Lunar Rover Navigation

8/3/2019 Lunar Rover Navigation

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A Robust Autonomous Navigation Strategy Of Planetary Rover In ThreeDimensional Terrain Using An Obstacle Map Generation And Hybrid Global

Potential Field

Abstract

Presented herein is an approach for effective obstacle map generation and path planning for anautonomous planetary rover in a general three dimensional environment. The map generation is done

with the help of grid abstraction. A hybrid, global potential field is used for path planning and efforts

have been made to solve the common problems of conventional potential field methodologies.

Introduction

Robotics has been a culmination of the human quest for emulating the nuances of biological behaviour

in intrinsically inanimate mechanical creations. The field has seen enormous growth with therealization of numerous applications for such biological mimicry. One of these contemporary advances

in the field has been in planetary robotics. Missions to extraterrestrial worlds present unique challengesand requirements with respect to the possibility of using planetary rovers to navigate across planetary

surfaces. The harshness of the inhospitable environments that exist on other extraterrestrial bodiesmakes it necessary for rovers to be available to navigate throughout the unknown terrain autonomously

in order to obviate the need for humans to do the same. Hence the need is for the rover to be capable of

perceiving the environment around it and realizing its details with respect to its traversability, thenplanning a possible and optimal path to its destination.

Such a task, although appearing mundane in human terms, is uniquely challenging for an autonomousrobot, more so if the robot is meant for applications hundreds of thousands of miles from the human

instructors. Space robotics also presents the problem of constraints in terms of resources, especially

computation cost and power. Thereby, any such process needs to take into account these considerations.The technique also needs to be robust enough to tackle with unforeseen terrains on the surface of themoon without human intervention. For generating the usable map of its surroundings, the kinematic

constraints of the rover have to be taken into account for the various points on the map to determine the

ease with which the rover could occupy a specific region on the map. Depending upon the suspectedinstability of certain regions of the map, they are duly marked as no-go regions and the path planned is

expected to avoid those regions absolutely. Besides regions where the rover cannot rest in a stable

fashion, there will also exist regions with high terrain irregularities where the rover will find it difficultbut not impossible to traverse. The path obtained should try to accommodate the intractability of these

regions and be changed accordingly, but only to the extent of not straying too far from the intended

path.

The methodology applied for data acquisition about the environment is the structured light scanning

technique. The resultant output of the scanning of the 180o region in front of the rover is a set of

scattered three dimensional points in terms of their X, Y and Z axis details with respect to the rover'sreference frame. These points are then subjected to the point analysis to find out the local gradients and

realize the traversability in the particular regions that is. grids, into which the point data is thereafter

reduced. The identification of forbidden regions stands for the marking of obstacles in the mappedspace. The rest of the region is perceived as a region of gradients, through which a path has to be

planned.


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The path planning technique involved is the potential field method. The potential field method is noted

for its heuristic and computationally less expensive nature. However, the nature of the potential field is

crucial in determining its robustness. The method is known to have a lot of drawbacks like the presence

of local minima in which case there is a complete failure. Ways to escape local minima, onceencountered, have been propounded and work to various degrees of success. The employment of

harmonic functions to prevent their occurrence in the first place is also prevalent, but there remains the

existence of saddle points and irregular motion even for simple obstacle regions. The proposed methodis meant to give a sub-optimal path without the problems of commonly occurring local minima, which

is robust enough to be applicable to real environments and maps. This method has shown promise in

overcoming all the common failure cases of normal potential field methods as well as being applicableto maps derived from actual structured light scans.

The path obtained from the potential field method can then be subjected to gradient considerations to

modify the path within permissible constraints to avoid highly irregular regions. Such a path then canbe communicated to the rover and the traversal toward the goal point can be undertaken.

Rover Features and constraints

The rover is based on a rocker-bogey mechanism which has been shown to be resilient in traversing

through tough terrains. The rover accordingly possessed six wheels, with six independent driving

motors and four sets of turning motors. This arrangement gives the rover the capability of tacklingdifficult terrain constraints. With safety considerations in mind, the maximum allowable vertical and

lateral gradient has been set at 35o. This determines the extent of continuous slopes it can traverse.

However, in real situations most terrain features are abrupt objects with slopes far greater than the

maximum allowable gradient but which are passable by virtue of their small dimension with respect tothe dimension of the rover and its wheels. Hence, another terrain parameter with regards to the

maximum instantaneous change in height the rover can go over, is considered as the maximum

allowable threshold height for detecting obstacles.

These considerations effectively demarcate the boundary conditions for the stability of the rover and

the violation of these entail the instability of a region, thereby necessitating those respective regions to be suitably marked as untraversable. It is evident that the realization of this is not a binary

consideration of just the two conditions. The demarcation also has to deal with cases that fall within the


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conditions and should be robust enough to work for all types of natural terrain features.

Structured Light Method for Terrain Mapping

The functioning of the lunar rover makes it imperative for a system to be in place to inform the rover of

the nature of its surroundings. The nature of the boulders, rocks, crevices and inclines around it is

supposed to influence all its path planning decisions. Such detailed information has to be effectivelyobtained as usable data that can be used further on. Its mapping system should be able to generate a

map that is capable of detecting, within acceptable error limits, the heights, depths and distances of theterrain features around it, within its sensing range.

This static preprocessing map data will then have to be updated with real time information about terrain

slippage, deformations and changes in the rover's position and orientation. However, the accuracy of allsubsequent processing will depend upon the accuracy of the mapping system.

The process used herein is the structured light scanning of the surrounding region using a diode laser.In this approach a known pattern of light is projected onto the three dimensional surface and the

deflection recorded from a perspective view of the pattern. This degree and direction of deflectiongives the height or depth and position of the points on the surface. Thereby, we get the data about thepositions of points in the three dimensional region around the point of sensing.

The advantages of this method is that it is relatively much more computationally less expensive whencompared to vision system based mapping methods. Hence, it presents applicability to the rover's use.

However, there are disadvantages to this technique as well. The accuracy is dependant on the

calibration of the sensing system. This problem can be minimized by careful calibration. Another

drawback is that the output is scattered point data. That means, we are liable to find no data between


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two points. If this blind region is large enough, it can cause problems. Another problem , intrinsic with

this method is caused by the scattering of light on various types of surfaces. Since the accuracy of the

data obtained depends upon the accuracy of the sensed deflection, the scattering of the deflected light

pattern is bound to give aberrations in the output. Also the presence of intense ambient light is alsoliable to interfere with the sensing. This problem is dealt with by using light of a nature that is less

prone to scattering, ie. a laser light. The use of photochrome glass as to filter the light can be used to

minimize the effects of ambient sunlight.

Moreover, the use of scattered light method is found to provide very detailed terrain features in the

form of scattered point data within its range of sensing.

Map Generation Process

The process of conversion of the rover's three dimensional environment into it's notion of traversableand untraversable spaces , can be represented by five steps, namely structured light data acquisition,

grid abstraction, traversability classification, obstacle region clustering, polygon fitting around

obstacles.

This representation of the three dimensional environment can further process for realizing a path that

navigates through the traversable regions, while avoiding the untraversable ones.

Structured light data acquisition

A 650 nm diode laser mounting for the production of a single strip of structured light is used to sense

the contours of the three dimensional environment. The calibration is done with respect to a flat ground

and vertical gradient. The deflections from such a reference laser line of the actual perceived laser linegives the measure of the height or depression of the surface at that point. 100 such readings at angular

steps of 1.8' are used to generate a contour estimation of the 180' field of view in front of rover. Every

reading consists of 480 discrete point data along the sensed laser line. Thereby, we obtain the heightdata about 48000 discretely identifiable points in front the rover.

The distribution of these 48000 points depends upon the radial distance from the rover as well as thenature of the surface. As the radial distance from the surface increases, the angular deviation of 1.8' is

expected to translate into farther separated laser lines, hence farther separated data points. The nature of

the surface also determines the regions the points represent. Since the points pertain to the points on the

laser line, and deflection of the laser line necessitates the absence absence of the data at regions wherethe laser line cannot go, that is the shadows behind obstructions to the laser line.

Hence the output of the structured light analysis is a set of 48000 discrete, scattered data points withcorresponding estimated heights.


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Fig: Experimental Set-up

Fig: Laser Line Data

Fig: Noisy contour data till radial distance of 150cm

Grid Abstraction

There are several reasons for the abstraction of the discrete point data obtained from the structured light

system, into grids, representing the three dimensional environment, with minimal loss of usefulinformation.

Firstly, a point, in itself has no importance whatsoever in realizing the traversability at that point or

region. For example, the height obtained at a point may be very high, that is , higher that the maximum

vertical obstacle the rover can climb. This information in itself may imply that the point is the peak ofsuch a vertical obstacle, in which case it is untraversable. Conversely, the point might as well be lying

on a long gentle slope that reaches that height at the point, in which case such a low slope, that is,

within the threshold gradient climbable by the rover, is evidently traversable. Hence, there is a need to


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look at more than a single point at a time for finding out necessary information about the space.

Secondly, the nature of the intended work environment and applications of planetary rovers impose

certain restrictions on the computational capabilities of the system. Remote operation from the missioncontrol station is not possible for real time functioning. Hence, a major part of the computation to be

done by the rover independently of the mission control station has to be computationally less

cumbersome. The processing of 48000 points, or even more if greater accuracy is desired, for thepurpose of terrain estimation is thereby not feasible to be performed at every step. Therein lies the need

to reduce the amount of data used to represent the environment, with minimum loss of useful

information, as mentioned before.

These problems have been resolved by reducing the discrete, scattered point data into a cell grid ,

representing the entire sensed environment. Let us consider that there are 50 cells in the X direction and

25 cells in the Y direction, of dimensions of about 10cm*10cm, subject to the calibration and range ofthe structure light set-up. Every grid cell contains the following information :

[xmax

, ymax

, zmax

, xmin

, ymin

, zmin

, zavg

, n, cell state, sd]

x max , y max, z max are the coordinates of the point in the cell with the maximum height

xmin

, ymin

, zmin

are the coordinates of the point in the cell with the minimum height

zavg

is the average height for all points in the cell

n is the number of points detected in the cell

cell state (0,1,2), 0 when the cell is free, 1 when the cell has obstacle, 2 when the cell can't be sensed.sd is the standard deviation of all the points that belong to the cell about z

avg.

Fig: Cellular gradient data, top view and 3-d perspective


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Traversability Classification

After the three dimensional environment has been successfully represented by the grid, the need arises

to somehow make the rover differentiate between the cells in the grid as far as traversability isconcerned. However, such a measure must be robust and must take into account not the traversability

with respect to the current position and orientation alone, but also the position and orientation with

respect to the surroundings of the cell.

Every cell records the maximum and minimum heights , thereby the we approximate the worst possible

gradient in the cell. Record the number of points encountered in the cell , so as to identify the shadowregions behind obstructions to the laser line, where there is no height data. The standard deviation and

mean measures are useful in noisy data isolation.

If the number of points detected in a cell is less than two, that is the minimum number of pointsnecessary to estimate the gradient, then the cell is marked as unknown. However, the behaviour of the

rover with respect to the unknown cells needs to be determined. Conservative estimates can mark

unknown cells as untraversable, that is worst case assumption, or traversable, that is best caseassumption. However, such a procedure is not desired, since both will lead to erroneous approximations

of the environment. It is found that, since such an unknown cell is the result of a obstruction that lies

directly between that cell and the initial (0,0) starting position, the traversability of the unknown celldepends directly on the traversability of the obstruction causing it. The existence of the shadow implies

that the contour in that cell lies behind the obstruction causing cell, and the height of the contour is also

less than the height of the obstruction causing cell. Since the height is bound to be less, if theobstruction causing cell is traversable then the shadow it causes is definitely traversable and if

untraversable then the shadow behind it may or may not be traversable. Since we have no data about

this region, and we cannot safely comment on the nature of the surface there, we mark it as

untraversable.

In cases where enough points are obtained in the cell to make reasonable estimations about the nature

of the surface in that cell, three restrictions pertaining to the motion of the rover have to be dealt with.

The rover cannot navigate in extended regions where slope angle is greater than 35'

The rover cannot navigate over vertical surfaces greater than a threshold height, here 10cm

The rover can navigate over surfaces which are inclined at angles greater than 35', and whose

height is less than 10 cm, but only when the size of the obstacle is small enough, that is less

than the dimension of the rover. Thereby, the entire rover will not rest on such an inclinedsurface.

Every cell is checked against these conditions and marked as traversable or untraversable accordingly.

As an output, distinct regions where the robot should and should not be are obtained.


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Fig: Grid cell classification- passable(white),impassable(red),unknown impassable(black)

Obstacle Region Clustering

The marked cells in the grid thereby obtained are still discrete data. In order to introduce the notion ofobstacles, in such an environment, the need arises to define obstacles aptly. An obstacle is a thing that

blocks the way or prevents or hinders progress. Till the cell marking in the grid, the rover's notion of

obstacles is still discrete cells as passable or not. However, therein lies a stark contrast with humaninterpretation of obstacles. Humans perceive entire objects as obstacles, not small regions of gradients

and heights. An aggregate region of such high heights and or gradients are perceived as one such

obstacle object. Such a view of the environment is necessary to plan strategies to avoid the entireobstacle, and not just one cell at a time. The dimension of such an obstacle, apart from the natural

grouping of the cells, is also dependent on the dimension of the rover, since narrow paths that cannot be

passed are effectively obstacles for the rover. Hence the identification of obstacles requires a clustering

of the cells in the grid.

Considering only the untraversable cells in a grid, they are clustered into clusters depending upon the

condition that any two cells belonging to two separate clusters can never be closer than the thresholddistance that permits the motion of the rover. The clusters so obtained represent the rover's notion of

obstacles in it's surrounding environment.

Fig: Clustering into obstacles(yellow and magenta)


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Polygon Fitting Around Obstacles

This step is necessitated by the path planning strategy adopted. The application of potential fields needs

the characterization of obstacle surfaces, for determining and applying the repulsive forces. Hence theboundary of the obstacle has to be approximated. For generalization, it is desirable to find an uniform

representation for all obstacles, here a polygon. Since the polygon can have intractable number of sides

to exactly represent the region, we have to find an optimum sided polygon that effectively representsthe region, for the current purpose. Application of a convex hull can approximate the nature of such a

region quite optimally, but with the immense drawback of efficient output only in case of convex

regions. Concave regions give rise to largely inaccurate approximations, in case of convex hulls.Thereby, the concept of alpha shapes is used to meet the current need. An alpha shape is obtained by

moving a circle of a pre-set radius, around the region and recording the points on the region it touches

tangentially. If we set the size of such a circle as the rover circle, then this will give an accurate

approximation of the region with respect to nooks and concavities of the regions where the rover can orcannot enter. Thereby we get an optimal polygon from the obtained points in the alpha shape.

Fig: Polygon fitting around obstacles

Error Management

In reality, sensor error exists and even in the visible portion of the environment there are points on thelaser line wherein no height information can be retrieved, in the image captured by the camera. Such

points are not blind spots, ie. Lying behind obstacles, but should be dealt with in order to differentiate

between sensor error and sensor blind spots. The points on the laser line which cannot be sensedproperly are replaced by the weighted average of the last correctly sensed point on the laser line and the

next correctly sensed point on the laser line, where the weight is determined by the proximity of that

sensed point from the point under consideration. Thereby we get a fairly accurate approximation of

regions where we cannot sense the laser line. Hence the output set of 48000 points does not deviate

arbitrarily from the real environment it tries to represent.

The outcome of the grid abstraction and subsequent classification does not pose the problem ofinaccurate representation because the classification abides by the stability constraints imposed on the

rover. Since this classification is not merely local in nature, because it checks the region around it for

considerations like the dimensions of the obstacle or length of the slope, the consequent clustering andpolygon fitting only leaves regions in space which are accessible as free. However, inside the regions

marked as obstacles, there may exist local free space regions, not big enough or continuous enough to

enable the rover to safely navigate. Thereby, these regions can be safely rejected as obstacles. Thereby,


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there is a global obstacle identification of regions which the rover should definitely avoid. The rest of

the region can be traversed by the rover in various degrees of difficulty, depending upon terrain

roughness and slope, but they are traversable nevertheless. These further non critical terrain features

should be dealt with during path planning, but the priority for the terrain negotiation is always given tothe marked obstacles, since they are definitely not traversable

Conventional Path Planning Strategies

The primary criterion for the effectiveness with which a mobile robot, including a planetary rover, can

tackle a new environment, with limited knowledge about it, deals with its navigational competence, ie.It should be able to navigate through the any environment and reach the goal state, if it is reachable

under its physical constraints. In specific cases, such results can be obtained using various techniques.

All possible paths can be explored and the best one chosen. This is sure to give the desired result.

However such complete scanning techniques can prove intractable in case of complex environmentswhere the number of paths is too high to compute efficiently. There are techniques that reduce the

environment into cells , of fixed or variable sizes, and then employ grassfire algorithm. Connectivity

approaches apply visibility graphs and Voronoi diagrams to determine the connectivity of free spacearound the rover and derive paths from visibility limits around obstacle corners or from the regions

farthest from the obstacles in Voronoi diagrams.

Other low level approaches are also present, that employ wall following tendencies. The Bug algorithm

and the tangent Bug algorithm give sure paths if possible but massively sacrifice completeness for

optimality and safety, The result is a path that clings to obstacles and fails to give any optimal results ingeneral cases.

However, Potential Field methods have proved very easy to implement and are extensively applied to

mobile robots. Because of the reduced computation and the versatility of its features, potential fieldcan be applied to the planetary rover to enable it to obtain a complete , competent navigation system.

Potential Field Approach

The potential field approach looks to emulate the effect of natural potential fields on particles that

experience a force when placed in that potential field. The potential field consists of forces on theparticle, here a rover. These forces should exist in all points in space and represent the gradient descent

of the potential function. Thereby, the effect of potential field is to move the particle down the gradient

of the potential function till it reaches a stationary point (necessarily a minima).

This resembles the action of the rover in starting at the start point and reaching the goal. The function

therefore has to be designed such that the potential has a minima at the goal and the field guides the

rover to the goal. For this purpose an attractive potential has to be maintained for the goal, ie. afunction whose functional value decreases with decreasing distance from the goal, and consequently,

whose gradient descent is towards the goal, where the distance to the goal becomes zero.

The other element in the potential field has to be the consideration of obstacles. The mere effect of the

attractive potential is to attract the rover towards the goal. However, it has no purpose in avoiding

obstacles, which might fall in its way towards the goal. When the rover encounters such obstacles, theobvious reaction should be a strongly repulsive one away from the obstacles. The repulsive force in a

direction denoted by 'away from the obstacle', is obtained by a force directed along the outward from

the surface of the obstacle nearest to the rover. The effect of this repulsive force should be strong


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enough to prevent the infiltration of the rover body inside the obstacle under all circumstances. This

implies that, whatever the attractive force trying to propel the rover into the obstacle, the repulsive

force should to strong enough to prevent this from happening. Thereby, ideally, the repulsive force

should have a value of infinity at the obstacle surface, so as to be independent of the value of theattractive force. However, this intense repulsion cannot persist as the distance from the obstacle

increases, since that would practically cripple any movement towards the goal, in the presence of any

obstacles in its path providing infinite repulsion. So the effect of the repulsive force should not be feltat large distances from the obstacle. Thus the desired nature of an ideal repulsive force for the obstacle

has functional value of infinity at the obstacle surface and zero at infinite distance, reaching

manageably low values after a particular distance from the obstacle.

Fig: Positions of obstacle point and goal points. Combined Repulsive and Attractive forces.

The combined effect of an attractive force towards the goal as well as a repulsive force that makes the

rover avoid obstacles, is a potential field that can possibly guide to the rover to the goal , avoiding

obstacles in its path. Since the force magnitude and direction can be calculated easily at every point, to

be translated into motion for the mobile rover, as the effect of the force, in order to the reach the nextpoint, cumbersome calculations can be avoided, and the simple force evaluation is enough to serve the

purpose of navigation.

Problems in potential field approach

The potential field approach, due to its inherent simplicity, also encounters some drawbacks that hinderits completeness and robustness. Firstly, due to the nature of the field as the summation of multiple

vectors representing different behaviours of the rover, ie. attraction to goal and repulsion to obstacles,

there can arise situations where the vector summation comes to zero. Such a situation is known as alocal minima, which lies in positions apart from the goal and hence attracts the rover in the region

around it, since the field follows the gradient descent. Thereby, if the rover reaches the local minima, it

is trapped. Such a situation may be caused by a particular configuration of obstacles, as well as by

special type of trapping obstacles like the box canyon.


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Fig: Situations of local minima: Single arrow represents Attractive force towards goal. Double headed arrow represents

repulsive force from the obstacle

There are ways to escape from local minima, which include backtracking along path and employing

evasive manoeuvres. Instead of backtracking, random movements may be undertaken with the intentionof escaping the attraction of the local minima. Some specific strategy, like wall following may be

employed when such a local minima is encountered. Much more complex potential fields like harmonic

potential functions can be used, so that the minima exists only at the goal. The potential field, in othercases is changed locally, when such a local minima is encountered, so that the local minima no longer

traps the rover.

However, the methods for avoiding or eliminating local minima pose certain compromises that have to

be made. In case of backtracking, random movements, wall following approaches, and altering the field

after the local minima has already trapped the rover, there is a huge loss in optimality as a far more

tedious and longer path is followed. Conventional harmonic potential fields, on the other hand, arecomplex and difficult to compute.

Progress towards the goal can also be made unstable in situations where oscillations are caused

between two obstacle surfaces. The surfaces can belong to the same concave obstacle or to differentobstacles, but the field causes the rover to go to and forth between the two surfaces, which can cause

highly irregular motion.

Fig: Inability to pass through obstacles placed close together. Oscillations in passages.

Another critical drawback of the potential field is its inability to guide the rover through closely spaced

obstacles in all cases. This prevents the field from being robust enough to tackle small gaps betweentwo obstacles in its way, since the attractive force might not be strong enough at all places to pull it

through the gap.


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These disadvantages associated with the potential field approach offers a serious impediment in

implementing this method in real world applications without any alterations that can circumvent these

problems.

Introduction of new scheme

Keeping in mind the inherent limitations of the potential field methodology, the need it to modify it so

that it becomes robust and complete, capable to handling any general obstacle filled environment.

However, the method has to retain the advantages of the potential field method, chiefly its simplicityand reduced computational complexity. The resultant path obtained should also be as optimal as

possible.

It is obvious that the potential field is incomplete in its present form because the two behaviours attraction towards goal and repulsion from obstacles sometimes are in direct conflict and the repulsive

forces impede the attainment of the goal state. There can be no further effect of the attractive force than

to attract the rover towards the goal. The basic purpose of the repulsive force as an impulse that pushesthe rover away from the obstacle surface is also justified with its specific behaviour. However, when

taken together, they fail to satisfy completeness. Thereby, we need to add a new force that takes a more

global view of the environment than just a single surface per obstacle and the goal point.

However, taking into account the computation constraints, the new force has to be easily calculable.

The intent of this force has to be to mitigate all the drawbacks posed by normal potential fields. Thisincludes the elimination or avoidance of local minima as well as prevention of the occurrence of

oscillations.

To go about exploring such a force, a method that is complete but far from optimal is looked into- theBug algorithm. There are two kinds of Bug algorithm that are commonly used. They are , the Bug 1

and the Bug 2. In the Bug 2 algorithm , the rover moves towards the goal until it encounters an

obstacle. It then initiates the wall following behaviour and starts going round the obstacle till it canmove in the direction of the goal again, at which point it leaves the obstacle wall and proceeds towards

the goal.

Fig: Paths generated by the Tangent Bug and the Bug 1 algorithm


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The Bug 1 approach includes the rule for leaving the obstacle it encounters and whose wall it follows,

as that the leaving point on the obstacle has to be the nearest to the goal. However we find that the path

followed is far more cumbersome than with the tangent Bug. This is because, the rover will make one

complete walk around the obstacle before it even determines which is the closest point of the obstaclefrom the goal. However, such an information might also be deduced from the global map. Still, we find

the path obtained, without the redundant walk around the obstacle, will still be less efficient than the

tangent Bug.

Fig: Bug 1 with a priori knowledge of closest point on obstacle from goal

Let the closest point on the obstacle from the goal be P.

Now, the consideration of the direction of travel has to be taken into account. In Bug 1, the only way to

determine the more efficient direction in which to follow the obstacle wall, in the first instance the

obstacle is encountered, is to find the shortest way along the wall to P. In Bug 1, that is the leavingpoint, hence P will be the intermediate goal for the rover. In tangent Bug, the vertex from which it can

leave will be such an intermediate goal. However, computing the nearest possible vertex from where

there is the direction of the goal is not impeded by the current wall of the obstacle would be laboriousto compute a priori.

However, the result of either of these is far from the most optimal path.

Fig: Shortest path with the same configuration of obstacles


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Since the initial intent is to represent the path in terms of a potential field, the analysis of the path has to

be done with this view. Throughout the path the general direction of motion indicates the presence of

an attractive force. As an inherent property of the potential field method, the present of the repulsive

force is also necessary, otherwise the rover is liable to penetrate obstacles. We assume such a force isalso present. At the Start, the force is not directed towards the goal. Since the repulsive force, by nature

is perpendicular to the surface, moment to the right has to be provided be a force that has a component

to the right, which is tangential to the surface of that obstacle closest to the rover. Hence, the nature ofthe missing force can be identified as tangential in nature, the direction of which is consistent with the

shortest along-wall distance to P(closest point on obstacle from goal), for the single obstacle. Thus, the

tangential force directed towards the right, combined with the attractive and repulsive forces willmanage to guide the rover from the start in the initial direction of the optimal path. If we add this

tangential force in the shortest direction towards P at every point, the second part of the path as well as

the portion of the path following the upper obstacle's wall can be achieved. For the portion of the path

where the rover travels between the two obstacles, the attractive force is trying to drag the rover intothe lower surface of the upper obstacle, and the repulsive force is radially outward from the closest

point, ie. lower right vertex of the upper obstacle, in this case. Thereby, the moment towards the left by

the attractive force has to be counteracted by a moment to the right in order to maintain the straightheading. The tangential force for the vertex possesses such a moment to the right. However the

tangential force for the lower obstacle's upper right corner will be generally to the left. So, for the

portion of the path between the two obstacles, the tangential force by the lower obstacle ceases toproduce any effect on the rover. Thereby, if the rover crosses the obstacle, it ceases to produce any

effect on it. It also implies conversely that during the motion of the rover, it should be acted upon by

one or more obstacles that lie towards the goal. By towards the goal, a specific region lying inscribedbetween the circles having radius of the distance between the point of the rover and the goal, and

centres at the goal and rover point respectively.

Fig: Region of influence in terms of the obstacles producing tangential force

Any obstacle lying in this region a much higher chance of interacting with the rover than the region

outside it. Hence, only the obstacles in this region influence the motion of the rover with the tangential

force. Since the position of the rover itself changes as a result of the field, the region of influence willalso shift and may take into account additional obstacles as and when they come inside the region of

influence towards the goal.


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The argument that all obstacles lying in this region have to be considered as affecting the rover motion

is justified because whereas the single closest obstacle will help it cross that single obstacle, but the

combined effect of multiple tangential forces will induce a sort of lookahead into the tangential

tendencies of the obstacles to come and the rover starts moving along the global tangential impulse.

Fig: A case which necessitates the consideration of multiple tangential force causing obstacles

In the above figure, the upper obstacle has to influence the rover as it turns from the bottom right vertexof the lower obstacle, in order to enable the rover to continue on the specified path.

Thereby, we add a tangential force component to the potential field, apart from the attractive and

repulsive forces. The results emulate the optimal results closely.

Fig: Implementation of the tangential force in the potential field, and the path obtained therein.

So, the missing force that holds the prospect to lending completeness and pseudo-optimality to the

potential field field method has been identified as a tangential force in the direction of the shortest

along-wall distance to the point on the obstacle closest to the goal.


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Analysis of the Forces

Fig: Attractive force, Repulsive force, Tangential force


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Attractive Force:

The nature of the attractive force is that it has the same magnitude at all points and is directed towards

the goal. The problem of approach error that results in overshooting of the goal point in real life

situations can be dealt with by setting a goal state acquisition threshold by virtue of which the rover istaken to have acquired the goal state when the distance from the goal becomes less than the threshold.

A constant attractive force alleviates the non-uniform interaction of the rover with obstacles at different

distances from the goal. If the attractive force is variable, then the rover potential field tends to push therover very close to the obstacles lying in into way to the goal, where the attractive force is strong. The

progress towards the goal becomes very slow where the attractive force is low in magnitude.

Moreover, since the behaviour of the rover near obstacles will be far more controlled in case of aconstant attractive force, there is a greater scope for easy manipulation of the other forces to alter the

nature of the path, chiefly the distance that is maintained from obstacle surfaces. The greater the

distance, the safer the path will be. However, there will be a compromise in terms of greater loss ofoptimality as the optimal path will always connect the vertices of obstacles tangentially, in case the

obstacle negotiation is necessary.

A reduced attractive force will lead to the rover keeping a greater distance from the obstacle surfaces.

Since the repulsive force increases as distance from the surface decreases, if a surface is directly in the

way of the rover, then the component of the attractive force that is opposite the direction of therepulsive force will counteract the repulsive force and the point at which they balance each other out

gives a measure of the closest the rover can go to the surface. In some cases this distance can be lesser,

especially in narrow passages. However, the nature of the repulsive force does not allow the rover to

foray into the obstacle.

Fattr= c force unitswhere c is constant

Thereby, the attractive force serves the purpose of a constant pull towards the goal, irrespective of

distance and orientation of the rover. In such a case when other behaviours of the rover act on it, likerepulsive or other forces, the attraction towards the goal remains the unchanging objective state

attraction.

Repulsive Force

Since mere attraction towards the goal is not sufficient to guide the rover along a safe path, thebehaviour of obstacle avoidance has to be compounded on the rover. This is represented by the

repulsive force that emanates from the nearest point on the obstacle surface. This force pushes the rover

away from the obstacle. Such magnitude of the force has to display the property of being very low andnegligible at distances greater than the intended field of influence of the rover, and it should also be

very high very close to the rover. The force should be prevent the rover from entering the obstacle

surface at all costs. Thereby, the force is expected to reach very high values, that can never tocounterbalanced by other oppositely directed forces. This necessitates the asymptotic convergence of

the force to infinity as the rover approaches the surface of the obstacle.


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The nature of the force beyond a certain distance signifying the region of influence is of a very low

magnitude, not comparable with the attractive force. Such a nature will enable the rover to take more

optimal paths because in case of obstacles in the rover's direct straight path to the goal, the optimal path

exists as coming infinitesimally close to its corners. The limit of the range of the repulsive force enablethe rover to come closer to the obstacles to enable possibly more optimal path wherever possible.

Moreover, the effect of the repulsive force should be felt only by rover at positions where it is in danger

of colliding with the obstacles because that is the elementary purpose of the repulsive force- to preventoccurrence of collisions.

However, for a robust potential field, and its corresponding repulsive force, the absolute prevention ofrover-obstacle collision will only be possible if the repulsive force is always greater than the sum of

components of all the forces directing the rover into the obstacle surface, when the rover reaches the

obstacle surface. Maintaining this is only possible by either somehow controlling the sum of the

magnitudes of the force components directed into the obstacle surface, or designing the repulsive forceto be always be always be greater than the sum at very low distances from the obstacle surface. The

latter can be implemented by representing the repulsive force by a distance dependant function that

asymptotically converges to a functional value of infinity as the rover approaches the obstacle surface.Since such high values of the repulsive force will only be reached at very low distances from the rover

surface, the possibility of an opposing force of comparable or greater magnitude is not possible unless

the opposing force itself is a repulsive force generated by a surface of another obstacle, since such highmagnitude of forces should only be a result of repulsion from obstacles. This scenario does not pose a

threat because the distance between the two surfaces, ie. the passage between them will be too narrow

to accommodate the rover. So the rover can never attain that state in the space.

Frep = k/d2

force units

where k is a constant, d is distance from rover to closest point on obstacle

Thereby the repulsive force acts to provide the safety consideration for the counteracting of the effect

of all other forces attempting to push the rover into unsafe regions, especially in states where the roverinfringes inside the obstacle. It therefore forms an integral part of the potential field methodology.

Tangential Force

As necessitated by the shortcomings of conventional potential field methodologies, an additional force

needs introduction- the tangential force. The tangential force shall serve the purpose of guiding therover along the contours of the obstacle , towards the goal, with global optimality considerations. The

direction of the force is such that a shorter path is taken to the point on the obstacle surface which is

closest to the goal. The attractive force gives the constant attraction towards the goal which is the

global objective. The tangential force, on the other hand, tries to move the rover towards theintermediate sub-goals, that is the point on the obstacle closest to the goal. The combined effect of

multiple such forces as well as the attractive force is meant to guide the rover along a robust, sub-

optimal path.

The nature of the tangential force has to be, as its purpose implies, along the tangent to the surface at

the point on the surface closest to the rover. Thereby there would be no component of the repulsiveforce along the direction of the tangential force for one obstacle and vice versa. The tangential force is

meant to guide the rover around the obstacle, not away from it.


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The tangential force is meant to act from every obstacle lying between the rover and the goal, in the

region of intersection of the circles having centres at the goal and the rover centre, and having the rover

centre and goal on the boundary, respectively. Since all the obstacles in this region influence the rover

with the tangential force, considering the magnitude of the repulsive force beyond a particular distanceto be negligible, beyond that distance, the tangential force has the opportunity to predominate. Thereby,

as the tangential force acts from multiple obstacles, their effect should be dependant on distance to the

closest obstacle surface in order to give relative importance to the effect of nearer importance, all thewhile taking into account the overall effect of the other obstacles.

The overpowering effect of the tangential force at greater distances, when compared to the repulsiveforce, results in the effect of the potential force having the tendency to guide the rover around obstacles

far before the obstacle's repulsive force, ie. safety constraint, comes into effect. So, instead to coming

inside obstacles, either concave or convex, and then escaping them, as Bug algorithm technique

followers, the potential field tends to make the rover avoid the obstacle in the first place. Thereby theconventional local minima problem of the box canyon concave obstacle doesn't pose a threat on

optimality as the rover will tend to avoid the obstacle in the first place, and not venture into it, even if it

is directly in the way to the goal.

The tangential force is effectively a manifestation of the wall following behaviour. When superimposed

on the repulsive and attractive force, it gives paths that aren't completely wall following, neither arethey completely conventional repulsive-attractive paths. Such a path is sub-optimal as in the presence

of obstacle lying in the way to the goal, the optimal path tends to follow directions towards a vertex for

surfaces lying in the way, and along the surface for obstacle surfaces lying parallel to its path. Bothbehaviours can be approximated by the hybrid field, with tangential force.

Ftan = q exp(d/r)

where q and r are a constant, d is the distance from the rover to the closest point on the obstacle

In order to maintain the smoothness of the potential field, the tangential force direction in situations

where the closest point on the obstacle from the rover it a vertex, is the tangent to the vertex that isperpendicular to the line joining the rover and the corresponding vertex.

Thus, we obtain the general features of the proposed potential field. The effect of the exact magnitudesof the forces depending upon the relative values of the constants is discussed further on.

Assumptions

The implication of the new potential field is that the tangential force should, at all times direct the rover

around the obstacle obstructing the rover's way. However, it might not be possible for the rover to go

around an obstacle in situations where the distance between surfaces of two different obstacles is lessthan the dimensions of the rover, ie. the situation where the rover cannot occupy region of free space

because it is too small to accommodate the rover. This problem will result in the rover getting stuck at

such a point, beyond which it cannot pass, into the obstacles.

This problem can be solved in the obstacle boundary identification process. In alpha shaping, since a

circle of radius equal to the radius of the rover circle is translated around obstacle grid cells, regions orpassages that are less than the rover dimensions would show up as parts of obstacles. Different groups

of untraversable grid cells that are separated by less than the rover circle diameter would be classified

as a single obstacle. Therefore, the eventuality that causes the problem does not arise.


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Also, in case of highly irregular obstacles with concavities that might distort the obstacle boundary to

such an extent that the along-wall distance measurement might not give the shorter distance direction

around the obstacle. In such a case the direction of the tangential force might oppose the tangentialforces from other obstacles. So, the tangential force methodology will not be a global moment but

contain local perturbations.

The solution to this problem is the to consider measure the distances along the surface of the convex

hull enclosing the obstacle, not the actual sides of the obstacle. Such a measurement will eliminate the

possibility of concavities in the obstacle surface affecting the general optimal direction of the tangentialforce.

Elimination of conventional potential field problems

The proposed hybrid, global potential field solves the common problems plaguing conventional

potential field methodologies.

All common types of local minima can be avoided by the proposed methodology.

Fig: Negotiation of local minima problem

Since the nature of the potential field has been altered, the common forms of local minima as illustrated

by the figure, do not occur. As the tangential force is guaranteed to direct the rover around the obstacle,

there is no instance in which there can be a local minima for a single obstacle in the environment. So,the special type of obstacles for which the conventional potential field methodologies fails, no longer

pose any threat.

Since the elementary method used is still a potential field method, there is still the chance of the sum of

all the forces becoming zero due to interaction between the attractive, repulsive and tangential forces

due to multiple obstacles. There are two specific instances for which are to b avoided.

The resultant of all three forces can become zero at some point in the environment. However, the

tangential force by nature is designed to navigate the rover around the obstacles. This implies, that

locally, it is always safe to follow the tangential force. Therefore, when the resultant of all the forcesbecome zero, the net tangential force is increased so that the position of the local minima can be

escaped. Since in the direction of the tangential force, the tangential force is expected to guide it further

onwards and not backtrack, the rover will not have a tendency to return to the local minima. Thisimplies, the local minima in this potential field are not locally converging.

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There is also the instance when the sum of the tangential forces can become zero. This can be caused

by when the rover is expected to follow a passage between two obstacles, such that one of the surfaces

does not fall in the way of the rover to the goal, while the other does. This situation will not arisebecause the region of influence with respect to obstacles producing the tangential force will prevent the

obstacle not falling in the rover's way to the goal, not applying any opposite tangential force.

Fig: Passing through narrow passages

The other problem with conventional potential field methodologies is the inability to guide the rover

through narrow gaps between obstacles. Also there is highly irregular, oscillating motion in narrow

passageways. Both these problems do not occur in the proposed method.

Since both the obstacles having the passage between them push the rover through the passage by virtue

of their tangential forces, when the path towards the goal lies through this passage. Thereby the roverwill pass through two obstacles even if the gap is narrow, because the gap implies that the rover can

pass through the space(due to the obstacle clustering and alpha shaping with the rover circle

consideration). Also, one inside the passage, there are no oscillations, because the tangential forces

from both the surfaces guide the rover through the passage. The oscillations become imperceptible.

Thereby, all the general problems faced by conventional potential field strategies are solved by thecurrent method.

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Fig: Potential field directions for different obstacle configurations. Goal at (250,500)

Fig: Potential field directions for different obstacle configurations. Goal at (500,500)


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Effect of constants

The calibration of the constants used in the equations representing the various forces in the potential

field is analysed. Since the most important force in terms of guiding the rover though the obstaclefilled environment is the tangential force. However, we find that the tangential force is not sufficient in

itself to guide the rover to the goal. Thereby, we have to keep the attractive force. However, the

attractive force is only the constant objective force that is not important for tackling the obstacleslocally. Therefore, if the maximum magnitude of the tangential force is set higher than the attractive

force, then there will always be a possible path generated. This is because, the tangential force, in some

instances, may be forced to direct the rover away from the goal, locally, in order to escape an obstacle.This will not be possible if the tangential force cannot overcome the attractive force. Therefore, the

attractive force is set a constant value that is lesser than the value of the tangential force.

In order to maintain a smooth movement and nullify any overbearing effect of the constants, amaximum net displacement with respect to the summed up net force, is set to a limiting value, here

5cm. Therefore, even if the forces are large and the displacement produces larger than 5cm, only the

direction of the net force is preserved, and the displacement never exceeds the limit. Thereby, the forcesare set to large values, ie. the maximum magnitude of the tangential force is set to a value many times

higher than the value of the attractive force. The nature of the repulsive force, since very definite, does

not allow much change in its magnitude at moderate distances from obstacles.

Thus, the exact magnitude of the forces become independent of the type of terrain and the relative

positions of the goal, the rover and the obstacles. A path will be possible irrespective of the forcemagnitudes, provided, the tangential force has higher maximum magnitude than the attractive force.

General mathematical Properties of the Potential Field

Components of the potential field -

Attractive : F attr

Repulsive : F repi such that i [1,number of obstacles]

Tangential : F tani such that i [1,number of obstacles]

F rep is along outward normal from the obstacle surface.

F tan is along tangent to obstacle surface.

F attr is along the line joining the current position to the goal.

They are dependent upon distances wrt the current position ie. f(d)

CASE 1: One obstacle, without considering tangential force.


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Minimum distance between surface and robot can occur when F rep and F attr are oppositely directed.

At equilibrium F attr (d) = F rep ()

should always be positive.

gives a measure of the safety threshold minimum distance to obstacles.

CASE 2: Multiple obstacles, without tangential force.

Equilibrium can be disturbed only when there is force component along F attr, in which case also can

decrease.

Therefore, there must be repulsive forces along F attr arising from other obstacles. For a robot of radius

r to be present at the position concerned, minimum distance between two facing surfaces of twoobstacles surrounding it has to be (2r+2).

Minimum distances between any two surfaces of different obstacles can be considered as

So,

F attr (d+r) + F rep2 (+r) + F rep

3 (+r+) + F rep4 (+r+2) + = F rep

1 (+r)

However, as F rep2 (+R) = F rep

1 (+r) , this approach fails to maintain distance from surface of

obstacle 1. Therefore, when a robot is present between two obstacles, the minimum distancesbetween facing surfaces is (2r+2+), where is very small.

So, when robot is closest to surface of obstacle 1,

F attr (d+r) + F rep2 (+r+) + F rep

3 (+r++) + F rep4 (+r++2) + = F rep

1 (+r)

Considering, F rep = , where n=2.


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F rep2 (+r+) + F rep

3 (+r++) + F rep4 (+r++2) +

= , where a= (+r+)

=

CASE 3: Tangential force is taken into account.

can decrease from previously attained equilibrium position only when tangential force has

component along F attr .

F attr(d+r) + F rep (+r++i) + F tan (+r++j) = F rep1 ()

Tangential force summation:

Considering F tan =

F tan (+r++j)

= ... , where a=(+r+)

=


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So,

F attr (d+r) + + = F rep ()

Terrain Interaction with rover

The terrain classification takes care of the untraversability conditions of the surrounding terrain. Thebinary classification of the environment is on the basis of the roughness, relative elevation, gradient

and size of free space or gradient. Therefore, the classification is a measure of whether the rover can

occupy a region safely or not. The path planned by the proposed hybrid potential field will generate apath that avoids all the untraversable regions and reaches the goal. However, within the traversable

regions marked in the environment grid, there may be regions of variable roughness and gradients.Further modification of the path can be made to avoid uneven regions. However these considerationsshould not hinder the attainment of the goal state. Since the path is traversable on its own, the

avoidance of the uneven terrain is a optimization that can be made or suppressed depending upon the

nature of the current objective.

The measure of terrain roughness is made by recording the standard deviations of the points found in

the grid. If this is high, that implies a rough or high gradient surface. The cells in front of the rover upto

a specified distance of lookahead are checked and the standard deviations of the cells lying on the leftand right sides are added. In case of the occurrence of an obstacle on either the left or the right side, the

corresponding side is strictly not considered for turning. If both sides contain obstacles, the behaviour

is suppressed. Depending upon which side the lesser summed standard deviation, or terrain roughnessmeasure lies, the rover is expected to turn towards that side. However, the degree of turn or the turning

angle has to be dependant on the relative roughness of the left and right sides. It is given by

90*(sdright~ sdleft)/( sdright~ sdleft ) degrees

where sdrightis the summed standard deviation on the right side, ahead of the rover, sdleftis on the left

The advantage of following such a path that avoids rough terrain is that lesser power will be consumedbecause the rover does not have to negotiate the uneven regions. However, optimality will be further

hampered in terms of the path obtained. Also, the path can be expected to be highly irregular.

Due to these reasons, the terrain interaction behaviour is suppressed during the field runs of the rover.


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Flowchart

Fig: Flow of control in the navigation process for the lunar rover


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Experiment

Fig: Experimental set-up

Fig: Grid abstraction, rover at origin, distances in cm

Fig: Grid classification, rover at (250,250), distances in cm

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Fig: Obstacle grid clustering, rover at (250,250), distances in cm

Fig: Polygon Fitting around obstacle clusters, rover at (250,250), distances in cm

Fig: Path planning with global hybrid potential field from (250,250) to (260,0), distances in cm

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Results

Fig: The run of the rover on the obstacle filled environment, to follow the path generated by the potential field methodology

Conclusions

The map generation process used can be applied to general three dimensional terrains with effectiveresults. When used in tandem with the proposed potential field methodology, the conventional

problems of potential fields are solved and the resulting system is robust enough to be applied to any

general environment. The complexity of the process is dependant on the number of sides of the

obstacle polygons obtained, not on the number of data points or cells. Therefore, the process isrelatively fast and can be implemented in planetary navigation where computation expenses are a

consideration. The addition of an efficient rough terrain avoidance behaviour can also be used to obtain

a path over smoother terrain. However, the proposed potential field method will work with theconsideration that all free space is interconnected, ie. it is dependant upon the map generation process.

It can be made more general by including the map generation approximation as pre-processing in the

path generation process. We have thus obtained a robust system for guiding the rover through the harshlunar terrain effectively and with a sub-optimal path. In the process, the conventional potential field

th d f th l i h b lt d d d l t

Lunar Rover Navigation

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