1

Time-Optimal Path Planning with Power Schedules for aSolar-Powered Ground Robot

Adam Kaplan, Nathaniel Kingry, Paul Uhing, and Ran Dai, Member, IEEE

Abstract—This paper examines an integrated path planningand power management problem for a solar-powered unmannedground vehicle (UGV). The proposed method seeks to minimizethe travel time of the UGV through an area of known energydensity by designing a smooth, heuristically optimized path andallocating the vehicle’s power among its electrical components,while the UGV harvests ambient energy along the designed pathto satisfy with the mission’s strict energy constraints. A scalarfield is first established to evaluate the solar radiation densityat discrete locations. A modified Particle Swarm Optimizationmethod is applied to search for a minimal time path wherein theenergy gathered is equal to or greater than the energy expended.The proposed modeling and optimization strategy is verifiedthrough computer simulation and experimental demonstration.

Note to Practitioners—With the advancement of autonomoustechnology, robotic systems have alleviated human operatorsfrom numerous tedious tasks where system endurance playsa crucial role. Current technology employed in solar-poweredrobotic systems is subject to design and power limitations andvarying environments. Intelligently harvesting energy from envi-ronments and scheduling power consumption to optimize desiredsystem performance will significantly improve the solar robot’sendurance. Therefore, the goal of realizing energy autonomy ofsolar-powered robotic systems motivates this work. Any roboticsystem which can access solar energy will potentially benefit fromthe results of this work. With solar panels integrated into roboticsystems applied to missions such as environmental monitoring,search and rescue, surveillance, and farming, the proposed pathplanning and power management strategy has the potential todramatically extend the operation time of a system.

Index Terms—Solar-Powered Robot, Path Planning, PowerManagement, Particle Swarm Optimization

I. INTRODUCTION

Unmanned robotic vehicles have proven to be uniquelyadept at dull, dirty, and dangerous tasks [1]. Dull,long-duration missions, such as environmental researchand analysis, communications, and information-surveillance-reconnaissance, stand to directly benefit from prolonged vehi-cle operation capabilities and increased loiter-times. In manyunmanned systems, an increased vehicle endurance is directlyproportional to a reduction in operating expenditures (power,fuel, etc.). Several approaches have been proposed to prolonga vehicle’s operating time, such as energy efficient pathplanning [2] and the application of power consumption modelsand saving strategies under different paths [3].

There is a great deal of interest in the robotics communityin developing long-duration solar-powered robotic systems for

Adam Kaplan, Nathaniel Kingry and Ran Dai are with the AerospaceEngineering Department, Iowa State University, Ames, IA, 50011 USAe-mail: akaplan@iastate.edu, nkingry@iastate.eduand dairan@iastate.edu

Paul Uhing is with the Electrical Engineering Department, Iowa StateUniversity, Ames, IA, 50011 USA e-mail:pfuhing@iastate.edu

UGVs [4]–[6] that utilize the energy from the environment andcharge storage batteries as backups for sustained operation.For example, the Mars Opportunity Rover, which persistentlyexplores unknown areas on Mars, has been working for tenyears using only solar power [7]. Another example is the“cool robot” designed to carry payloads during summer in theAntarctic, where the solar radiation distribution is assumedto be uniform. These examples demonstrate the unique roleof long-duration solar-powered robots in both civil and ex-traterrestrial applications where a constant power supply isnot readily available.

While operational areas such as Mars or the Antarctic maybe considered to have a uniform solar energy distribution forsome period of time, many environments feature nonuniformdistributions which require careful planning, including bothpower schedule and movement plan, to make the most oftheir available ambient energy. Despite this, research on solar-powered vehicles has focused primarily on their design as-pects, i.e., reasonable dimensions, payload, weight [8], and etc.On the other hand, the energy efficient deployment and oper-ation of UGVs using non-renewable energy in surveillance,monitoring or tracking missions has received much attention.Examples include power allocation among UGVs to maintainarea coverage [9], path planning to explore unknown areaswith limited power [10], power consumption models and con-servation strategies in path searching [11], among others [12]–[14]. However, little attention has been paid to simultaneousenergy harvesting and power management of solar-poweredunmanned vehicles. Although dynamic power managementmicrocontrollers have been investigated and implemented inunmanned vehicles [15], [16], these controllers simply allocatepower according to the solar energy available to the system ateach moment, where available power is first allocated to themotors to maintain motion, and the remainder is used to chargea backup battery. Without long-term mission planning, suchcontrollers cannot guarantee persistent operation and globaloptimal performance in nonuniform solar radiation distributionarea.

For example, consider a vehicle equipped with a solarpanel which must traverse an area with a nonuniform solarenergy distribution through a subregion which offers littleavailable solar power. For missions that require minimal traveltime while constraining the vehicle’s net energy change, it isinsufficient to travel at a constant energy-efficient speed. Thevehicle would instead benefit from using knowledge of thearea’s solar energy density to generate a motion plan such thatthe vehicle would travel quickly through low-energy regionsand slowly enough through high-energy areas so that it wouldsatisfy its energy constraint.

This paper describes an integrated path planning and power

2

management method to minimize the travel time of a solar-powered UGV though an area of known energy distribution,subject to a net energy constraint, by optimizing the UGV’spath and power allocation schedule against a scalar fieldinterpolation of solar radiation distribution. Characterizingsolar radiation distributions is a nontrivial problem; previousresearch has emphasized the importance of precise estimationof the solar radiation distribution and its impact on missionsuccess for solar-powered UGVs [17]. Existing software suchas ArcGIS [18] can build digital elevation models (DEMs)of ground surfaces from two-dimensional geographic images,and transform a DEM into a solar radiation map. Ratherthan addressing how to best obtain a solar energy densitydistribution, the work contained in this paper focuses on howto use such a solar radiation map to improve the performanceof an unmanned vehicle.

The motion planning method discussed in this paper, amodified Particle Swarm Optimizer (PSO), is used to generatea modified form of the classical Dubins curve [19], known as aPseudo-Dubins curve, which substitutes the traditional uniformminimum turning radii with individually designed turningradii. This modified PSO is advantageous over conventionalmotion planning methods for a multitude of reasons. Solarradiation distributions are often highly sporadic, making itinfeasible to generate continuous mathematical representationsto precisely estimate radiation values with respect to theirlocation. While this attribute precludes the use of continuousoptimizers, such as nonlinear programming [20], PSO canprecisely interpolate data from discrete samples. Furthermore,the problem posed in this paper requires only that the UGV’snet energy change be greater than or equal to zero, anddoes not necessarily prevent the vehicle from expending someamount of energy from an on-board battery.

Conventional heuristic motion planning methods such asA* [21], Probabilistic Road Maps (PRM) [22], and Rapidly-exploring Random Trees (RRT) [23], [24] have been usedto identify optimal paths constructed from a series of pathsegments, particularly in obstacle-avoidance scenarios. To gen-erate a time-optimal path which satisfies our problem’s netenergy constraint, each possible path segment identified bythese continental planners must satisfy the constraint, as anyof them may be used to construct path. However, the modifiedPSO proposed in this paper is capable of generating solutionsin which individual path segments are permitted to violate theenergy constraint so long as the path meets the net energyconstraint.”

The organization of the paper is as follows: the problem for-mulation with UGV dynamics, energy harvesting, and powerconsumption models are introduced in §II. We then presenta modified PSO algorithm for integrated path planning andpower scheduling in §III. The robot design and simulationenvironments are presented in §IV, followed by virtual andexperimental simulation results in §V. We conclude the paperwith a few remarks in §VI.

II. PROBLEM DESCRIPTION

A. Path Segments and Design Variables

Consider a UGV with well-defined performance character-istics, equipped with a solar panel, which is retasked withtraveling through a two-dimensional area of known solar-energy density in such a way that the UGV’s total expendedenergy is less than or equal to what it gained in transit.The vehicle’s initial coordinate W1(x1, y1) and desired finalcoordinate Wn(xn, yn) are known, and the initial headingangle is set as zero.

The UGV’s movement is governed by two differential-driveprimitives including straight lines, where both drive wheelsmove with same speed and direction, and circular-arc turns,where both drive wheels move in the same direction but atdifferent speeds. These two types of motion comprise thepreviously introduced Pseudo-Dubins curve. While a classicalDubins curve is characterized by circular turns using thevehicle’s minimum turning radius, imposing such a constrainton vehicle capable of performing zero radius turns yields aBalkcom-Mason curve [25]. Allowing the radii of the UGV’sturns to vary not only allows the principles of the Dubins curveto be applied to a differential drive vehicle, but also allows theUGV to preserve variable portions of its translational velocitywhile changing its orientation, and in the case of our UGV,require less power than the zero-radius turns employed underthe Balkcom-Mason curve.

The path from the initial point W1 to the final point Wn

is composed of n− 2 interception points and n− 1 turn andline pairs. To simplify the path planning problem, the designvariables for the Pseudo-Dubins curve are comprised of theintersection points, denoted as Wi(xi, yi), i = 2, . . . , n − 1,the linear speed along the line segment, denoted as Vi, i =1, . . . , n−1, and angular speed along the turn segment, denotedas ωi, i = 1, . . . , n−1. The path segments and their geometryrelationship are demonstrated below in Fig. 1.

Fig. 1: Path segments of the Pseudo-Dubins curve.

The shape of the planned path can be determined from theselected design variables. For example, the starting point Si

and ending point Ei of circular-arc i are determined by

xSi = xi −Ri cos θi, ySi = yi −Ri sin θi, (2.1)xEi = xi +Ri cos θi, yEi = yi +Ri sin θi, (2.2)

∀ i = 2, . . . , n− 1,

where Ri is the turn radius of circular-arc i, and is predefinedby a polynomial function with respect to the angular velocity,

3

denoted as fR(ωi). The heading angle θi along line segmenti is found from

θi = tan−1 yi+1 − yixi+1 − xi

, ∀ i = 2, . . . , n− 1. (2.3)

Equations 2.1 - 2.3 may be used for circular-arcs i, i =2, . . . , n − 1. For the first circular-arc, i = 1, S1 coincideswith the specified initial point W1, such that xS1

= x1 andyS1

= y1. The coordinate of E1 are determined by

xE1= x2 − sign(x2 − x1)d1 cos(tan−1 y2 − yr

x2 − xr

− sin−1 R1

d1), (2.4)

yE1 = y2 − sign(y2 − y1)d1 sin(tan−1 y2 − yr

x2 − xr

− sin−1 R1

d1), (2.5)

where xr = x1 and yr = y1 + sign(y2 − y1)R1 are thecoordinates for the center of the first circular-arc, R1 can againbe determined from the polynomial function fR(ω1), and d1 =√

(y2 − yr)2 + (x2 − xr)2.The time required for a UGV to complete the arc turns and

line segments of the entire path is expressed as

Ttotal =

n−1∑i=1

(T r(i) + T l(i)), (2.6)

where the superscripts ‘r’ and ‘l’ denote variables associatedwith the circular-arc turn and line segment, respectively. Withthe corresponding design variables, ωi and Vi, the entiretransition time can be reformulated as

Ttotal =

n−1∑i=1

(∆θiωi

+Li

Vi), (2.7)

where ∆θi = θi+1 − θi, i = 1, . . . , n − 1, is thechanged heading angle along circular-arc i and Li =√

(xSi+1− xEi

)2 + (ySi+1− yEi

)2 is the length of line-segment i.

B. Energy Harvesting and Consumption Models

The gained energy along each turn and line segment isan integral function of the energy collection rate, Pin, overrelative time span, where Pin is dependent on the solarradiation strength over the concerned area and the solar panelarea of the UGV, denoted as As. To more accurately evaluatethe solar radiation map, a scalar field interpolation, Rin(a, b) isconstructed from a number of discrete solar energy samples torepresent the solar energy distribution for a given area. Alongeach turn and line segment i, Q equidistantly spaced samplesacross the scalar field from the starting to the ending pointsare used to evaluate the average solar radiation strength along

the line segment. Under these parameters, the gathered energyis determined by

Elin(i) =

Q∑q=1

Rin(aq, bq)AsLi/(QVi) (2.8)

Erin(i) =

Q∑q=1

Rin(aq, bq)As∆θi/(Qωi). (2.9)

The energy consumed by the UGV to complete each seg-ment is determined by the engine consumption rate P l

e(i) andP re (i), e = 1, . . . , Ne, where Ne is the number of engines, the

power required by the on-board components (microcontroller,wireless modem, voltage/current sensors, etc.), and time re-quired to finish the corresponding segment. Each engine con-sumption rate under both linear and rotational movements isa predefined polynomial function of corresponding linear andangular speed, denoted as P l

e(Vi) and P re (ωi), e = 1, . . . , Ne,

respectively. In order to simplify the expression, we assumethe power controller will always allocate the necessary powerto the microcontroller, wireless sensor, and idle power. Thesummation from the three consumption units is denoted asthe vehicle’s passive power consumption, Pa. Mathematically,the consumed energy is represented by

Elout(i) = (

∑Ne

e=1 Ple(Vi) + Pa)T l(i) (2.10)

Erout(i) = (

∑Ne

e=1 Pre (ωi) + Pa)T r(i) (2.11)

To satisfy the energy constraint at the final point, Wn, suchthat the net energy change ∆Etotal at the end of the vehicle’smovement is above zero, we have

∆Etotal =

n−1∑i=1

(Elin(i) + Er

in(i)

−Elout(i)− Er

out(i)) ≥ 0. (2.12)

The engine power may be supplied from two resources, theharvested energy from environment and the backup battery.At each movement segment, the power controller allocatesrequested power amount to each engine to achieve the de-sired UGV motion. Therefore, the power among all electriccomponents is balanced by

Ne∑e=1

P le(Vi) + Pa + P l

b(i) =

Q∑q=1

Rin(aq, bq)As/Q, (2.13)

Ne∑e=1

P re (ωi) + Pa + P r

b (i) =

Q∑q=1

Rin(aq, bq)As/Q, (2.14)

where the battery consumed/supplied power, P lb(i) and P r

b (i)is constrained by its capacity, Pbmax and Pbmin , such thatPbmin

≤ P lb(i) ≤ Pbmax

and Pbmin≤ P r

b (i) ≤ Pbmax.

C. Formulation of Path Planning Problem

With the path, energy harvesting, and consumption modelsdescribed above, the integrated path planning and power

4

management problem can be summarized as

minX∑n−1

i=1 (T l(i) + T r(i)) (2.15)

s.t.,∑n−1

i=1 (Elin(i) + Er

in(i)− Elout(i)− Er

out(i)) ≥ 0∑Ne

e=1 Ple(Vi) + Pa + P l

b(i) =∑Q

q=1Rin(aq, bq)As/Q∑Ne

e=1 Pre (ωi) + Pa) + P r

b (i) =∑Q

q=1Rin(aq, bq)As/Q

Pbmin≤ P l

b(i) ≤ Pbmax

Pbmin≤ P r

b (i) ≤ Pbmax,

where the variable set X to be optimized includes the coordi-nates of n − 2 interception points (xi, yi), i = 2, . . . , n − 1,the n−1 linear speed Vi, and the n−1 angular speed ωi alongeach segment i, i = 1, . . . , n− 1. The path planning problemdescribed above attempts to use the minimum number ofdesign variables to find the time optimal path while satisfyingthe geometry and power constraints.

III. MODIFIED PARTICLE SWARM OPTIMIZATION METHOD

A. Introduction of PSO Method

Particle Swarm Optimization, first introduced by Kennedyand Eberhart [26], is a heuristic optimization method whichiteratively evaluates and adjusts a number J of possiblesolutions to the optimization problem, represented as particles.These “particles” exist in a space consisting of as manydimensions d as there are optimization variables, such thateach particle represents a distinct solution

P j = [xj1, xj2, . . . , x

jd], ∀ j = 1, . . . , J. (3.16)

These particles are iteratively perturbed a varying distance Min each dimension i in the direction of the particle’s personalbest performing location, Pbest, and the group’s single bestperforming location, Gbest, influenced by the local and globalbest weights C1 and C2, which are typically set equal to2. Both movements are multiplied by a random number Rbetween 0 and 1, and each of the particle’s subsequent move-ments in a given dimension are influenced by the previousmovement M j

i (h − 1), multiplied by an inertia factor I .Mathematically, at each iteration step h, M j

i (h) is expressedas

M ji (h) = IM j

i (h− 1) + (C1R(P jbesti− P j

i ))

+(C2R(Gbesti − Pji )). (3.17)

Each particle is reevaluated after its movement, and if ap-propriate, replaces its previous Pbest location and the group’sGbest. In a typical minimization problem, these update evalu-ations take the form of

P jbest = pj , if v(P j

best) < v(pj) (3.18)Gbest = pj , if v(Gbest) < v(pj), (3.19)

where v(pj), v(P jbest), and v(Gbest) represent the values

produced by the design variables in the jth particle, thejth particle’s best location, and the group’s best location,respectively. Subsequent updates to Gbest will identify aprogressively optimal solution set, eventually leading to a localor global extrema. Upon initialization, each particle’s Pbest

must be its present position. As a result the initial movementexpression Minitial may be simplified as

M ji initial = C2R(Gbesti − P

ji ). (3.20)

The particle which is found to be the initial global, bestsuch that P j

i = Gbesti , will experience no movement, i.e.M j

i initial = 0.

B. A Modified PSO for UGV Motion Planning

PSO is an iterative process, as each particle must be reeval-uated between movements and compared to its previous valuein order to gradually shift the swarm towards an extrema. Sucha mechanism makes it an appropriate method to be applied inconjunction with our solar energy density scalar field interpo-lation. In the integrated path planning and power managementproblem formulated in (2.15), each particle P consists of n−2interception points Wi = (xi, yi), i = 2, . . . , n − 1, andn − 1 pair of speed settings Ui = (Vi, ωi), i = 1, . . . , n − 1,summarized as

P j = [W j2 , . . . ,W

jn−1, U

j1 , . . . , U

jn−1]

= [xj2, yj2, . . . , x

jn−1, y

jn−1, V

j1 , ω

j1, . . . , V

jn−1, ω

jn−1].

Because the dimension of each particle directly correlatesto the number of intersection points n, a small n will decreasethe PSO’s computation time but also may lead to an infeasiblesolution. Conversely, specifying a large n will increase thecomputation time in exchange for a greater likelihood ofa feasible solution with an improved performance index.Therefore, the dimension n must be carefully chosen basedoff of the physical dimensions of the problem area and thesparsity of the available solar energy.

In our previous work [27], the Pbest and Gbest updates occuronly when a particle’s Ttotal is less than the previous Pbest

or Gbest value and when the particle’s ∆Etotal satisfies theenergy constraint, which are expressed as

P jbest = P j , if [Ttotal(P

jbest) > Ttotal(P

j)]

&[∆Etotal(Pj) > 0], (3.21)

Gbest = P j , if [Ttotal(Gbest) > Ttotal(Pj)]

&[∆Etotal(Pj) > 0]. (3.22)

Due to the strict constraints in the above update conditions, itis difficult to generate random particles in the initialization stepto guarantee that each particle satisfies the energy constraints.In the method employed by this paper, we utilize a simplerinitialization method and particularly impose a new Pbest

update relationship to incentivize the particles’ gradual energy-constraint.

At the initialization step, each Pj is pseudorandomly pop-ulated such that x1 < xj2 < . . . < xjd−1 < xd and all W andU values fall within permissible ranges originally defined bythe problem. Each particle’s location, P j , and correspondingobjective value, Ttotal(P j), are saved regardless of the valuesof ∆Etotal(P

j). If any particle exists with a ∆Etotal > 0,Gbest will be selected from that group by looking for thelowest Ttotal. If none of the initialized particles meet the

5

energy constraint, then Gbest will be selected as the particlewith the ∆Etotal closest to zero.

Once the particles have been initialized, they are iterativelyperturbed as described in § III-A. After each individual particlehas been perturbed, it is evaluated against its Pbest and thegroup’s Gbest, and if necessary, they are replaced by

P jbest = P j , if [Ttotal(P

jbest) > Ttotal(P

j)]

&[∆Etotal(Pj) > 0], (3.23)

or

P jbest = P j , if [∆Etotal(P

j) > ∆Etotal(Pjbest)]

&[∆Etotal(Pjbest) < 0], (3.24)

and

Gbest = P j , if [Ttotal(Gbest) > Ttotal(Pjbest)]

&[∆Etotal(Pj) > 0]. (3.25)

By imposing this dual Pbest update relationship, we allowparticles which are not necessarily power consistent uponinitialization to gradually move towards satisfying the prob-lem’s energy constraint. Once a particle has achieved energyconsistence, the first update relationship continues to driveit towards a minimal Ttotal. The modified PSO method issummarized in Fig. 2.

Fig. 2: Modified PSO method.

IV. VEHICLE DESIGN AND SIMULATION ENVIRONMENT

A. Vehicle Design and Power System Control

For a vehicle to be capable of both executing and validatingthe prescribed motion plan, it must be capable of effectivelygathering ambient solar energy and converting it to the op-erating voltage level, while also measuring the incoming andoutgoing energy of the system. The demonstration vehicle,pictured in Fig. 3, is based on a Dagu 5 robot chassis [28]controlled by an Arduino Uno [29] and a wireless modem. Theambient solar energy is gathered through a top-mounted 18Wsolar panel, which indicates the approximate intake power forthis solar panel is 18W under bright sun light, i.e., a solarradiation density of 1 kW/m2.

Recall that the efficiency of a given solar panel is char-acterized by a nonlinear current-voltage curve, and for thisreason the resistance which yields the maximum power point

Fig. 3: Demonstration vehicle with 18W solar panel.

of a solar panel varies. Subsequently, any energy gathered bythe UGV’s solar panel is directed to an on-board MaximumPower Point Tracking (MPPT) solar charger, which adjustsits resistance to the solar panel in order to produce theoptimal load. The MPPT may then pass the energy on toeither the vehicle’s onboard battery or load (onboard computer,motors, etc). Recovered ambient energy is first delegated to thevehicle’s load, and any remaining energy is then used to chargethe battery. In the event that the recovered ambient energy isinsufficient to supply the required load power, the difference issupplied by the battery. Two voltage/current (V/C) sensors areplaced on either side of the MPPT’s outgoing lines, as shownin Fig. 4. These sensors monitor the vehicle’s power flow andprovide experimental data in real time.

Fig. 4: Vehicle power system block diagram.

B. Simulation Environment

Solar panels generate their advertised power output under asolar radiation density of 1 kW/m2, which is the approximatevalue produced during a bright, cloudless day [30]. In orderto simulate such sunlight within the confines of a researchlab located underground, we employ high pressure sodiumlights which are commonly used in industrial agriculturegreenhouses, shown in Fig. 5. This indoor solar simulator cangenerate a solar radiation output around 0.6 kW/m2 underthe center of the light. This indoor solar environment producesapproximately half of the outdoor average solar radiation; thus,a verified indoor solar robotic system is likely to achievehigher performance in outdoor tests during sunny or partlycloudy days.

For the purpose of our experiments we designed a mappingvehicle to gather the discrete energy samples necessary toconstruct the ambient energy scalar field, Rin(a, b), used foroptimization. The mapping vehicle is based off of a simpleArduino tank-style robot, and features a 2W solar panel. Thissolar panel is connected to a voltage/current sensor alongwith a resistor chosen to create the optimal load for thearrays maximum power point. This allows for consistent, albeitslightly suboptimal, energy density sampling. Furthermore, the

6

Fig. 5: Indoor Test Environment.

lab is equipped with Vicon [31] camera system that can capturethe motion of the robot in a volume of 6 × 9 × 6 m3. Withthe constant solar radiation output using the high pressuresodium lamps and precise sampling of the solar radiation map,simulations in the created environment are used to calibratethe UGV performance under the given radiation map. Thesesimulation results allow precise evaluation of the proposedpath planning and power management algorithm.

V. RESULTS

We now present a number of examples using both virtualand experimental scenarios. The linear and angular speed ofthe UGV is controlled by adjusting the pulse-width modulation(PWM) of the voltage supplied to the motors. For straightline motion, the PWM of both drive-wheels are set the samenumber. For a circular-arc turn, the interior drive-wheel’sPWM is held constant at a predefined value while PWM of theexterior drive-wheel is adjusted to achieve the desired motion.Polynomial functions of the linear speed, angular speed, andconsumed power in terms of PWM are interpolated from largeamount of sampled data and are illustrated in Fig. 6. Withthe following polynomial functions, one can control the UGVspeed settings at permissible regions and accurately evaluatethe UGV performance under corresponding settings.

Fig. 6: Polynomial functions of the linear speed, angular speed,and consumed power in terms of PWM for the testbed UGV.

Each scenario present solutions for a naive straight-lineplan, a Balkcom-Mason plan, and a Pseudo-Dubins. Addi-tionally, §V-A also features a plan adapted from the Bellman-Ford algorithm [32] to help demonstrate the advantages of themodified PSO over conventional motion planning methods forthis type of problem. Both of the Balkcom-Mason and Pseudo-Dubins plans were generated from the modified PSO describedin §III using n = 5 waypoints, J = 2500 particles over 5cycles, and PSO settings of I = 1 and C1 = C2 = 2. Each

section has three solutions characterized by the UGV’s path,the anticipated intake power, PIN ( ), the consumed power,POUT ( ), and the change of battery energy, ∆Battery( ). In Scenario 2, the corresponding recorded experimentaldata is denoted as PIN ( ), POUT ( ), and ∆Battery( ). Please note that energy gained by the system is de-noted as positive, while energy expended from the system isnegative.

A. Scenario 1

We begin by examining a simulated 5.5 × 5.5 m2 sparse-energy area which yields a maximum power intake of lessthan 7W for the aforementioned UGV. A simple straight-linesolution, using the greatest possible uniform-velocity whichcan satisfy our energy constraint, is pictured below in Fig. 7and Fig. 8. This solution features a total travel time of 32.27seconds and yields a net energy gain of 0.43 J.

Fig. 7: Straight-Line Path for Virtual Scenario

Fig. 8: Simulated Straight-Line Power & Energy Schedule

We then generate a 5-waypoint Balkcom-Mason motionplan [27], presented below in Fig. 9 and Fig. 10. This planrequires the vehicle to move for 26.86 seconds and predicts a1.05 J net energy gain.

Next, we generate a 5-waypoint Pseudo-Dubins motionplan, summarized below in Fig. 11 and Fig. 12. The Pseudo-

7

Fig. 9: Balkcom-Mason Path for Virtual Scenario

Fig. 10: Simulated Balkcom-Mason Power & Energy Schedule

Dubins solution predicts a 26.03 second travel time accompa-nied by a 0.26 J net energy increase.

Fig. 11: Pseudo-Dubins Path for Virtual ScenarioWhile both the Balkcom-Mason and Pseudo-Dubins plans

offer a significant time reduction from the naive straight-line solution (16.76% and 19.34%, respectively), the Pseudo-Dubins offers only a 3.09% reduction from the Balkcom-Mason motion plan.

To demonstrate the advantages of the modified PSO overconventional motion-planning methods in solving this type ofproblem, we conclude by examining a motion plan generated

Fig. 12: Simulated Pseudo-Dubins Power & Energy Schedule

from a fixed-grid Bellman-Ford algorithm. Like other heuristicmotion planners discussed in §I, the fixed-grid Bellman-Fordalgorithm works by identifying the optimal path from a set ofmultiple path segments. The simulated environment is dividedinto a set of grids and path segments, which are denoted by theedges connecting any node (indicated as circles in Fig. 13) ina given column to any other node in the adjacent columns.The cost of each segment is determined by the minimumamount of time required for the vehicle to travel each edgewhile satisfying the net energy gain. Thus, the net energy gainconstraint for the entire path is enforced upon each samplesegment. For any edge that cannot satisfy the net energy gainconstraint, a arbitrarily large cost is assigned. Grids comprisedof 5× 5, 7× 7, . . ., 25× 25 nodes were designed and solved,tabulated below in Table I, while the best performing motionplan identified by the Bellman-Ford algorithm is presentedbelow in Figs. 13 and 14. This plan requires a 38.52 secondtravel time, and a net energy increase of 0.13 J.

TABLE I: Travel time t and net energy change ∆E for theBellman-Ford algorithm using 5 × 5 through 25 × 25 fixedgrids, where n denotes the number of nodes in the horizontaland vertical line.

n t seconds ∆E J5 40.27 0.35047 38.71 0.43069 44.70 0.317511 39.83 0.327813 42.76 0.257415 51.43 0.244017 45.51 0.370219 51.03 0.188321 40.14 0.298123 56.16 0.217525 50.76 0.2712

The fixed-grid Bellman-Ford solution is slower than theBalkcom-Mason (43.41%) and Pseudo-Dubins (46.50%) dueto the constraint on each path segment. The net energyconstraint on each edge does not allow trade-off of the energygain/loss between the edges. Conversely, the modified PSOallows some segments to lose energy while the others willcompensate the loss to maintain the overall net energy gain.When high density grids are generated, samples at low solarradiation area may require more penalty on time to meet

8

Fig. 13: Path from Bellman-Ford Algorithm for Virtual Sce-nario

0 5 10 15 20 25 30 35 40

Power (W)

0

5

10

15

Time (s)

0 5 10 15 20 25 30 35 40

∆ Battery (J)

0

2

4

6

Fig. 14: Simulated Power & Energy Schedule from Bellman-Ford Algorithm

its energy constraint. And accumulation of the time penaltyfrom increasing number of edges leads to extended traveltime. Another important reason leads to poor performance ofBellman-Ford solution comes from the varying travel speedof each edge. The travel speed of each edge is a designvariable that determines the travel time of corresponding edgeand it is also coupled with the power consumption. Again,compared to the modified PSO which evaluates all of the speedvariables together, determining each edge speed separatelyloses flexibility to trade-off these values. Therefore, adaptingconventional motion planners to this type of problem cannotyield results with similarly improved performance.

B. Scenario 2

In the second scenario, we employ the real-world test envi-ronment described in §IV B. Three high-pressure sodium lightsare distributed within a 3.8×1.31 m2 area. The resulting solarenergy density distribution is mapped and used to generatea naive straight-line solution, a 5-waypoint Balkcom-Masonsolution, and a 5-waypoint Pseudo-Dubins solution. The re-sulting motion plans were executed by the demonstrationvehicle, previously detailed in §IV A, five separate times.

The naive straight-line solution, including path, predictedvariable history, and the recorded experimental data, is shownin Figs. 15-16. The anticipated travel time for the straight-line

solution is 57.71 seconds with a net energy increase of 1.19J, while the average experimental travel time is 57.13 secondswith a minimum record of 56.42 seconds and a maximumrecord of 57.79 seconds, and an average experimental netenergy increase 1.89 J with a minimum net energy increaseof −0.17 J and a maximum of 3.97 J. The average standarddeviation of the experimental ∆Battery from the prediction is1.68 J, with a minimum of 1.46 J and a maximum of 1.97 J.

Fig. 15: Straight-Line Path for Real-World Scenario

Fig. 16: Experimental Straight-Line Power & Energy DataThe straight-line solution requires the vehicle to travel at

the speed of 0.066 m/s which is very close to the UGV’sminimum speed of 0.055 m/s. Even if a straight-line pathfrom start to finish were broken up into smaller line segmentswith nonuniform velocities, the vehicle would be unable totravel slower through the high-energy regions, and would haveinsufficient energy to travel more quickly in the sparse regions.A straight line, in this case, cannot yield the fastest route fromstart to finish while satisfying the net energy gain constraint.

The path and predicted variable history of the Balkcom-Mason solution [27] can be found in Figs. 17-18. The an-ticipated travel time for the straight-line solution was 25.01seconds with a net energy increase of 0.25 J, while the averageexperimental travel time was 24.78 seconds with a minimumof 24.41 seconds and a maximum of 25.25 seconds, andan average experimental net energy increase −0.03 J with aminimum net energy increase of −1.30 J and a maximum of1.72 J. The average standard deviation of the experimental∆Battery from the prediction is 1.48 J, with a minimum of0.92 J and a maximum of 1.96 J.

The pseudo-Dubins solution, including its path, predictedvariable history, and the recorded experimental data, arepresented in Figs. 19-20. The anticipated travel time for the

9

Fig. 17: Balkcom-Mason Path for Real-World Scenario

Fig. 18: Experimental Balkcom-Mason Power & Energy Data

pseudo-Dubins solution is 20.22 seconds with a net energyincrease of 3.39 J, while the average experimental travel timeis 20.20 seconds with a minimum of 19.90 seconds and amaximum of 20.41 seconds, and an average experimental netenergy increase 4.17 J with a minimum net energy increaseof 1.53 J and a maximum of 7.99 J. The average standarddeviation of the experimental ∆Battery from the prediction is1.66 J, with a minimum of 1.13 J and a maximum of 2.24 J.

Fig. 19: Pseudo-Dubins Path for Real-World ScenarioMuch like Scenario 1, both the Balkcom-Mason and

Pseudo-Dubins plans resulted in decreased transit time fromthe naive straight-line case (56.66% & 64.97% predictedand 56.59% & 64.63% average experimental, respectively).More significantly, the Pseudo-Dubins solution constituted amarked improvement from the Balkcom-Mason plan, offeringpredicted travel time reduction of 19.16% and an averageexperimental time reduction of 18.54%. The Balkcom-Masonsolution requires that the UGV moves slowly underneath theleftmost energy source and pass close to the rightmost sourcein order to meet the energy constraint. In contrast, the Pseudo-Dubins solution is planned to gather sufficient energy from themiddle source alone and consumes less energy to complete itsturns by preserving some of the vehicle’s translational motion

Fig. 20: Experimental Pseudo-Dubins Power & Energy Data

towards the goal during the turn. The small discrepanciesobserved between the anticipated and experimental energytrends may be contributed by small deviations in the vehicle’spath from its planned path, due to interference between ourtest environment’s motion capture system and the infraredlight emitted by the high-pressure sodium lamps. In testing,the demonstration vehicle averaged a 1.06% error in distancetraveled and 0.80% error in angle turned. Measurement noisein the UGV’s voltage/current sensors may also contribute tothese small experimental discrepancies.

VI. CONCLUSIONS

This paper investigated an integrated path planning andpower management method for a solar powered unmannedground vehicle (UGV) which minimized the vehicle’s traveltime subject to net energy and power constraints. Virtual andreal-world experiments, conducted in an indoor test environ-ment, show that a differential-drive, power-constrained UGVoperating under an optimized Pseudo-Dubins curve traveledacross a known area more quickly than if conducted accordingto a naive straight line or an optimized Balkcom-Masoncurve. In the future we intend to generate a modified particleswarm optimization method for more efficient energy efficientpaths, explore alternative solar energy mapping methods, andinvestigate the applicability of adapting the method for time-variant environments.

VII. ACKNOWLEDGMENT

This work is sponsored by National Science FoundationECCS-1453637.

REFERENCES

[1] L. Takayama, W. Ju, and C. Nass, “Beyond dirty, dangerous and dull:what everyday people think robots should do,” in Proceedings of the 3rdACM/IEEE international conference on Human robot interaction, 2008,pp. 25–32.

[2] T. Wang, B. Wang, H. Wei, Y. Cao, M. Wang, and Z. Shao, “Staying-alive and energy-efficient path planning for mobile robots,” in AmericanControl Conference, 2008, pp. 868–873.

[3] Y. Mei, Y.-H. Lu, Y. C. Hu, and C. G. Lee, “Energy-efficient motionplanning for mobile robots,” in 2004 IEEE International Conference onRobotics and Automation, vol. 5, 2004, pp. 4344–4349.

10

[4] I. Heng, A. S. Zhang, and A. Harb, “Using solar robotic technologyto detect lethal and toxic chemicals,” in IEEE Global HumanitarianTechnology Conference, 2011, pp. 409–414.

[5] A. Sulaiman, F. Inambao, and G. Bright, “Solar-hydrogen energy: Aneffective combination of two alternative energy sources that can meet theenergy requirements of mobile robots,” in Robot Intelligence Technologyand Applications 3. Springer, 2015, pp. 819–832.

[6] J. H. Lever, A. Streeter, and L. Ray, “Performance of a solar-poweredrobot for polar instrument networks,” in Robotics and Automation, IEEEInternational Conference on, 2006, pp. 4252–4257.

[7] Mars exploration rover - spirit. Retrived Dec 2015. [Online]. Available:http://www.jpl.nasa.gov/missions/details.php?id=5917

[8] F. Vaussard, P. Retornaz, M. Liniger, and F. Mondada, “The autonomousphotovoltaic marxbot,” in Intelligent Autonomous Systems. Springer,2013, pp. 175–183.

[9] P. Martin, R. Galvan-Guerra, M. B. Egerstedt, and V. Azhmyakov,“Power-aware sensor coverage: An optimal control approach,” in 19thInternational Symposium on Mathematical Theory of Networks andSystems, Budapest, Hungary, July, 2010.

[10] A. Yazici, G. Kirlik, O. Parlaktuna, and A. Sipahioglu, “A dynamic pathplanning approach for multirobot sensor-based coverage consideringenergy constraints,” Cybernetics, IEEE Transactions on, vol. 44, no. 3,pp. 305–314, March 2014.

[11] J. Kim, F. Zhang, and M. Egerstedt, “Battery level estimation ofmobile agents under communication constraints,” in Sensor Networks,Ubiquitous, and Trustworthy Computing (SUTC), IEEE InternationalConference on, 2010, pp. 291–295.

[12] W. Sheng, Q. Yang, S. Zhu, and Q. Wang, “Efficient map synchroniza-tion in ad hoc mobile robot networks for environment exploration,” inIntelligent Robots and Systems, IEEE/RSJ International Conference on,2005, pp. 2297–2302.

[13] D. Michel and K. McIsaac, “New path planning scheme for completecoverage of mapped areas by single and multiple robots,” in Mechatron-ics and Automation (ICMA), IEEE International Conference on, 2012,pp. 1233–1240.

[14] Y. Mei, Y.-H. Lu, Y. C. Hu, and C. G. Lee, “Deployment of mobilerobots with energy and timing constraints,” Robotics, IEEE Transactionson, vol. 22, no. 3, pp. 507–522, 2006.

[15] L. E. Ray, J. H. Lever, A. D. Streeter, and A. D. Price, “Design andpower management of a solar-powered cool robot for polar instrumentnetworks,” Journal of Field Robotics, vol. 24, no. 7, pp. 581–599, 2007.

[16] T. de J Mateo Sanguino and J. E. Gonzalez Ramos, “Smart hostmicrocontroller for optimal battery charging in a solar-powered roboticvehicle,” Mechatronics, IEEE/ASME Transactions on, vol. 18, no. 3, pp.1039–1049, 2013.

[17] P. A. Plonski, P. Tokekar, and V. Isler, “Energy-efficient path planningfor solar-powered mobile robots*,” Journal of Field Robotics, vol. 30,no. 4, pp. 583–601, 2013.

[18] An overview of the solar radiation tools. Retrived Dec 2015. [Online].Available: http://resources.arcgis.com/en/communities/arcgis-content/

[19] L. E. Dubins, “On curves of minimal length with a constraint onaverage curvature, and with prescribed initial and terminal positions andtangents,” American Journal of mathematics, pp. 497–516, 1957.

[20] H. W. Kuhn, “Nonlinear programming: a historical view,” in Traces andEmergence of Nonlinear Programming. Springer, 2014, pp. 393–414.

[21] W. Zeng and R. Church, “Finding shortest paths on real road networks:the case for a*,” International Journal of Geographical InformationScience, vol. 23, no. 4, pp. 531–543, 2009.

[22] L. E. Kavraki, P. Svestka, J.-C. Latombe, and M. H. Overmars, “Prob-abilistic roadmaps for path planning in high-dimensional configurationspaces,” Robotics and Automation, IEEE Transactions on, vol. 12, no. 4,pp. 566–580, 1996.

[23] S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimalmotion planning,” The International Journal of Robotics Research,vol. 30, no. 7, pp. 846–894, 2011.

[24] N. R. Lawrance and S. Sukkarieh, “Autonomous exploration of awind field with a gliding aircraft,” Journal of Guidance, Control, andDynamics, vol. 34, no. 3, pp. 719–733, 2011.

[25] D. J. Balkcom and M. T. Mason, “Time optimal trajectories forbounded velocity differential drive vehicles,” The International Journalof Robotics Research, vol. 21, no. 3, pp. 199–217, 2002.

[26] J. Kennedy, “Particle swarm optimization,” in Encyclopedia of MachineLearning. Springer, 2010, pp. 760–766.

[27] A. Kaplan, P. Uhing, N. Kingry, and R. Dai, “Integrated path planningand power management for solar-powered unmanned ground vehicles,”in IEEE International Conference on Robotics and Automation (ICRA),Seattle, WA, 2015.

[28] “Dagu 5 robot chassis,” retrived Dec 2015. [Online]. Available:http://www.pololu.com/product/1551

[29] “What arduino can do,” http://arduino.cc/, 2016.[30] H. Garg and S. Garg, “Prediction of global solar radiation from bright

sunshine hours and other meteorological data,” Energy Conversion andManagement, vol. 23, no. 2, pp. 113–118, 1983.

[31] “Affordable motion capture for any application,” retrived Dec 2015.[Online]. Available: http://www.vicon.com/System/Bonita

[32] R. Bellman, “On a routing problem,” Quarterly of Applied Mathematics,vol. 16, pp. 87–90, 1958.