The optimization of mobile robots energy efficiency

Annals of Warsaw University of Life Sciences – SGGW Agriculture No 70 (Agricultural and Forest Engineering) 2017: 79–88 (Ann. Warsaw Univ. Life Sci. – SGGW, Agricult. 70, 2017) DOI 10.22630/AAFE.2017.70.20

The optimization of mobile robots energy efficiency ANDRZEJ CHOCHOWSKI1, IHOR BOLBOT2, VITALII LYSENKO2, VOLODYMYR RESHETIUK2 1 2

Department of Fundamentals Engineering, Warsaw University of Life Sciences – SGGW Department of Automation and Robotic Systems, National University of Life and Environmental Sciences of Ukraine in Kyiv – NUBIP of Ukraine

Abstract: The optimization of mobile robots energy efficiency. The paper presents an analysis of the factors affecting on the resource cost of the accumulator battery of mobile robot for phytomonitoring plants in the greenhouse and usage of the method of variations calculus for optimization energy consumption. Key words: energy consumption, optimization, linear speed, variation calculus

INTRODUCTION Modern robotic systems have many advantages over traditional: high efficiency, reduced cost of human resources, the possibility of modernization. Their main drawback, especially mobile, is limited energy resource, because of relatively small capacity battery. The above-mentioned problem can be solved by increasing the battery life or reducing the energy consumption. Our research aimed at the reducing of energy consumption in the process of robot moving [Mei et al. 2004]. Energy saving is achieved if the robot moves with optimal speed (the speed

with constant acceleration, infrequent speed changes in the conditions of the robot straight line moving). Delivery of work equipment to the place of manufacturing operations conducted under conditions of uncertainty, when previously unknown routes mobile robotics and location of obstacle objects, with which it can interact during movement. In this regard, a crucial role in pressing mobile robotics start playing its terrain parameters (movement and maneuvers on surfaces with different coatings, poverty rises, the threshold noise, etc.) provided by its system of movements [Maslov et al. 2005, Lysenko et al. 2010, Shvorov et al. 2012]. Different aspects of development and use mobile robotics special purpose involved in a number of national scientific and industrial organizations and foreign companies operating in the field of mobile robotics firm such as REMOTEC, Foster-Miller, Cybermotion (USA), Alvis Logistics, Lockheed Martin (UK), Cybernetix, Giat Industries (France), Telerob (Germany)

80 A. Chochowski et al.

and others [Borzenkov 2002, Maslov 2005]. Despite of significant advances in development theory and practice of building special-purpose robots, some of the challenges relating with the creation and organization of operation mobile robotics extra-light class, settled in full. The analysis shows that the actual robotics is in a special issue of development and research of wheeled transportation mobile robotics extra-light class. Conduct an analysis of the factors affecting on the resource cost of the accumulator battery of mobile robot for phytomonitoring plants in the greenhouse and usage of the method of variations calculus for optimization energy consumption. RESEARCH TOPICS AND METHODS During the research we reviewed the theoretical question of approach, construction and operation of robotic systems, studied scientific problem of optimal design and use of intelligent mobile robots.

§ x · ¨ ¸ ¨ y ¸ = Tp ¨ θ ¸ © ¹

§v· § cos θ sin θ 0 · ⋅ ¨ ¸ , Tp = ¨ ¸ 0 1¹ © w¹ © 0

T

(1)

To simplify the calculations, we assume that both engines have identical: armature resistance (Ra), anti-EMF (Kb), torque (Kt) and gear number (n). If the voltage of the accumulator battery mark (Vs), then the balance equation for engines will look like: Ra · i = Vs · u – Kb · n · w

(2)

where: i = [i ^ R i ^ L] ^ T – vector armature current; u = [u ^ R u ^ L] ^ T – input normalized vector of controlling; w = [w ^ R w ^ L] ^ T – vector angular speed of the wheels. Indices R, L correspond to the left and to the right motor respectively. The dynamic relationship between the angular speed and motor current, given the inertia and friction for the engine can be written as:

RESULTS AND DISCUSSION To formulation the optimization problem we accept that mobile robotic electrical system for phytomonitoring is a nonholonomic system with symmetric structure, as set in motion two identical DC motors [Martynenko 2005]. Mark the robot position (coordinates and angle) as P (t) = [x (t) y (t) θ (t)] ^ T, linear speed (v), angular speed (w). Then the kinematic equations for describing will be an expression:

J

dw + Fv ⋅ w = K t ⋅ n ⋅ i dt

(3)

where: J – matrix moments of inertia motors; Fv – friction coefficient. From the expressions (2) and (3) you can get the differential equation: w + A ⋅ w = B ⋅ u

(4)

The optimization of mobile robots energy efficiency

where: §a a · A = ¨ 1 2 ¸ = J −1 © a2 a1 ¹

81

§ K ⋅ Kb ⋅ n2 · ⋅ ¨ Fv + t ¸; Ra © ¹

§b b · V ⋅ Kt ⋅ n B = ¨ 1 2 ¸ = J −1 ⋅ s . Ra © b2 b1 ¹

Define a state vector as z = [v w]T, associate w and v with wR and wL in the expression: §v· z = ¨ ¸ = Tq © w¹

§ wR · ⋅ ¨ L ¸ = Tq ⋅ w, ©w ¹

r/2 · § r/2 Tq = ¨ ¸ © r / 2 ⋅ b −r / 2 ⋅ b ¹

(5)

Using the similarity conversion with expressions (4) and (5) we get: z + A ⋅ z = B ⋅ u

(6)

where:

0 · §π 0 · § a1 + a2 A = Tq ⋅ A ⋅ Tq−1 = ¨ v ¸ = ¨ ¸; a1 − a2 ¹ © 0 πw ¹ © 0

§β B = Tq ⋅ B = ¨ 1 © β2

§ r ⋅ ( b1 + b2 )

β1 · ¨ 2 ¸ =¨ − β 2 ¹ ¨ r ⋅ ( b1 − b2 ) ¨ ©

2

r ⋅ ( b1 + b2 ) · ¸ 2 ¸. r ⋅ ( b1 + b2 ) ¸ − ¸ 2 ¹

The energy passes from the battery power is converted into mechanical energy of motion and heat loss. Heat loss causes the internal resistance of the accumulator battery, resistance of the control device (driver) of the motor, resistance of the armature engine and viscous friction in motion [Sve Liin Khtu Aung 2011]. On the Figure 1 is shown a simplified diagram of the electrical system of a mobile robot. To reducing of heat loss in the engine control device the PWM controller is used because it has less energy and generates less FIGURE 1. The simplified scheme of an electrical part of the mobile robot


heat than linear voltage regulator. Therefore, we can determine the resistance amplifier (Ramp) and duty cycle PWM (uR, uL). To simplifing the calculations, we assume that the heat loss through the internal resistance of the accumulator battery and the resistance of the amplifier control device engine low and therefore not counted. Thus, the energy supplied from the accumulator battery in a scheme of a drive part of the mobile robot is a function for minimization and can be represented as [Barili et al. 1995]: Ew =

³

tf

t0

tf

iT ⋅ V ⋅ dt = Vs ³ iT ⋅ u ⋅ dt

(7)

t0

where: V = [VRVL]T – input voltage passes to the engines from the accumulator battery (Vs – battery voltage); V R L T u = = ª¬u u º¼ . Vs The battery capacity is limited, and the voltage will be limited too: –umax ≤ uR, uL ≤ umax

(8)

From the expressions (2), (5), the optimization function (Ew) through the speed can be written as: Ew =

³ (k

⋅ u T ⋅ u − k2 ⋅ z T ⋅ T −T ⋅ u ) dt

tf

t0

1

(9)

where: k1 =

Vs2 K ⋅ n ⋅ Vs T ; k2 = b ; z = [v w] . Ra Ra

Thus, taking into account the expressions (2), (3) the general optimization function can be represented as follows: tf

Ew = Ra ⋅ ³ iT ⋅ i ⋅ dt + Fv t0

K + b Kt

³

tf

t0

−T q

z ⋅ T T

Kb Kt

³

t0

z T ⋅ Tq−T ⋅ Tq−1 ⋅ z ⋅ dt +

(10) −1 q

⋅ J ⋅T T

tf

⋅ z ⋅ dt

(

tf

)

In expression (10) the first term Er = Ra ⋅ ³ iT ⋅ i ⋅ dt is the energy that dissit0

pated by armature resistance in the engine [Kusko and Galler 1983]; the second term § · Kb t f T z ⋅ Tq−T ⋅ Tq−1 ⋅ z ⋅ dt ¸ corresponds to the loss of energy to over¨ EF = Fv ³ t Kt 0 © ¹


83

§ · K tf come friction; the last term ¨ EK = b ³ zT ⋅ Tq−T ⋅ J T ⋅ Tq−1 ⋅ z ⋅ dt ¸ – the kinetic K t t0 © ¹ energy of a mobile robot that has zero average value when the speed is constant or terminal speed is starting. This means that the contribution of the last term in energy consumption is zero. As the robot moves mostly straightforward, just consider this variant. The speed of rotation (w) in this period is zero, and P(t) = [x(t) 0 0]T – position, z(t) = [v(t) 0]T – the speed at the moment of the time (t). Then the task of minimizing energy consumption can be formulated as follows: to find the value of the linear speed [v(t)] and the control value [u(t)], which minimizes the function: Ew =

³ (k tf

1

t0

⋅ u T ⋅ u − k2 ⋅ z T ⋅ Tq−T ⋅ u ) dt

(11)

under the following conditions: • the start and the final position: P(t0) = [x0 0 0]T and P(tf) = [xf 0 0]T; • the start and the final speed: z(t0) = [v0 0]T and z(tf) = [vf 0]T; • satisfactory value of battery charge where t0 and tf is the start and the final moment of the robot moving. We think that the start and the final robot speed is zero, and its movement starts from the first position. Then the energy minimization task can be written as [Egami et al. 1990]: min Ew =

³ (k tf

t0

1

⋅ u T ⋅ u − k2 ⋅ z T ⋅ Tq−T ⋅ u ) dt

z = − Az + Bu

(13)

z ( 0 ) = z ( t f ) = [ 0 0]

T

Pf =

³

tf

0

(12)

Tp ⋅ z ⋅ dt = ª¬ x f 0 0º¼

(14) T

§ u max · § u R · § u max · − ¨ max ¸ ≤ u = ¨ L ¸ ≤ ¨ max ¸ ©u ¹ ©u ¹ ©u ¹

(15)

(16)

Searching of optimal speed based energy costs, which minimizes the equation (12) satisfying constraints (14)–(16) for the system of equations (13), will implement the usage of variation calculus method. Based on the research that resulted in [Mei et al. 2004], the Lagrange multiplier for expression (15) is a = [ax ay aθ]T. Defined multiplier function for expression (13) λ = [λv λw]T, where the Hamiltonian function would be: aT ⋅ Pf (17) H = k1 ⋅ u T ⋅ u − k2 ⋅ z T ⋅ Tq−T − aT ⋅ Tp ⋅ z + + λ T ( − Az + Bu ) tf


Necessary conditions for optimal speed (z*) and input signal (u*): ∂H = 2k1 ⋅ u − k2 ⋅ Tq−1 ⋅ z + B T ⋅ λ = 0 ∂u

(18)

∂H = −k2 ⋅ Tq−T ⋅ u − TpT ⋅ a + AT ⋅ λ = −λ ∂z

(19)

+ Bu z = − Az

(20)

From the expressions (18)–(20) we obtain the following differential equation: § · k z − ¨ B ⋅ B T ⋅ AT ⋅ B −T ⋅ B −1 ⋅ A − 2 ⋅ B ⋅ B T ⋅ Tq−T ⋅ B −1 ¸ z + k1 © ¹ 1 + ⋅ B ⋅ B T ⋅ TpT ⋅ a = 0 2 ⋅ k1

(21)

When B ⋅ B T and A – the diagonal matrix equation (21) is reduced to: z − QT ⋅ Q ⋅ z + R ⋅ TpT ⋅ a = 0

(22)

where: k QT ⋅ Q = AT ⋅ A − 2 ⋅ B ⋅ B T ⋅ Tq−T ⋅ B −1 ⋅ A ; k1 §1 · 0¸ ¨τ T n 0· v ¸ ; R = B ⋅ B = §¨ v Q =¨ ¸; ¨ 1¸ 2 ⋅ k1 © 0 nw ¹ 0 ¨ τ w ¸¹ ©

τv = τw =

J1 + J 2

Fv ⋅ ( Fv + K t ⋅ K b ⋅ n 2 / Ra ) J1 − J 2

Fv ⋅ ( Fv + K t ⋅ K b ⋅ n 2 / Ra )

– electromechanical time constant to moving;

– electromechanical time constant for turning of the mobile robot.

When the energy loss due to rotating the mobile robot is not taken into account (consider moving in straight line), optimal linear speed (z*) can be expressed as: § v * ( t ) · § C ⋅ e t / τ v + C2 ⋅ e − t / τ v + K v · z * ( t ) = ¨¨ * ¸¸ = ¨ 1 ¸ 0 ¹ © w (t ) ¹ ©

(23)


85

where: C1 = C2 = Kv =

e e

−t f /τ v

t f /τ v

−1 ⋅ Kv ; −t /τ −e f v t /τ v

1− ef t /τ v

ef

−e

−t f /τ v

⋅ Kv ;

(

t /τ v

xf ⋅ e f

−e

−t f /τ v

)

§ · t /τ −t /τ 2 ⋅ τ v ⋅ ¨ 2 − eτ v − eτ v ¸ + t f ⋅ e f v − e f v ¨ ¸ © ¹ tf

tf

(

)

.

Then the equation (13) gives the equation optimal control signal (u*): u * ( t ) = B −1 ⋅ ( z* + A ⋅ z * ) =

1 2 ⋅ β1 ⋅ τ v

t − § · τv τv ¨ C1 ⋅ (1 + τ v ⋅ π v ) ⋅ e − C2 ⋅ (1 − τ v ⋅ π v ) ⋅ e + τ v ⋅ π v ⋅ K v ¸ ¨ ¸ t t − ¨ ¸ τv τv © C1 ⋅ (1 + τ v ⋅ π v ) ⋅ e − C2 ⋅ (1 − τ v ⋅ π v ) ⋅ e + τ v ⋅ π v ⋅ K v ¹ t

(24)

Speed [m/s]

To drawing of dependencies of optimal speed of relative time (from the beginning of the movement to its end) (Fig. 2) used specifications as given in the Table.

Time [s]

FIGURE 2. Speed depends on the relative distance of moving at different values meaning of k (mechanical time constant before the time of displacement k = tv / tf )

86 A. Chochowski et al. TABLE. Technical characteristics of mobile robot Parameter

Value 0.71 Ω

Resistance of anchor, Ra

0.0230 N·m/А

Torque, Kt

0.0230 V/(rad·s)

Anti-EMF, Kb The radius of the wheels, r

0.15 m

Base

0.206 m 12 V

Battery voltage, Vs max

1V

Limitation of voltage, u

0.054 N·m/(rad·s)

Coefficient of friction, Fv Gear ratio, n

§J The inertia of engines, J = ¨ 1 © J2

49.8

J2 · ¸ J1 ¹

§ 0.1241 0.0098 · ¨ ¸ © 0.0098 0.1241¹

The results of materials are shown in Figure 2 can make the following conclusions – minimal energy consumption rate graphic has a symmetrical shape, thus: • if k ≈ 0 (k = фv/tf) then graphic of optimal veloсity is trapezoidal; • if k > 0.2 then graphic takes the form of a parabola. Basing on equations (12)–(15) we can receive: §t · §t − t· § tf · sin h ⋅ ¨ f ¸ − sin h ⋅ ¨ f ¸ − sin h ⋅ ¨ ¸ xf ©τv ¹ © τv ¹ ©τv ¹ v* ( t ) = ⋅ τv § § t ·· t §t · 2 ⋅ ¨¨1 − cos h ⋅ ¨ f ¸ ¸¸ + f sin h ⋅ ¨ f ¸ ©τv ¹¹ τv ©τv ¹ ©

(25)

Energy consumption are defined by losses because of armature resistance [the first component of expression (10)] – minimization of energy of an armature were investigated; losses because of resistance of an armaturer and losses on friction [the first and second parts of expression (10)] – minimization of expenses of energy. In a parabolic rate schedule energy consumption to some extent offset by the partial engine operation in generator mode. For comparison, a graph of simulation experiment (Fig. 3) for such data tf = 10 s, xf = 5 m. Comparing the two ways of optimization energy consumption, we conclude that minimize energy consumption armature expedient when moving work to a distance of 5 m (friction in this movement essentially no effect). Otherwise reasonable to have a full energy minimization.

a

0,8 Linear speed [m/s]

0,7 0,6 0,5 0,4 0,3 0,2 0,1 0

0

b

1

2

3

4

5 Time [s]

6

7

8

9

10

Energy consumption [J]

30 25 20 15 10 b)

5 0 0

c

1

2

3

4 5 Time [s]

6

7

8

9

10

1,5

Current [A]

1 0,5 0

0

1

2

3

4

5

6

7

8

9

10

-0,5

FIGURE 3. Graphs of comparing two power saving modes: a – schedule of optimal speed; b – battery consumption; c – change of current at anchor (i = iR = iL)

-1 -1,5 Time [s] - - complete minimization of energy consumption .... minimize of energy anchor

180

Full minimizaƟon of energy (modeling)

160

Full minimizaƟon of energy (experiment)

Power [Wt]

140 120

MinimizaƟon of power consumpƟons of an anchor (modeling)

100

MinimizaƟon of power consumpƟons of an anchor (experiment)

80 60 40 20 0

0,5

1

2,5 5 Distance [m]

10

15

FIGURE 4. Comparison of data of modeling to the data received during experiment


CONCLUSIONS 1. According to the analytical drawing of regulators, the expression of the optimal linear speed of the mobile robot, which minimizes energy costs is received. 2. The choice of moving speed and parameter of optimization depends on the distance at which the robot must move, at a distance of 5 m using parabolic rate schedule, while more – trapezium. REFERENCES BARILI A., CERESA M., PARISI C. 1995: Energy-saving motion control for an autonomous mobile robot. Proc. IEEE Int. Symp. Indust. Electr., Athens, Vol. 2: 674–676. BORZENKOV V.V. 2002: The use of modular elements in the construction of mobile robots ultralight class of special-purpose non-toggling movers. Drive Techn. 4: 48–52. EGAMI T., MORITA H., TSUCHIYA T. 1990: Efficiency optimized model reference adaptive control system for a DC motor. IEEE Trans. Ind. Electron. 37 (1): 28–33. KUSKO A., GALLER D. 1983: Control means for minimization of losses in AC and DC motor drives. IEEE Trans. Ind. Appl. 19 (4): 561–570. LYSENKO V., BOLBOT I. 2010: Roboty ta robototechnichni systemy v agropromyslovomu kompleksi. Naukovyy Visnyk NUBiP Ukrainy 153: 105–110. MEI Y., LU Y-H., HU Y.C., LEE C.S.G. 2004: Energy-efficient motion planning for mobile robots. Proc. IEEE Int. Conf. Robot. Aut., New Orleans, Vol. 5: 4344–4349. MARTYNENKO Yu.G. 2005: Upravleniye dvizheniyem mobilnykh kolesnykh robotov. Fund. Prikl. Mat. 11 (8): 29–80. MASLOV O.A. 2005: Mobile robots to detect and eliminate VU. Special Vehicle 5: 18–21. MASLOV O., PUZANOV A., KUVANOV K., PLATOV A. 2005: Design and fabrication of

all-terrain mobile robots for special purposes with the use of modern CAD. CAD/CAM/CAE Observer 2: 61–64; 3: 53–55. SVE LIIN KHTU AUNG 2011: Navigatsiya i upravleniye dvizheniyem mobilnogo robota v gorodskikh usloviyach. Avtoref. Kand. Tekh. Nauk. [manuscript]. SHVOROV S., RESHETIUK V., BOLBOT I., SHTEPA V., CHIRCHENKO D. 2012: Theoretical issues construction and operation of agricultural mission robotic system. Ann. Warsaw Life Sci. – SGGW, Agricult. 60: 97–102. Streszczenie: Optymalizacja efektywności energetycznej mobilnego robota. Celem badań była identyfikacja parametrów i analiza ich wpływu na koszty żywotności baterii ruchomych robotów do fitomonitoringu roślin w szklarni. W celu optymalizacji zużycia energii zastosowano metodę różnicową. Zgodnie z wynikami badań analitycznych otrzymano zależność dla optymalnej prędkości liniowej ruchomego robota, która minimalizuje koszty zużycia energii. Wybór prędkości jazdy i parametrów optymalizacji zależy od odległości robotów: do 5 m należy korzystać z parabolicznego wykresu prędkości, a przy większych odległościach trzeba skorzystać z wykresu trapezowego.

MS received July 2017 Authors’ addresses: Andrzej Chochowski Wydział Inżynierii Produkcji SGGW Katedra Podstaw Inżynierii 02-787 Warszawa, ul. Nowoursynowska 164 Poland e-mail: [email protected] Vitalii Lysenko, Volodymyr Reshetiuk, Ihor Bolbot National University of Life and Environmental Sciences of Ukraine Department of Automation and Robotic Systems Heroiv Oborony 12, Kyiv 03041 Ukraine e-mail: [email protected]