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Power Consumption Modeling of Skid-Steer Tracked Mobile Robots on Rigid Terrain Jesús Morales, Jorge L. Martínez, Anthony Mandow, Alfonso J. García-Cerezo, and Salvador Pedraza Departamento de Ingeniería de Sistemas y Automática University of Málaga, 29071 Málaga, Spain e-mail: [email protected] Abstract - Power consumption is a key element in outdoor mobile robot autonomy. This issue is very relevant in skid-steer tracked vehicles on account of their large ground contact area. In this paper, the power losses due to dynamic friction have been modeled from two different perspectives: 1) the power drawn by the rigid terrain and 2) the power supplied by the motors. Comparison of both approaches has provided new insight on skid steering on hard flat terrains at walking speeds. Experimental power models, which also include traction resistance and other power losses, have been obtained for two different track widths over marble flooring and asphalt with Auriga-, which is a full-size mobile robot. To this end, various internal probes have been set at different points of the power stream. Furthermore, new energy implications for navigation of these kinds of vehicles have been deduced and tested. Keywords: dynamic; outdoor mobile robot autonomy; power consumption modeling; rigid terrain; skid-steer tracked mobile robots; skid-steer tracked vehicles; mobile robots; power consumption; robot dynamics. __________________________________________________________________________________________ This document is a self-archiving copy of a copyrighted publication. The published article is available in: http://dx.doi.org/10.1109/TRO.2009.2026499.

How to Cite: Morales, J., Martínez, J.L., Mandow, A., García-Cerezo, A., Pedraza, S. Power consumption modeling of skid-steer tracked mobile robots on rigid terrain (2009) IEEE Transactions on Robotics, 25(5), pp.1098 -1108. @ARTICLE{Morales2009TRO, author={Morales, J. and Martínez, J. L. and Mandow, A. and García-Cerezo, A. J. and Pedraza, S.}, journal={IEEE Transactions on Robotics}, title={Power Consumption Modeling of Skid-Steer Tracked Mobile Robots on Rigid Terrain}, year={2009}, month=oct. , volume={25}, number={5}, pages={1098 -1108}, doi={10.1109/TRO.2009.2026499}, ISSN={1552-3098}, }

_____________________________________________________________________________________ © 2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

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IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 5, OCTOBER 2009

Power Consumption Modeling of Skid-Steer Tracked Mobile Robots on Rigid Terrain Jesús Morales, Jorge L. Mart´ınez, Anthony Mandow, Member, IEEE, Alfonso J. Garc´ıa-Cerezo, Member, IEEE, and Salvador Pedraza

Abstract—Power consumption is a key element in outdoor mobile robot autonomy. This issue is very relevant in skid-steer tracked vehicles on account of their large ground contact area. In this paper, the power losses due to dynamic friction have been modeled from two different perspectives: 1) the power drawn by the rigid terrain and 2) the power supplied by the motors. Comparison of both approaches has provided new insight on skid steering on hard flat terrains at walking speeds. Experimental power models, which also include traction resistance and other power losses, have been obtained for two different track widths over marble flooring and asphalt with Auriga-β, which is a full-size mobile robot. To this end, various internal probes have been set at different points of the power stream. Furthermore, new energy implications for navigation of these kinds of vehicles have been deduced and tested. Index Terms—Friction, mobile robots, motion control, power consumption, skid-steer, tracked vehicles.

I. INTRODUCTION

M

OBILE robots are increasingly being developed for outdoor missions that demand an extended degree of autonomy. These include applications such as search and rescue, disaster response, agriculture, military, forestry, mining, and planetary exploration. A key aspect of vehicle autonomy is power consumption, which has become particularly relevant in applications with critically limited energy sources [1]. However, this issue has not been usually perceived as a major problem because most mobile robots use wheels under the nonslipping and nonskidding conditions [2], which are more power-efficient than legged or treaded traction systems on hard smooth terrains [3]. Power consumption has been considered at different levels in robot system design. From a mechanical standpoint, passive locomotion systems have been proposed to reduce energy use [1]. In small and light robots, nonmechanical components (e.g., sensing, communications, or computations) may be responsible for most of the power consumption; therefore, efficient scheduling [4], as well as specific energy conservation techniques, can have a major impact [5].

Manuscript received October 28, 2008; revised April 2, 2009 and June 17, 2009. First published July 28, 2009; current version published October 9, 2009. This paper was recommended for publication by Associate Editor K. Iagnemma and Editor J.-P. Laumond upon evaluation of the reviewers’ comments. This work was supported in part by the Spanish Project DPI2008-00533 and Andalusian Project TEP-01379. J. Morales, J. L. Mart´ınez, A. Mandow, and A. J. Garc´ıa-Cerezo are with the Escuela Técnica Superior de Ingenieros Industriales, Universidad de Málaga, Málaga 29071, Spain (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). S. Pedraza is with the Optimi Corporation, Institutos Universitarios, Málaga 29590, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TRO.2009.2026499

In the case of heavier mobile robots, tractive power has a dominant share in power consumption. In wheeled vehicles with no slippage, the focus has been on minimizing energy objective functions ultimately related to the path length [6], without explicitly addressing actual power consumption. In fact, this has been the traditional goal of path planning in 2-D space [7]. More generally, information about orography and terrain types can be used to plan an optimal path that minimizes the energy expended due to gravity and soil–wheel friction [8]. Motor resistances have been identified as the main source of power dissipation in the traction system of wheeled robots, which can be minimized with an appropriate velocity profile [9]. In this sense, motion control methods have also been proposed to reduce the loss of kinetic energy [10], steering actuations [11], and accelerations in potential field strategies [12]. Moreover, an energy model of motor losses has been considered to test the performance of different motion patterns in searching open areas [13]. Power consumption has also been studied for alternative locomotion mechanisms, such as wheeled vehicles with redundant actuators [14], limbed robots [15], or snake-like robots [16]. Nevertheless, we find that power efficiency of tracked robots has not been specifically treated in the technical literature. Tracked locomotion offers a large contact area with the ground, which provides better traction than wheels on natural terrains [17]. Because of this, power consumption due to track–soil interactions can be very relevant. The skid-steer principle is based on controlling the relative velocities of both tracks (see Fig. 1). For steering, one track pushes the vehicle, while the other drags it, which results in a turning torque. This causes dynamic friction, as the linear motions of the tracks on the ground do not agree with their motor velocities [18]. A kinematic equivalence between skid-steer mobile robots (both tracked and wheeled) and differential drive wheeled vehicles has been established for velocities below 2 m/s, i.e., human walking speed [19] [20]. Despite the kinematic similarities, power requirements differ. In the latter, power demanded by one motor is almost independent of the speed commanded to the other, whereas in skid-steer, the power required by one motor heavily depends on the speed of the other [21]. In this paper, we model power consumption of skid-steer tracked mobile robots at walking speeds on flat hard terrains. In particular, the power losses due to dynamic friction have been modeled from two different perspectives: 1) the power drawn by the terrain and 2) the power supplied by the motors. We have analyzed this issue for Auriga-β, which is a 286-kg mobile robot with rubber tracks, by using several probes at different points of

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MORALES et al.: POWER CONSUMPTION MODELING OF SKID-STEER TRACKED MOBILE ROBOTS ON RIGID TERRAIN

Fig. 2. Fig. 1.

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Track ICRs on the motion plane represented as virtual wheels.

Skid-steer mechanism.

the power stream. Novel energy implications for navigation are also deduced based on this study. The rest of the paper is organized as follows. The next section is devoted to model power consumption in skid-steer tracked vehicles. Section III presents the Auriga-β case study. In Section IV, energy-related navigation guidelines are deduced and tested. Conclusions and future work are discussed in the last section. II. POWER CONSUMPTION MODELING OF TRACKED VEHICLES Three main locomotion configurations have been proposed for tracked vehicles: articulated steering, curved track steering, and skid steering [22]. The latter is the most widely used, since it is simpler from the mechanical standpoint, and it achieves a faster response. Another feature of skid steering is that its turning radius is not bounded, although the maximum forward speed of the vehicle is limited proportionally with angular speed. This section first summarizes a kinematic approach for tracked skid-steer vehicles. Then, it models the power losses associated with slippage. Finally, other power losses are considered to derive the total power demanded by the locomotion system. A. Kinematic Approximation This section briefly reviews the work presented in [19] and [20], where we proposed an approximate kinematic model of skid-steer vehicles as a function of the instantaneous centers of rotation (ICRs) of treads on the 2-D ground plane. These ICRs, which are different from the vehicle’s ICR, represent the position of equivalent differential drive ideal wheel contact points, as illustrated in Fig. 2. Let us assume that the local frame of the vehicle has its origin in the geometrical center of the convex area spanned by the tracks’ contact points and its Y-axis is aligned with the forward motion direction. The XY plane is parallel to the ground plane. " r for the left and " l and C Local ICR vectors can be defined as C " l,r = (C l,r , right tracks, respectively. Their coordinates are C x Cy , 0), where l, r denotes any of both tracks. Both ICRs have the

same Y coordinate Cy since they lie beyond their corresponding track centerlines on a line that is parallel to the local X-axis. The ICRs coordinates with respect to the local frame of the vehicle are dynamics-dependent but remain within a bounded area at walking speeds. Therefore, optimized constant values for ICR positions can be obtained from experimental identification [19]. Track ICR positions depend on terrain type and vehicle design, especially on the position of the center of gravity and on the track type. If the center of mass of the vehicle does not lie on the Y-axis, then the closest track will slip less due to pressure. This circumstance results in track ICRs that are not symmetrical with respect to the Y-axis (see Fig. 2). The relationship between track speeds and vehicle speed can be expressed as a function of track ICRs in the following way: vx = vy = ωz =

Vyr − Vyl Cy Cxr − Cxl

Vyr − Vyl Vyr + Vyl − r 2 Cx − Cxl

(1) !

Cxr + Cxl 2

Vyr − Vyl Cxr − Cxl

"

(2) (3)

where Vyl and Vyr are the longitudinal speeds for the left and right tracks, respectively, vx and vy are the components of translational velocity with respect to the local frame of the vehicle "v = (vx , vy , 0), and ωz is the angular velocity of the vehicle ω " = (0, 0, ωz ). Note that the sign of the angular speed only depends on the difference between track speeds and is independent of track ICR positions, as stated by (3). The track speed inputs for motion control can be obtained from the desired longitudinal and angular speeds vysp and wzsp , respectively, as follows: Vyl,r = vysp + Cxl,r ωzsp

(4)

where the vxsp set point cannot be addressed due to the nonholonomic restriction of the locomotion system [23].

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Similarly to differential drive, it is necessary to consider that every ωzsp set point has an associated maximum longitudinal speed vymax  sp z  sign(vysp ) Vymax − Cxr ωzsp , if ωsp >0 vy vymax (ωzsp ) = (5)  sign(vysp ) Vymax − Cxl ωzsp , otherwise

where Vymax is the top track speed. Thus, the highest longitudinal speed of the vehicle, i.e., vymax (0) = sign(vysp ) Vymax , can only be achieved in straight-line motion. Conversely, the top angular speed of the vehicle ωzmax =

sign(wzsp ) 2Vymax Cxr − Cxl

(6)

is reached only when turning on spot with vysp = 0. B. Power Losses Due to Dynamic Friction The turning resistance of one track is due to the dominant longitudinal component of its contact surface, which tends to straight-line motion. Then, the vehicle can turn only if one track counteracts with the other, thus generating slippage between the ground and the tracks. These effects result in relevant power losses. The power lost due to slippage can be modeled from two different approaches: first by considering the power drawn by the terrain and second by taking into account the power supplied by the motors, both caused by dynamic friction only. From the first standpoint, the power drawn by the terrain from each track PSl,r can be modeled as follows: & " PSl,r = − f"(a) · ϑ(a) ds (7) Ωl , r

where ds is the differential of the surface integral, f" is the dynamic friction applied to any point a of the track contact area " is the corresponding slipping velocity vector. Ωl,r , and ϑ In the case of hard uniform surface soils, such as asphalt, concrete, or pavement, a general anisotropic friction model can be assumed with Coulomb’s law as follows: " ϑ(a) (8) f"(a) ≈ −µ p(a) " #ϑ(a)#

This implies that instantaneous power losses due to dynamic friction are positive and proportional to the absolute value of the angular velocity. Then, the integral of the absolute value of the angular velocity & ψ=

(11)

|ωz | dt

is related to the energy spent due to slippage. According to (10), tread length increases dynamic friction losses, since points in the longitudinal extremes of the tread yield " l,r #. Then, given a particular higher values for the term #"r − C l,r tread shape, PS strongly depends on the track ICR positions. Consequently, farther ICRs result in lesser power efficiency. Based on this effect, an efficiency index χ can be defined as the inverse of the normalized distance between the track ICRs L (12) χ= r Cx − Cxl

where L is the distance between track centerlines (see Fig. 2). Index χ is a positive real value less than one. In the case of ideal differential drive wheels, χ = 1, and the integral part " l,r . of (10) corresponds to the one contact point where "a = C l,r Therefore, no power losses due to slippage occurs, i.e., PS = 0. It must be noted that instantaneous power consumption can depend on the sign of ωz because of ICR asymmetries. Then, the efficiency index χ would represent an average value for the vehicle, thus assuming that chances that the mobile robot turns to the left or to the right are the same. The total power drawn by the terrain due to slippage PSt results from the contribution of the left and right tracks PSt = PSl + PSr .

(13)

Then, using (10) !& & " l # ds+ p(a)#"a − C PSt ≈ µ |ωz |

" " r # ds . p(a)#"a − C

Ωl

Ωr

(14) On the other hand, from the viewpoint of the power supplied by the motors, PSt can also be expressed as & & " l ds − " r ds PSt = − (15) f"(a) · V f"(a) · V Ωl

Ωr

where p is the pressure under each point a of Ωl,r , and µ is the friction coefficient that depends both on track and terrain types. " of a point a in Ωl,r can be Besides, the slipping velocity ϑ considered as the result of turning around its corresponding track ICR

" l,r = (0, −Vyl,r , 0) is the velocity vector of the track where V contact surface with respect to the vehicle. This longitudinal vector is the same for all points in the contact surface. Therefore, (15) can be rewritten as & & PSt = Vyl fy (a) ds + Vyr fy (a) ds (16)

" " l,r ) ϑ(a) =ω " × ("a − C

PSt = Vyl Fyl + Vyr Fyr

(9)

where "a is the coordinate vector of the point a relative to the local frame (see Fig. 2). Using (8) and (9) in (7), PSl,r can be approximated by & " PSl,r ≈ µ p(a) #ϑ(a)# ds Ωl , r

= µ|ωz |

&

Ωl , r

" l,r # ds. p(a) #"a − C

(10)

Ωl

Ωr

which is

(17)

where Fyl,r stands for the overall longitudinal dynamic friction of a track. Note that for turning, different signs for both Fy ’s are necessary. Specifically, the sign of Fy of the inner track is opposed to vy , whereas Fy of the outer track has the same sign. As stated, (13) and (17) offer different approaches to obtain PSt . Even though both expressions add two terms associated with the left and the right tracks, the corresponding terms can



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First, the motors have to provide power for other traction resistances apart from dynamic friction: 1) frictions caused by the deformation of the tracks and by soil shearing; they depend on the weight of the vehicle and its payload, as well as on track and ground types; 2) internal frictions of the belt around its track train and in the gearheads. The power drawn due to these factors PR can be modeled approximately as proportional to the absolute value of the track speeds as follows: PR ≈ K (|Vyl | + |Vyr |)

(18)

PM = PSt + PR .

(19)

where K denotes the proportional constant. Note that if one track is dragged by the other, then the traction resistances of this track are assumed by the other motor. Therefore, the total mechanical power provided by the motors PM can be expressed as the sum of (14) and (18) Fig. 3. Dynamic friction cases. (a) Straight-line motion. (b) Different positive track speeds. (c) Left track speed null. (d) Tracks with opposite speeds.

take different values. Thus, the comparison of these equations provides more insight for power demanded by skid steering. In particular, the following four cases can be considered. 1) If the two track velocities are equal, the vehicle exhibits straight-line motion with ωz = 0. Thus, PSt in (13) is zero. Therefore, since Vyl = Vyr %= 0, Fyl,r in (17) are both null (see Fig. 3(a)). 2) If both tracks move in the same direction (i.e., Vyl Vyr > 0), then the slowest one is dragged by the other. This provokes a turning torque; therefore, both terms in (13) are greater than zero. However, the sign of Fy for the slowest track is the opposite of its track velocity Vy . Hence, the corresponding term in (17) is negative, which means that the slowest motor is drawing mechanical power (i.e., it is acting as an electric generator for its driver). This implies an extra power consumption for the fastest track motor apart from the power supplied for all the dynamic friction (see Fig. 3(b)). 3) If only one track speed is zero, dynamic friction occurs on both track surfaces [i.e., both terms in (13)], because ωz %= 0. However, the power is only supplied by the other track’s term in (17) (see Fig. 3(c)). 4) When tracks move with opposite speeds (i.e., Vyl Vyr < 0), their overall longitudinal dynamic frictions Fyl,r oppose to their respective motions; therefore, both terms of (17) are positive. In this case, both motors contribute to dynamic friction losses in (13) (see Fig. 3(d)). These cases are illustrated in Fig. 3, where the right track speed has been considered constant positive, and the left track speed varies from Vyr to −Vyr . Thus, ωz increases and vy decreases as the left track speed varies from case 1) to 4). C. Other Power Losses Other relevant power losses of the locomotion system need to be taken into account. These are the power to overcome traction resistance PR and that consumed by the motor drivers PD .

Second, power consumption of the drivers PD results from summing up the following contributions. 1) Power-up: This is a constant value to maintain the driver circuitry active. 2) Braking power: When one track speed is either zero or opposed to the other, i.e., cases (c) and (d) in Fig. 3, its driver requires a control effort for braking the motion induced by the other track. 3) Nonregenerated power: As discussed in the previous section, when both tracks move in the same direction, i.e., case (b) in Fig. 3, an extra power consumption is present for the fastest track motor to provide the mechanical power drawn by the other. Depending on the driver characteristics, it is possible that this mechanical power is regenerated into electrical power. In this case, the extra consumption is somewhat compensated. However, if the drivers do not allow power regeneration, this mechanical power will be lost. To sum up, the total power PT demanded by the locomotion system of a skid-steer tracked vehicle can be expressed as PT =

PM + PD η

(20)

where η is the electrical efficiency rate that represents electrical power dissipation due to resistances in the motors and their drivers (0 < η < 1). III. POWER CONSUMPTION OF AURIGA-β A. Auriga-β Mobile Robot The tracked mobile robot Auriga-β has been designed as a member of a group of robots for fire extinction tasks (see Fig. 4). Its dimensions are 0.7 m width, 1.2 m length, 0.96 m height, and 286 kg weight. It is equipped with a small self-stabilized landing platform for radio-controlled minihelicopters (which are employed for fire detection [24]) and a small fire extinguisher. Skid steering is based on two independent brushless ac motors with resolvers for dead reckoning. The maximum speed of each


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Fig. 6.

Two rubber belt types.

TABLE I MAXIMUM POWER CONSUMPTION OF THE COMPONENTS OF AURIGA-β

Fig. 4. Mobile robot Auriga-β and a minihelicopter landing on its selfstabilized platform.

Fig. 5.

belt contact length is 0.72 m, and the distance between track centerlines is L = 0.42 m. Apart from the resolvers, the robot includes an inertial measurement unit with gyroscopes. It is also equipped with a GPS receiver that accepts local-area differential corrections. This differential GPS (DGPS) provides positioning errors around 0.02 m with good sky visibility. The computation system is based on a Pentium IV industrial computer and a cFieldPoint, which is a programmable logic controller. The computer acts as the user interface, whereas the cFieldPoint interfaces with the motor drivers and sensors through several input–output modules. The cFieldPoint executes a real-time LabVIEW program for autonomous navigation, which consists of several concurrent processes. The main process implements a finite-state machine that initializes the robot and changes the operation mode (standby, manual operation, and autonomous navigation). The other processes interact with the sensors and the drivers to gather information and to update motion references.

Auriga-β track train with a punctual pressure distribution model.

B. Power System of Auriga-β Vymax

track is = 0.86 m/s. The track train consists of a sprocket and an idler wheel, with two rollers in between (see Fig. 5). These are mounted on a rigid suspension system, i.e., without springs or shock absorbers. Track belt tension is adjusted by a spring that shifts the idler in the longitudinal direction of the track. Gearheads between each motor and its sprocket provide a mechanical efficiency of about 93%. Two different rubber belt sets have been employed. They only differ on their width, which is 0.16 m and 0.11 m for the wide and narrow tracks, respectively (see Fig. 6). In both cases, the

The power needed by the vehicle is provided by an onboard 3.8-kW petrol-fed 220 V ac generator. The maximum power consumption for the different components of the mobile robot is shown in Table I. Note that in this kind of vehicles, the locomotion system can be responsible for most of the power consumption (up to 80% for Auriga-β). Motors are fed by two independent ac drivers that do not allow regenerated electric power to be transferred from one driver to the other, nor to the onboard ac generator. Instead, regenerated power exceeding the internal capacitor bank of each driver is



Fig. 7.

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Internal power measurements and power losses in the mobile robot Auriga-β.

shunted into an external resistor. According to data sheets, the electrical efficiency of the driver–motor set is η ≈ 0.95. Power has been probed at different points of the power stream (see Fig. 7). First, PT is measured from a high-resolution watthour meter that monitors the ac power supplied by the generator to the drivers. This sensor can also integrate power for a given period of time to measure the total energy consumption. l,r is Second, mechanical power delivered by each motor PM estimated from the motor currents il,r and the motor speeds σ l,r , which are measured by the drivers and the resolvers, respectively. In this way l,r = σ l,r τ l,r = σ l,r kτ il,r PM

(21)

where τ l,r is the motor torque and kτ is the current–torque relation constant. l,r estimations and the total Then, the relationship between PM mechanical power PM is l r + PM . PM = PM

(22)

C. Experimental Power Model This section presents the experimental estimation of the parameters discussed in Section II for the Auriga-β robot. Experiments have consisted of a stationary power analysis on a dry hard horizontal flat terrain. Two different types of flooring have been tested: marble and asphalt. For each belt and terrain type, all possible combinations of track speeds (with steps of 0.2 m/s) have been considered. Each experiment has started with zero velocity. Then, the goal speed is reached with an acceleration ramp of 2.5 s. This value is maintained for 5 s, where the average consumption is recorded l from both the watthour meter (PT ) and the motor drivers (PM r and PM ), before going back to zero with a deceleration ramp of 2.5 s. l r and PM for two of these experiments is The evolution of PM illustrated in Fig. 8 for wide belts on marble flooring. When track speeds are equal but opposite, the mechanical power contributed by both motors is almost the same, as shown in Fig. 8(a). The case of different speeds of the same sign is presented in Fig. 8(b). It can be observed that the right motor power is negative because it is being dragged by the other track. Hence, this motor is acting as a generator.

Fig. 8. Mechanical power of the motors with V yr = 0.4 m/s and (a) with V yl = −0.4 m/s and (b) with V yl = 0.8 m/s (vehicle on marble flooring and with wide belts).

Power model parameters µ and K have been estimated by optimization from the aforementioned experimental data. Particularly, the downhill simplex method [25] has been applied to minimize the sum of the absolute value of the difference


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TABLE II EFFICIENCY INDEX χ OF AURIGA-β ON DIFFERENT TERRAINS AND TRACK WIDTHS

between measured and modeled power PM in (22) and (19), respectively. Note that the friction coefficient only depends on the flooring since the rubber of both track belts is the same. Conversely, the parameter K depends on the employed track set (through the contact surface size), but it is independent of the type of the hard terrain. The friction coefficient for marble flooring is approximately µ ≈ 0.63 and increases to µ ≈ 0.81 for asphalt. The proportional constant in (18) has been estimated as K ≈ 342 N for wide belts and K ≈ 324 N for the narrow set. To approximately evaluate (14) in (19) for Auriga-β, the following assumptions have been considered. 1) Constant symmetric track ICRs: The efficiency index has been experimentally estimated for each terrain type and track width (see Table II) by measuring the total rotated angle when equal opposite track speeds are applied [19]. According to (12), these values mean that the distance between ICRs on the local X-axis grows on the asphalted terrain and with the wide belt set. 2) Punctual model for pressure distribution p on the track contact area: This model assumes that pressure is concentrated on the contact points of the sprockets, rollers, and idlers (see Fig. 5). Considering the moderate speed range of the vehicle, its rigid suspension system, and that its center of gravity is close to the local frame origin, the vehicle’s weight can be assumed to be evenly distributed for all the contact points [26]. The power consumption of the drivers PD has been also obtained by the simplex method. The cost function has been defined as the sum of the absolute value of the difference between measured PT from the watthour meter and the total modeled power in (20). The following values have been estimated: 1) power-up ≈ 58W; 2) braking power ≈ 164 W; 3) nonregenerated power ≈ (162 N) (|Vyl | + |Vyr |), where speeds are expressed in meters per second. Actual power PT of the locomotion system of Auriga-β, as measured by the watthour meter, for the wide and narrow belt sets over marble flooring are shown in Fig. 9(a) and (b), respectively. For the sake of clarity, only a half range of possible track speed combinations is presented (the other half is almost symmetrical with respect to the origin). It can be observed that power reaches maximum values when both tracks have the opposite top speeds (i.e., turning around the vehicle’s center at maximum angular speed), whereas minimum values of power occur when both track speeds are the same (i.e., straight-line motion with ωz = 0). Comparison of Fig. 9(a) and (b) also reveals that the narrow belt set requires less power when turning due to a smaller ground contact surface than the wide set.

Fig. 9. Experimental power consumption of Auriga-β over (a) marble flooring with wide tracks, (b) marble flooring with narrow tracks, and (c) hoisted.

The same experiments have been reproduced in Fig. 9(c) without ground friction (i.e., µ = 0) by hoisting the vehicle. Then, the estimated traction resistance constant is K ≈ 282 N. The relatively small reduction of this constant with respect to the



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Fig. 11. Estimated contribution of dynamic friction losses in P T (vehicle on marble flooring and with wide belts).

D. Validation of the Power Model

Fig. 10. Experimental power consumption of Auriga-β over asphalt with (a) wide and (b) narrow tracks.

on-the-ground values reveals that internal friction is the main factor in traction resistance for Auriga-β. Actual power PT for the wide and narrow belt sets over asphalt are shown in Fig. 10(a) and (b), respectively. It can be observed that power requirements are considerably larger than for marble flooring due to a greater friction coefficient. Contribution of dynamic friction losses to PT can be estimated through (14). This has been represented in Fig. 11 for marble flooring and wide belts. The dots correspond to the different track speed combinations presented in Fig. 9(a). These are enveloped by the estimations for the maximum longitudinal speed, as stated by (5), and for turning on spot, which has the maximum dynamic friction loss for every ωz . It is zero with ωz = 0, but it achieves high percentages when the absolute value of angular speed increases (close to 70% with ωzmax ). This figure illustrates the relevant role of dynamic friction on the total power consumption of the locomotion system. Note that in a nonslipping wheeled vehicle, this contribution is zero, regardless of the angular speed.

To validate the static power model, its response has been compared with mechanical power measurements PM . These have been recorded during manually operated spiral-like paths. Fig. 12(a) corresponds to a turn to the left with wide belts, while Fig. 12(b) represents a turn to the right with the narrow belt set, both on marble flooring. Five stages can be distinguished in both experiments, which have been labeled in Fig. 12 as the cases presented in Fig. 3. The first and the last stages correspond to straight-line motion (i.e., case 1 of Section II-B). The second and fourth parts (case 2) have different positive track speeds. Finally, the third part (case 4) corresponds to tracks with opposite speeds. Note that transitions between cases 2 and 4 require that one track speed is null. The previously identified parameters K and µ are employed to simulate power with (19). Note that the only inputs to the model are the instantaneous track speeds, i.e., no dynamics are considered. Hence, power fluctuations due to dynamic effects of the traction system are not simulated by the model. Nevertheless, it can be observed that modeled power requirements are similar to experimental data. Particularly, the mean absolute error and the standard deviation are e¯ = 84.5 W and σ = 91.6 W, and e¯ = 63.3 W and σ = 96.6 W for the wide and narrow belt experiments, respectively. IV. ENERGY IMPLICATIONS FOR NAVIGATION This section discusses how the preceding theoretical and experimental power analysis could be used to minimize the energy spent for navigation by skid-steer tracked vehicles. In particular, unlike nonholonomic vehicles with nonslipping wheels, two general guidelines can be considered.


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Fig. 13.

Heading control for a goal point.

Fig. 14. Two alternative paths from A to B. The presented tracked paths are for the case of narrow tracks and speed of 0.3 m/s.

reach the current goal from actual robot pose in straight-line motion [11], [21]. The motion control law compensates heading error φe (see Fig. 13) by actuating over the desired angular speed ωzsp as follows: ωzsp = G φe

Fig. 12. Real and simulated mechanical power during spiral-like paths on marble flooring with (a) wide and (b) narrow belts.

1) The navigation strategy should minimize the total amount of steered angle [i.e., ψ in (11)] as much as the robotic task can admit. This way, dynamic friction is minimized, which is a dominant factor for power requirements of full-size skid-steer tracked mobile robots. 2) Smooth trajectories may not be necessarily energyefficient. Instead, when turns are concentrated in a sharp way, straight-line motion is favored, which is the most energy-efficient case, especially for long distances. Regarding sharp turns, (5) implies that every angular speed has a maximum longitudinal speed. Therefore, the centrifugal acceleration of the vehicle (ωz vy ) is always kept bounded. A simple navigation strategy that complies with these guidelines is heading-error correction with way points. The navigation plan is composed of a list of goal points with connecting line segments that are assumed to be obstacle free. The aim of the controller is not to follow exactly the line segments but to

(23)

where G is the control gain. When the distance d between the vehicle and the goal point is less than a threshold radius dth , the next way point in the path becomes the current goal point. This may provoke a sharp turn to compensate for the new heading error. A. Navigation Results The aforementioned heading control has been implemented for Auriga-β. Experiments have taken place on a flat asphalted parking lot. Heading error is computed from DGPS. The next goal point is commanded when the distance of the vehicle to the current goal point is below dth = 1 m. The robotic task consists of avoiding an obstacle when going from point A to B (see Fig. 14). Two alternative solutions with similar path lengths (about 52 m) have been evaluated: one and two intermediate way points. These experiments have been repeated for different track widths, speeds, and control gains, as summarized in Table III. The value of ψ in (11) has been computed based on the onboard gyroscope. Path length has been estimated with the DGPS. It can be observed that the energy spent is always greater with the wide belts than with the narrow set. This result agrees with the values for the efficiency index presented in Table II.



TABLE III DATA OF THE TRACKED PATHS

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Future work includes the real-time estimation of the dynamic friction and traction resistance coefficients during navigation. Moreover, we are also interested on power requirements of wheeled skid-steer vehicles. REFERENCES

Energy consumption also increases for the case of four way points than with three way points because the total amount of steered angle is always greater, as the values of ψ indicate. This result agrees with the first navigation guideline. In Table III, it is noticeable that the tracked vehicle always needs more energy when moving at a reference longitudinal speed of vysp = 0.3 m/s than at 0.5 m/s. This is mainly because motor drivers need to be powered up during more time (about 68 s longer). The followed path, as recorded by DGPS, is closer to the line segments between the goal points in the case of G = 0.18, and smoother transitions are obtained with G = 0.12 (see Fig. 14). Data in Table III reveals that G = 0.18 always requires less energy than G = 0.12. This result agrees with the second navigation guideline. V. CONCLUSION Tracked robots have a considerable ground contact area to improve traction on natural terrains, but power losses increase due to slippage. This issue has not been specifically studied in robotics literature. The paper proposes a static power model for skid-steer tracked vehicles moving at walking speeds on hard plane terrains that only depends on instantaneous track speeds. Dynamic friction losses have been modeled as the power drawn by the terrain and also as the power supplied by the motors. The comparison of both approaches has provided useful insight in this kind of locomotion system. Traction resistance and other power losses complete the model. The mobile robot Auriga-β has been presented as a case study. Experimental power parameters have been obtained and validated. The relevance of dynamic friction losses with respect to total power consumption has been confirmed. Furthermore, some general energy-related navigation guidelines for skid-steer tracked vehicles have been deduced and tested. These take into account that consumed energy heavily depends on the total turned angle.

[1] S. Michaud, A. Schneider, R. Bertrand, P. Lamon, R. Siegwart, M. Winnendael, and A. Schiele, “Solero: Solar-powered exploration rover,” presented at the 7th ESA Workshop Adv. Space Technol. Robot. Autom., Noordwijk, The Netherlands, 2002. [2] D. Wang and C. B. Low, “Modeling and analysis of skidding and slipping in wheeled mobile robots: Control design perspective,” IEEE Trans. Robot., vol. 24, no. 3, pp. 676–687, Jun. 2008. [3] P. F. Muir and C. P. Neuman, “Kinematic modeling of wheeled mobile robots,” Robot. Inst., Carnegie Mellon Univ., Pittsburgh, PA, Tech. Rep. CMU-RI-TR-86-12, 1986. [4] J. Liu, P. Chou, N. Bagherzadeh, and F. Kurdahi, “Power-aware scheduling under timing constraints for mission-critical embedded systems,” in Proc. ACM IEEE Des. Autom. Conf., Las Vegas, NV, 2001, pp. 840–845. [5] J. Brateman, C. Xian, and Y.-H. Lu, “Energy-efficient scheduling for autonomous mobile robots,” in Proc. Int. Conf. Very Large Scale Integr., Nice, France, 2006, pp. 361–366. [6] I. Duleba and J. Z. Sasiadek, “Nonholonomic motion planning based on Newton algorithm with energy optimization,” IEEE Trans. Control Syst. Technol., vol. 11, no. 3, pp. 355–363, May 2003. [7] Y. K. Hwang and N. Ahuja, “Gross motion planning—A survey,” ACM Comput. Surveys, vol. 24, no. 3, pp. 219–291, Sep. 1992. [8] Z. Sun and J. H. Reif, “On finding energy-minimizing paths on terrains,” IEEE Trans. Robot., vol. 21, no. 1, pp. 102–114, Feb. 2005. [9] C. H. Kim and B. K. Kim, “Energy-saving 3-step velocity control algorithm for battery-powered wheeled mobile robots,” in Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005, pp. 2375–2380. [10] A. Barili, M. Ceresa, and C. Parisi, “Energy-saving motion control for an autonomous mobile robot,” in Proc. IEEE Int. Symp. Ind. Electron., Athens, Greece, 1995, pp. 674–676. [11] B. M. Leedy, J. S. Putney, C. Bauman, S. Cacciola, J. M. Webster, and C. F. Reinholtz, “Virginia Tech’s twin contenders: A comparative study of reactive and deliberative navigation,” J. Field Robot., vol. 23, no. 9, pp. 709–727, 2006. [12] S. Ancenay and F. Maire, “A time and energy optimal controller for mobile robots,” Lecture Notes Comput. Sci., vol. 3339, pp. 1181–1186, Dec. 2004. [13] Y. Mei, Y. Lu, Y. Hu, and C. Lee, “Energy-efficient motion planning for mobile robots,” in Proc. IEEE Int. Conf. Robot. Autom., New Orleans, LA, 2004, pp. 4344–4349. [14] K. Iagnemma and S. Dubowsky, “Traction control of wheeled robotic vehicles in rough terrain with application to planetary rovers,” Int. J. Robot. Res., vol. 23, no. 10, pp. 1029–1040, Oct. 2004. [15] F. Silva and J. Tenreiro-Machado, “Energy analysis during biped walking,” in Proc. IEEE Int. Conf. Robot. Autom., Detroit, MI, 1999, pp. 59–64. [16] M. Saito, M. Fukaya, and T. Iwasaki, “Serpentine locomotion with robotic snakes,” IEEE Control Syst. Mag., vol. 22, no. 1, pp. 64–81, Feb. 2002. [17] J. Y. Wong and W. Huang, “Wheels vs. tracks—A fundamental evaluation from the traction perspective,” J. Terramech., vol. 43, no. 1, pp. 27–42, Jan. 2006. [18] D. Endo, Y. Okada, K. Nagatani, and K. Yoshida, “Path following control for tracked vehicles based on slip-compensation odometry,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., San Diego, CA, 2007, pp. 2871– 2876. [19] J. L. Mart´ınez, A. Mandow, J. Morales, S. Pedraza, and A. Garc´ıa-Cerezo, “Approximating kinematics for tracked mobile robots,” Int. J. Robot. Res., vol. 24, no. 10, pp. 867–878, Oct. 2005. [20] A. Mandow, J. L. Mart´ınez, J. Morales, J. L. Blanco, A. J. Garc´ıa Cerezo, and J. González, “Experimental kinematics for wheeled skid-steer mobile robots,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., San Diego, CA, 2007, pp. 1222–1227. [21] J. Morales, J. L. Mart´ınez, A. Mandow, A. J. Garc´ıa-Cerezo, J. M. GómezGabriel, and S. Pedraza, “Power analysis for a skid-steered tracked mobile robot,” in Proc. IEEE Int. Conf. Mechatron., Budapest, Hungary, 2006, pp. 420–425. [22] J. Y. Wong, Theory of Ground Vehicles, 3rd ed. New York: Wiley, 2001, ch. 6.


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[23] Z. Shiller, W. Serate, and M. Hua, “Trajectory planning of tracked vehicles,” in Proc. IEEE Int. Conf. Robot. Autom., Atlanta, GA, 1993, pp. 796– 801. [24] L. Merino, F. Caballero, J. R. Mart´ınez-Dios, J. Ferruz, and A. Ollero, “A cooperative perception system for multiple UAVs: Application to automatic detection of forest fires,” J. Field Robot., vol. 23, no. 3/4, pp. 165– 184, 2006. [25] J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J., vol. 7, no. 4, pp. 308–313, 1965. [26] M. Kitano, K. Watanabe, and N. Nagatomo, “Stability and controllability of high speed tracked vehicles: Linear model and vehicle response,” in Proc. 10th Conf. Int. Soc. Terrain-Vehicle Syst., Kobe, Japan, 1990, pp. 659–670.

´ Morales received the M.Sc. degree in electriJesus cal engineering and the Ph.D. degree with European Mention, both from the University of Málaga, Málaga, Spain, in 2001 and 2007, respectively. In 2002, he joined the System Engineering and Automation Research Group, University of Málaga, where he is currently an Assistant Professor with the Engineering School. His current research interests include mobile robotics and medical robot applications. He has authored or coauthored six international journal papers and nine conference papers.

Jorge L. Mart´ınez received the Ph.D. degree in computer science from the University of Málaga, Málaga, Spain, in 1994. Since 1998, he has been an Associate Professor with the Department of Systems Engineering and Automation, University of Málaga. He has authored or coauthored 13 international journal papers, 33 conference papers, and two book chapters on different aspects of mobile robotics.

Alfonso J. Garc´ıa-Cerezo (M’94) received the Ind. Electr. Eng. and the Doctoral Eng. degrees from the Escuela Tecnica Superior de Ingenieros Industriales of Vigo, Vigo, Spain, in 1983 and 1987, respectively. From 1983 to 1988, he was an Associate Professor with the Department of Electrical Engineering, Computers, and Systems, University of Santiago de Compostela, Santiago de Compostela, Spain, where he was an Assistant Professor from 1988 to 1991. Since 1992, he has been a Professor of system engineering and automation with the University of Málaga, Málaga, Spain, where he was the Head of the Escuela Tecnica Superior de Ingenieros Industriales de Málaga from 1993 to 2004, is currently the Head of the Department of System Engineering and Automation, and is also responsible for the Instituto de Automática Avanzada y Robótica de Andaluc´ıa. He has authored or coauthored about 150 journal articles, conference papers, book chapters, and technical reports. His current research interests include mobile robots and autonomous vehicles, surgical robotics, and mechatronics and intelligent control. He has also been involved in more than 15 research projects over the past ten years. Prof. Garc´ıa-Cerezo is a member of the International Federation of Automatic Control, the Spanish Production Technology Automation and Robotics Association, and the Comité Español de Automática. He was the General Chair of the 2009 IEEE International Conference of Mechatronics. Since September 2008, he has been a Coordinator of the Spanish Thematic Group of Robotics.

Salvador Pedraza received the M.S. degree in electrical engineering and the Ph.D. degree in control engineering from the University of Málaga, Málaga, Spain, in 1995 and 2000, respectively. He was engaged in research on modeling and control of mobile robots. He was with Nokia Networks and, since 2003, has been with Optimi Corporation, Institutos Universitarios, Málaga, where he is involved in the mobile telephony research arena. His current research interests include mobile robots modeling and optimization techniques for mobile network performance.

Anthony Mandow (M’08) received the Engineering and Ph.D. degrees in computer science from the University of Málaga, Málaga, Spain, in 1992 and 1998, respectively. He is currently an Associate Professor with the Department of Systems Engineering and Automation, University of Málaga. He has been engaged in several robotics and automation projects. He has authored or coauthored more than 30 conference and journal papers. His current research interests include field robotics, robot learning, search and rescue operations, 3-D perception, and vehicle motion control. Dr. Mandow was the Organizing Chair of the 2009 IEEE International Conference on Mechatronics.
