i and the other M -1 experts by:

Conf(i, E) = 1 M -1 M j=1,i =j Conf(i, j), (19)
where E = {1, . . . , M } is the set of experts in conflict with i.

Another definition is given by:

Conf(i, M ) = d(m i , m M ), (20)
where m M is the bba of the artificial expert representing the combined opinions of all the experts in E except i.

We use the distance defined in [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF], which is for us the most appropriate, but other distances are possible. See [START_REF] Florea | Crisis Management using Dempster Shafer Theory: Using dissimilarity measures to characterize sources' reliability[END_REF] for a comparison of distances in the theory of belief functions. This distance is defined for two basic belief assignments m 1 and m 2 on G Θ by:

d(m 1 , m 2 ) = 1 2 (m 1 -m 2 ) T D(m 1 -m 2 ), (21)
where D is an G |Θ| × G |Θ| matrix based on Jaccard distance whose elements are:

D(A, B) =        1, if A = B = ∅, |A ∩ B| |A ∪ B| , ∀A, B ∈ G Θ . (22)
However, this measure is a total conflict measure. In order to define a contradiction measure we keep the same spirit. First, the contradiction of an element X with respect to a bba m is defined as the distance between the bba's m and m X , where m X (X) = 1, X ∈ G Θ , is the categorical bba:

Contr m (X) = d(m, m X ), (23)
where the distance can also be the Jousselme distance on the bba's. The contradiction of a bba is then defined as a weighted contradiction of all the elements X of the considered space

G Θ : Contr m = c X∈G Θ m(X)d(m, m X ), ( 24
)
where c is a normalized constant which depends on the type of distance used and on the cardinality of the frame of discernment in order to obtain values in [0, 1] as shown in the following illustration.

A. Illustration

Here the value c in the equation ( 24) is equal to 2. First we note that for all categorical bbas m Y , the contradiction given by the equation (23) gives Contr mY (Y ) = 0 and the contradiction given by the equation (24) brings also Contr mY = 0. Considering the bba m 1 (θ 1 ) = 0.5 and m 1 (θ 2 ) = 0.5, we have Contr m1 = 1. That is the maximum of the contradiction, hence the contraction of a bba takes its values in [0, 1].

θ 3 0 m 1 : θ 1 0.6 θ 2 0.3 θ 3 0.1 m 2 :
Taking the Bayesian bba given by: m 2 (θ 1 ) = 0.6, m 2 (θ 2 ) = 0.3, and m 2 (θ 3 ) = 0.1. We obtain:

Contr m2 (θ 1 ) ≃ 0.36, Contr m2 (θ 2 ) ≃ 0.66, Contr m2 (θ 3 ) ≃ 0.79
The contradiction for m 2 is Contr m2 = 0.9849. Take m 3 ({θ 1 , θ 2 , θ 3 }) = 0.6, m 3 (θ 2 ) = 0.3, and m 3 (θ 3 ) = 0.1; the masses are the same than m 2 , but the highest one is on Θ = {θ 1 , θ 2 , θ 3 } instead of θ 1 . We obtain:

Contr m3 ({θ 1 , θ 2 , θ 3 }) ≃ 0.28, Contr m3 (θ 2 ) ≃ 0.56, Contr m3 (θ 3 ) ≃ 0.71
The contradiction for m 3 is Contr m3 = 0.8092. We can see that the contradiction is lowest thanks to the distance taking into account the imprecision of Θ. If we consider now the same mass values but on focal elements of cardinality 2: m 4 ({θ 1 , θ 2 }) = 0.6, m 4 (θ 1 , θ 3 ) = 0.3, and m 4 (θ 2 , θ 3 ) = 0.1. We obtain:

Contr m4 ({θ 1 , θ 2 }) ≃ 0.29, Contr m4 ({θ 1 , θ 3 }) ≃ 0.53, Contr m4 ({θ 2 , θ 3 }) ≃ 0.65
The contradiction for m 4 is Contr m4 = 0.80.

Fewer of focal elements there are, smaller the contradiction of the bba will be, and more the focal elements are precise, higher the contradiction of the bba will be.

IV. DEGREES OF UNCERTAINTY

We have seen in the section II that the measure nonspecificity given by the equation ( 6) take its values in a space dependent on the size of the discernment space Θ. Indeed, the measure of non-specificity takes its values in [0, log 2 (|Θ|)].

In order to compare some non-specificity measures in an absolute space, we define a degree of non-specificity from the equation ( 6) by:

δ NS (m) = X∈G Θ , X ≡∅ m(X) log 2 (|X|) log 2 (|Θ|) = X∈G Θ , X ≡∅ m(X) log |Θ| (|X|). (25)
Therefore, this degree takes its values into [0, 1] for all bba's m, independently of the size of discernment. We still have δ NS (m Θ ) = 1, where m Θ is the categorical bba giving the total ignorance. Moreover, we obtain δ NS (m) = 0 for all Bayesian bba's.

From the definition of the degree of non-specificity, we can propose a degree of specificity such as:

δ B (m) = 1 - X∈G Θ , X ≡∅ m(X) log 2 (|X|) log 2 (|Θ|) = 1 - X∈G Θ , X ≡∅ m(X) log |Θ| (|X|). (26)
As we observe already the degree of non-specificity given by the equation ( 26) does not really measure the specificity but the Bayesianity of the considered bba. This degree is equal to 1 for Bayesian bba's and not one for other bba's.

Definition The Bayesianity in the theory of belief functions quantify how far a bba m is from a probability.

We illustrate these degrees in the next subsection.

A. Illustration

In order to illustrate and discuss the previous introduced degrees we take some examples given in the table I. The bba's are defined on 2 Θ where Θ = {θ 1 , θ 2 , θ 3 }. We only consider here non-Bayesian bba's, else the values are still given hereinbefore.

We can observe for a given sum of basic belief on the singletons of Θ the Bayesianity degree can change according to the basic belief on the other focal elements. For example, the degrees are the same for m 2 and m 3 , but different for m 4 . For the bba m 4 , note that the sum of the basic beliefs on the singletons is equal to the basic belief on the ignorance. In this case the Bayesianity degree is exactly 0.5. That is conform to the intuitive signification of the Bayesianity. If we look m 5 and m 6 , we first note that there is no basic belief on the singletons. As a consequence, the Bayesianity is weaker. Moreover, the bba m 5 is naturally more Bayesian than m 6 because of the basic belief on Θ.

We must add that these degrees are dependent on the cardinality of the frame of discernment for non Bayesian bba's. Indeed, if we consider the given bba m 1 with Θ = {θ 1 , θ 2 , θ 3 }, the degree δ B = 0.75. Now if we consider the same bba with Θ = {θ 1 , θ 2 , θ 3 , θ 4 } (no beliefs are given on θ 4 ), the Bayesianity degree is δ B = 0.80. The larger is the frame, the larger becomes the Bayesianity degree.

V. DEGREE OF SPECIFICITY

In the previous section, we showed the importance to consider a degree instead of a measure. Moreover, the measures of specificity and non-specificity given by the equations ( 7) and ( 6) give the same values for every Bayesian bba's as seen on the examples of the section II.

We introduce here a degree of specificity based on comparison with the bba the most specific. This degree of specificity is given by:

δ S (m) = 1 -d(m, m s ), (27)
where m s is the bba the most specific associated to m and d is a distance defined onto [0, 1]. Here we also choose the Jousselme distance, the most appropriated on the bba's space, with values onto [0, 1]. This distance is dependent on the size of the space G Θ , we have to compare the degrees of specificity for bba's defined from the same space. Accordingly, the main problem is to define the bba the most specific associated to m.

A. The most specific bba

In the theory of belief functions, several partial orders have been proposed in order to compare the bba's [START_REF] Dubois | A set-theoretic view on belief functions : logical operations and approximations by fuzzy sets[END_REF]. These partial ordering are generally based on the comparisons of their plausibilities or their communalities, specially in order to find the least-committed bba. However, comparing bba's with plausibilities or communality can be complex and without unique solution.

The problem to find the most specific bba associated to a bba m does not need to use a partial ordering. We limit the specific bba's to the categorical bba's: m X (X) = 1 where X ∈ G Θ and we will use the following axiom and proposition:

Axiom For two categorical bba's m X and m Y , m X is more specific than m Y if and only if |X| < |Y |.
In case of equality, the both bba's are isospecific. Proposition If we consider two isospecific bba's m X and m Y , the Jousselme distance between every bba m and m X is equal to the Jousselme distance between m and m Y :

∀m, d(m, m X ) = d(m, m Y ) (28) if and only if m(X) = m(Y ).
Proof The proof is obvious considering the equations ( 21) and (22). As the both bba's m X and m Y are categoric there is only one non null term in the difference of vectors m-m X and m-m Y . These both terms a X and a Y are equal, because m X and m Y are isospecific and so according to the equation

(22) D(Z, X) = D(Z, Y ) ∀Z ∈ G Θ . Therefore m(X) = m(Y ), that

proves the proposition

We define the most specific bba m s associated to a bba m as a categorical bba as follows: m s (X max ) = 1 where

X max ∈ G Θ .
Therefore, the matter is now how to find X max . We propose two approaches:

First approach, Bayesian case If m is a Bayesian bba we define X max such as:

X max = arg max(m(X), X ∈ Θ). (29)
If there exist many X max (i.e. having the same maximal bba and giving many isospecific bba's), we can take any of them. Indeed, according to the previous proposition, the degree of specificity of m will be the same with m s given by either X max satisfying (29).

First approach, non-Bayesian case

If m is a non-Bayesian bba, we can define X max in a similar way such as:

X max = arg max m(X) |X| , X∈ G Θ , X ≡ ∅ . (30)
In fact, this equation generalizes the equation (29). However, in this case we can also have several X max not giving isospecific bba's. Therefore, we choose one of the more specific bba, i.e. believing in the element with the smallest cardinality. Note also that we keep the terms of Yager from the equation [START_REF] George | Quantification of conflict in Dempster-Shafer framework: a new approach[END_REF].

Second approach

Another way in the case of non-Bayesian bba m is to transform m into a Bayesian bba, thanks to one of the probability transformation such as the pignistic probability. Afterwards, we can apply the previous Bayesian case. With this approach, the most specific bba associated to a bba m is always a categorical bba such as: m s (X max ) = 1 where X max ∈ Θ and not in G Θ .

B. Illustration

In order to illustrate this degree of specificity we give some examples. The table II gives the degree of specificity for some Bayesian bba's. The smallest degree of specificity of a Bayesian bba is obtained for the uniform distribution (m 1 ), and the largest degree of specificity is of course obtain for categorical bba (m 8 ).

The degree of specificity increases when the differences between the mass of the largest singleton and the masses of other singletons are getting bigger: δ S (m 3 ) < δ S (m 4 ) < δ S (m 5 ) < δ S (m 6 ). In the case when one has three disjoint singletons and the largest mass of one of them is 0.45 (on θ 1 ), the minimum degree of specificity is reached when the masses of θ 2 and θ 3 are getting further from the mass of θ 1 (m 6 ). If two singletons have the same maximal mass, bigger this mass is and bigger is the degree of specificity: δ S (m 2 ) < δ S (m 3 ).

In the case of non-Bayesian bba, we first take a simple example:

m 1 (θ 1 ) = 0.6, m 1 (θ 1 ∪ θ 2 ) = 0.4 (31) m 2 (θ 1 ) = 0.5, m 2 (θ 1 ∪ θ 2 ) = 0.5. (32)
For these two bba's m 1 and m 2 , both proposed approaches give the same most specific bba: m θ1 . We obtain δ S (m 1 ) = 0.7172 and δ S (m 2 ) = 0.6465. Therefore, m 1 is more specific than m 2 . Remark that these degrees are the same if we consider the bba's defined on 2 Θ2 and 2 Θ3 , with Θ 2 = {θ 1 , θ 2 } and Θ 3 = {θ 1 , θ 2 , θ 3 }. If we now consider Bayesian bba m 3 (θ 1 ) = m 3 (θ 2 ) = 0.5, the associated degree of specificity is δ S (m 3 ) = 0.5. As expected by intuition, m 2 is more specific than m 3 .

We consider now the following bba:

m 4 (θ 1 ) = 0.6, m 1 (θ 1 ∪ θ 2 ∪ θ 3 ) = 0.4. (33)
The most specific bba is also m θ1 , and we have δ S (m 4 ) = 0.6734. This degree of specificity is naturally smaller than δ S (m 1 ) because of the mass 0.4 on a more imprecise focal element.

Let's now consider the following example:

m 5 (θ 1 ∪ θ 2 ) = 0.7, m 5 (θ 1 ∪ θ 3 ) = 0.3. ( 34
)
We do not obtain the same most specific bba with both proposed approaches: the first one will give the categorical bba m θ1∪θ2 and the second one m θ1 . Hence, the first degree of specificity is δ S (m 5 ) = 0.755 and the second one is δ S (m 5 ) = 0.111. We note that the second approach produces naturally some smaller degrees of specificity.

C. Application to measure the specificity of a combination rule

We propose in this section to use the proposed degree of specificity in order to measure the quality of the result of a combination rule in the theory of belief functions. Indeed, many combination rules have been developed to merge the bba's [START_REF] Martin | Toward a combination rule to deal with partial conflict and specificity in belief functions theory[END_REF], [START_REF] Ph | Analyzing the combination of conflicting belief functions[END_REF]. The choice of one of them is not always obvious. For a special application, we can compare the produced results of several rules, or try to choose according to the special proprieties wanted for an application. We also proposed to study the comportment of the rules on generated bba's [START_REF] Osswald | Understanding the large family of Dempster-Shafer theory's fusion operators -a decision-based measure[END_REF]. However, no real measures have been used to evaluate the combination rules. Hereafter, we only show how we can use the degree of specificity to evaluate and compare the combination rules in the theory of belief functions. A complete study could then be done for example on generated bba's. We recall here the used combination rules, see [START_REF] Martin | Toward a combination rule to deal with partial conflict and specificity in belief functions theory[END_REF] for their description.

The Dempster rule is the normalized conjunctive combination rule of the equation ( 16) given for two basic belief assignments m 1 and m 2 and for all X ∈ G Θ , X ≡ ∅ by:

m DS (X) = 1 1 -k A∩B=X m 1 (A)m 2 (B). ( 35
)
where k is either m c (∅) or the sum of the masses of the elements of ∅ equivalence class for D Θ . The Yager rule transfers the global conflict on the total ignorance Θ:

m Y (X) =    m c (X) if X ∈ 2 Θ \ {∅, Θ} m c (Θ) + m c (∅) if X = Θ 0 if X = ∅ (36)
The disjunctive combination rule is given for two basic belief assignments m 1 and m 2 and for all X ∈ G Θ by:

m Dis (X) = A∪B=X m 1 (A)m 2 (B). (37)
The Dubois and Prade rule is given for two basic belief assignments m 1 and m 2 and for all X ∈ G Θ , X ≡ ∅ by:

m DP (X) = A∩B=X m 1 (A)m 2 (B)+ A∪B=X A∩B≡∅ m 1 (A)m 2 (B). ( 38
)
The PCR rule is given for two basic belief assignments m 1 and m 2 and for all X ∈ G Θ , X ≡ ∅ by:

m PCR (X) = m c (X) + Y ∈G Θ , X∩Y ≡∅ m 1 (X) 2 m 2 (Y ) m 1 (X)+m 2 (Y ) + m 2 (X) 2 m 1 (Y ) m 2 (X)+m 1 (Y ) , (39)
The principle is very simple: compute the degree of specificity of the bba's you want combine, then calculate the degree of specificity obtained on the bba after the chosen combination rule. The degree of specificity can be compared to the degrees of specificity of the combined bba's.

In the following example given in the table III we combine two Bayesian bba's with the discernment frame Θ = {θ 1 , θ 2 , θ 3 }. Both bba's are very contradictory. The values are rounded up. The first approach gives the same degree of specificity than the second one except for the rules m Dis , m DP and m Y . We observe that the smallest degree of specificity is obtained for the conjunctive rule because of the accumulated mass on the empty set not considered in the calculus of the degree. The highest degree of specificity is reached for the VI. CONCLUSION First, we propose in this article a reflection on the measures on uncertainty in the theory of belief functions. A lot of measures have been proposed to quantify different kind of uncertainty such as the specificity -very linked to the imprecision -and the discord. The discord, we do not have to confuse with the conflict, is for us a contradiction of a source (giving information with a bba in the theory of belief

Figure 1 .

 1 Figure 1. Bayesian bba's

Figure 2

 2 Figure 2. Non-dogmatic bba

Figure 3 .

 3 Figure 3. Focal elements of cardinality 2

Figure 4 .

 4 Figure 4. Two non-Bayesian bba's

Table II ILLUSTRATION

 II OF THE DEGREE OF SPECIFICITY ON BAYESIAN BBA.

		θ 1	θ 2	θ 3	δ S
	m 1	1/3	1/3	1/3	0.423
	m 2	0.4	0.4	0.2	0.471
	m 3	0.45	0.45	0.10	0.493
	m 4	0.45	0.40	0.15	0.508
	m 5	0.45	0.3	0.25	0.523
	m 6	0.45	0.275	0.275	0.524
	m 7	0.6	0.3	0.1	0.639
	m 8	1	0	0	1

Table III DEGREES

 III OF SPECIFICITY FOR COMBINATION RULES ON BAYESIAN BBA'S.Yager rule, for the same reason. That is the only rule given a degree of specificity superior to δ S (m 1 ) and to δ S (m 2 ). The second approach shows well the loss of specificity with the rules m Dis , m Y and m DP having a cautious comportment. With the example, the degree of specificity obtained by the combination rules are almost all inferior to δ S (m 1 ) and to δ S (m 2 ), because the bba's are very conflicting. If the degrees of specificity of the rule such as m DS and m PCR are superior to δ S (m 1 ) and to δ S (m 2 ), that means that the bba's are not in conflict.Let's consider now a simple non-Bayesian example in table IV.

		m 1	m 2	mc	m DS	m Y	m Dis	m DP m PCR
	∅	0	0	0.76	0	0	0	0	0
	θ 1	0.6	0.2	0.12	0.50	0.12	0.12	0.12	0.43
	θ 2	0.1	0.6	0.06	0.25	0.06	0.06	0.06	0.37
	θ 3	0.3	0.2	0.06	0.25	0.06	0.06	0.06	0.20
	θ 1 ∪ θ 2	0	0	0	0	0	0.38	0.38	0
	θ 1 ∪ θ 3	0	0	0	0	0	0.18	0.18	0
	θ 2 ∪ θ 3	0	0	0	0	0	0.20	0.20	0
	Θ	0	0	0	0	0.76 0.619	0.619 0.497
	δ S 2-0.639	0.655 0.176 0.567 0.457	0.478	0.478 0.497

Table IV DEGREES

 IV OF SPECIFICITY FOR COMBINATION RULES ON NON-BAYESIAN BBA'S.

		m 1	m 2	mc	m DS	m Y	m Dis	m DP m PCR
	∅	0	0	0.47	0	0	0	0	0
	θ 1	0.4	0.2	0.2	0.377	0.2	0.08	0.2	0.39
	θ 2	0.1	0.3	0.17	0.321	0.17	0.03	0.17	0.28
	θ 3	0.3	0.1	0.12	0.226	0.12	0.03	0.12	0.24
	θ 1 ∪ θ 2 0.2	0.1	0.04	0.076	0.04	0.31	0.18	0.06
	θ 1 ∪ θ 3	0	0	0	0	0	0.1	0.1	0
	θ 2 ∪ θ 3	0	0.2	0	0	0	0.18	0.1	0.03
	Θ	0	0.1	0	0	0.47	0.27

Acknowledgments

The authors want to thanks BREST METROPOLE OC ÉANE and ENSTA BRETAGNE for funding this collaboration and providing them an excellent research environment during spring 2010.


Paul Backes

Max Bajracharya

Daniel Helmick

Oussama Khatib

Vincent Padois

James Warren

Rover-arm Based Coring with Slip: Technological Development and Control Approach

Technology to enable core sample acquisition from a low-mass rover on slopes is being developed. A rotary percussive coring tool was integrated onto a low-mass rover and sensing, control, and simulation technologies are being developed. Initial results indicate that coring with modest rover slippage will be feasible. 1

I. INTRODUCTION

Technology to enable core sample acquisition from a lowmass rover on slopes is being developed. The current stateof-the-art in coring and drilling from a planetary rover armmounted tool, represented by the Mars Science Laboratory mission, assumes that the rover is a stationary platform. Future missions could benefit by enabling arm-mounted drilling or coring where the rover may experience modest slippage during the drilling or coring operation. To enable such a scenario, a rotary percussive coring tool is being integrated onto a rover and sensing, control, and simulation technologies are being developed. The rotary percussive coring tool has mass and impact energies similar to expected future planetary coring tools and it is being integrated onto an 85 kg six wheel rocker-bogie rover. Sensing includes sensing to detect slippage and sensing for control to accommodate slippage during the coring operation. Slip will be detected using measurements of filtered force and torque values, coring tool current, and visually detected rover motion relative to the environment. It is assumed that the rover slippage will be slow relative to the speed at which the arm can be reconfigured to accommodate the slip. If the rover slips at a greater rate, then a separate recovery action will be taken. The arm will be automatically reconfigured to accommodate the rover slippage using control based on force-torque and motor current feedback. If the arm reaches the edge of its workspace, then the rover will autonomously drive to place the arm back into its operational workspace and the arm, with tool remaining in the rock, with be reconfigured accordingly. A simulation environment is being developed to simulate the coring operation. An example of the Rocky 8 rover on a slope in the JPL Marsyard is shown in Figure 1.

Figure 1. Rover Slope Slip Test

II. OPERATIONS SCENARIO

For the work described in this paper, it is assumed that any rover slippage will be slow compared to the rate at which the arm can be reconfigured to accommodate the slip, to allow the coring tool to remain in the coring hole during reconfiguration. For example, the rover might slip 20 cm during an hour long coring operation. The operational scenario for coring and accommodation of rover slippage during coring is described below.

1. The rover drives onto a slope so that a science target is within the workspace of the rovermounted manipulator. The rover could approach the target from any direction.

2. The arm is deployed and the coring tool is placed on the target and coring is initiated.

3. While the coring operation is active, sensing is active to detect rover slippage.

4. If rover motion relative to the coring hole is detected, then the arm is automatically reconfigured to accommodate the slip if the rover slip is within a safe bound. The rover is not actively moved. If the slip is beyond a safe bound, then the coring operation is stopped and the tool is removed from the hole.

5. If the arm approaches the edge of its workspace due to accommodation of rover slip, then the coring tool is turned off and the rover is moved so that the target is back near the center of the arm workspace.

Then the coring operation is continued.

III. SLIP SENSING

Several sensing approaches will be active during the coring operation to detect motion of the rover relative to the coring hole. A six-axis force-torque sensor will detect binding of the coring tool as lateral forces between the coring tool and hole and as torques on the tool perpendicular to the tool axis. Coring tool current and rotation speed will be monitored to detect binding of the tool -increased current or reduced speed will be indicators of tool binding. Absolution Motion Visual Odometry (AMVO), described below, will measure absolute motion of the rover relative to its pose at the start of the coring operation.

2

IV. ABSOLUTE MOTION VISUAL ODOMETRY

Absolute Motion Visual Odometry (AMVO) is a technique for measuring the motion of a rover to sub-millimeter precision for relatively small motions of the vehicle (< 50 cm). We use this technique to accurately detect and measure the slippage that a rover undergoes while collecting a sample from the environment. AMVO uses the exact same algorithm as visual odometry (VO) [START_REF] Helmick | Slip-Compensated Path Following for Planetary Exploration Rovers[END_REF][START_REF] Maimone | Two Years of Visual Odometry on the Mars Exploration Rovers[END_REF] but it uses the concept of a "reference image pair" to avoid the well established measurement drift associated with VO. We are able to use the technique because we expect that the rover will not slip a significant amount (up to ~50 cm) before we choose to stop sampling or to accommodate the slippage.

The "reference image pair" is taken before the sampling is started. Then the each subsequent image pair taken for motion estimation is processed along with the reference image pair using the visual odometry algorithm. We will briefly describe the visual odometry algorithm here, for a more in depth explanation see [START_REF] Helmick | Slip-Compensated Path Following for Planetary Exploration Rovers[END_REF][START_REF] Maimone | Two Years of Visual Odometry on the Mars Exploration Rovers[END_REF]. VO takes two image pairs as inputs (in our case, the reference image pair and the motion image pair). Initially, it finds a set of features in the first image pair based on an interest operator (i.e. Forstner). Then it localizes each of the features in 3D using stereo triangulation. Next, it finds a set of features in the second image pair (using an estimate of their image location from an estimated motion input) and performs stereo triangulation on these as well. It then attempts to match features from each of the image pairs with one another. Finally it estimates 6-DOF motion using a two step process. The first step is a rough motion estimate using a singular value decomposition technique called Schonemann motion estimation. The second step is a maximum likelihood motion estimation that refines the first estimate.

By using the reference image pair for each motion estimate we get an estimate of the "absolute motion" of the rover and we are able to avoid the drift that occurs from integrating the relative estimates associated with standard VO, thus enabling extremely precise motion estimation.

In the first experiment, a sequence of 500 image pairs were taken with the rover hazcams without any rover motion to find 'minimum bound' of slippage estimation during sampling using the AMVO technique (see Figure 2). Submm (potentially <0.1 mm) motion detection can be achieved using this technique (the 3σ variance of this signal is 0.089 mm). For MER class manipulators this would correspond to <5 Newtons of tool force which is of a similar accuracy as most force sensing techniques. A second experiment was to command the rover the smallest possible motion. This resulted in a motion estimate of 0.54 mm (see Figure 3), which was easily detected by AMVO.

A third experiment was to simulate sample collection on a sandy slope by attaching a vibration motor to the rover while on a slope of 16.2° (pitch) and [START_REF] Khatib | A Unified Approach to Motion and Force Control of Robot Manipulators: The Operational Space Formulation[END_REF].2° (roll). With AMVO, a slip of 1.6 mm was detected. A picture of the slope taken by the front hazcam is shown in Figure 4.

Results of the experiment are shown in Figure 5.

V. DRIVING

Driving the rover back into the arm workspace will require measurement and accommodation of rover slip during the drive. The goal of the rover drive subsystem is to drive the errors of the three controllable DOFs of the rover to zero [START_REF] Helmick | Slip-Compensated Path Following for Planetary Exploration Rovers[END_REF]. The three controllable DOFs of the rover are x, y, and yaw. When the rover slips to the edge of the arm workspace, the sampling process will pause and this driving with slip correction algorithm will be invoked. Simultaneously, the arm will accommodate the rover motion using force control.

VI. CORING TEST

A coring tool was attached to an 85kg rover and coring tests were performed. The coring test setup is shown in Figure 6. The coring bit and resulting hole and core are shown in Figure 7. The LSAS coring tool [START_REF] Stanley | The Low-force Sample Acquisition System[END_REF] was attached to the Pluto rover in the JPL Planetary Robotics laboratory for the tests. The LSAS tool is a rotary percussive coring tool and weighed 400 grams and required 15 N average weight on bit. Maximum impact forces felt by the arm were about 60 N. A 2 cm core was acquired at a rate of about 2 cm/hour in limestone. A longer core could have been acquired. Cuttings were removed by a vacuum (an alternative approach would be needed for a Mars application). Coring started easily without a centering bit. An intact core was acquired. This test was significant in that it demonstrated that coring from a low mass rover is feasible.

VII. SIMULATION RESULTS

The use of a high fidelity simulation environment allows for reduction in the cost of experimentation associated with hardware and software rover design. SAI [START_REF] Khatib | Robotics and Interactive Simulation[END_REF] (Simulation and Active Interface) is the simulation tool developed at Stanford University. It is used within the framework of this project to assist the design of the physical rover. It is also a valuable tool to analyze the behavior of a low mass rover coring rocks on slopes. The controller design stage can also benefit from such a simulation tool since it provides a low cost means to validate controllers developed for the rover.

4

SAI is unique in that it is a real-time interactive environment that allows the user to apply and sense forces within the virtual world via haptic devices. SAI features multi-contact resolution for multi-body systems, efficient algorithms for articulated body dynamics, and simulated friction and ground reaction forces. SAI is based upon a general framework [START_REF] Ruspini | Collision / Contact Models for Dynamic Simulations and Haptic Interaction[END_REF] for the resolution of multi-contact between articulated multi-body systems in the context of operational space control for robots [START_REF] Khatib | A Unified Approach to Motion and Force Control of Robot Manipulators: The Operational Space Formulation[END_REF]. Using this framework, the dynamic relationships between all existing contact points can be described. These relationships are characterized by the masses as perceived at the contact points. A force exerted at a contact point, whether from a collision with another object or from interaction with a user, can be translated into forces at all related contact points. The necessary computations can be performed with an efficient recursive algorithm.

The contact space representation allows interaction between groups of dynamic systems to be described easily without having to examine the complex equations of motion of each individual system. A collision model can be developed with the same ease as if one were considering interaction only between simple bodies. Impact and contact forces between interacting bodies can then be solved efficiently. SAI was used to simulate the slippage occurring while coring on a terrain with slope (pitch: 16.2°, roll: 6.2°). A model of the forces induced by the coring tool operation was input to SAI and simulations were run where the manipulator configuration was held fixed using a stiff joint controller. The resulting slippage of the wheeled platform as well as the wheels' orientation were shown to be similar to empirical results obtains with the Rocky 8 rover. This simulation however required an adaptation of the friction model in SAI. SAI utilizes rigid-body dynamics and cannot utilize friction models that include wheel sinkage and deformable terrain. To model this behavior, SAI represents friction as a combination of static and Coulomb friction where the friction coefficients change as the rover slips. Although this is a general model, for initial testing we use a linear relationship relating the friction coefficients to distance, to represent wheels digging in. We also use SAI to develop an algorithm that computes the optimal coring configuration given the coring conditions: terrain slope, coring point location, ground type. This requires the definition of the criterion use to characterize the optimality of a given configuration. We used different manipulability measures [START_REF] Walker | Impact configurations and measures for kinematically redundant and multiple armed robot systems[END_REF] so that the force capabilities of the robot along the coring tool axis are maximized and the overall manipulability is maximized in all operational direction. We are still working on incorporating a third requirement associated to the fact that the effects of disturbance forces at the end-effector should be minimized in terms of slippage.

Finally, SAI is used to test different control strategies for the coring task. In order to provide the system with a model that is generic enough for force analysis purposes, we chose to use the free-floating representation of the system. This representation does not assume the contact of the wheels on the ground as a granted fact. The system is rather represented as a floating system subjected to gravity and ground reaction forces. This is a very general description framework which is an interesting characteristic for analysis. However this induces the necessity to describe which constraints have to be enforced for the wheel/ground contact to be maintained. This constraint can for example be expressed as: "null acceleration for the wheel/ground contact point in the contact normal direction". Given this virtual task, any other task assigned to the rover has to be realized without violating this highest priority task, i.e. has to be realized in the null space of the wheel/ ground contact task. Within this framework, a controller for the coring task is being integrated and tested in SAI.

VIII. CONCLUSIONS

Technology to enable coring from a slow-mass rover is being developed.

Absolute motion visual odometry (AMVO) was tested to show that small slip motion of a rover can be measured. A coring test was performed that demonstrates the feasibility of coring from a low-mass rover. The SAI simulation environment is being used to simulate the coring system. Coring control is being further developed with results anticipated during the summer of 2007 that will show that coring with modest rover slippage will be feasible.

Figure 2 :

 2 Figure 2: AMVO Minimum Bound Test

  : 'Absolute Motion' VO Test (motion estimate = 0.54mm)

Figure 3 :

 3 Figure 3: AMVO Small Motion Test

Figure 4 :Figure 6 :Figure 5 :

 465 Figure 4: Picture of Slope Taken by Front Hazcam

Figure 7 :

 7 Figure 7: Coring Bit and Resulting Hole and Core

IX. ACKNOWLEDGEMENTS

The research described in this publication was carried out at the Jet Propulsion Laboratory, California Institute of Technology, and at Stanford University under contract from the National Aeronautics and Space Administration (NASA), with funding from the Mars Technology Program, NASA Science Mission Directorate.