Practical Model for Energy Consumption Analysis of Omnidirectional Mobile Robot

sensors

Article Practical Model for Energy Consumption Analysis of Omnidirectional Mobile Robot

Linfei Hou 1 , Fengyu Zhou 1, Kiwan Kim 2 and Liang Zhang 3,*

1 School of Control Science and Engineering, Shandong University, Jinan 250061, China; [email protected] (L.H.); [email protected] (F.Z.) 2 Department of Electrical & Electronics Engineering, Chungnam State University, Cheongyang 33303, Korea; [email protected] 3 School of Mechanical, Electrical and Information Engineering, Shandong University, Weihai 264209, China * Correspondence: [email protected]; Tel.: +86-130-6118-7255

Abstract: The four-wheeled Mecanum robot is widely used in various industries due to its maneu- verability and strong load capacity, which is suitable for performing precise transportation tasks in a narrow environment. While the Mecanum wheel robot has mobility, it also consumes more energy than ordinary robots. The power consumed by the Mecanum wheel mobile robot varies enormously depending on their operating regimes and environments. Therefore, only knowing the working environment of the robot and the accurate power consumption model can we accurately predict the power consumption of the robot. In order to increase the applicable scenarios of energy consumption modeling for Mecanum wheel robots and improve the accuracy of energy consumption modeling, this paper focuses on various factors that affect the energy consumption of the Mecanum wheel robot, such as motor temperature, terrain, the center of gravity position, etc. The model is

derived from the kinematic and kinetic model combined with electrical engineering and energy ﬂow principles. The model has been simulated in MATLAB and experimentally validated with the

Citation: Hou, L.; Zhou, F.; Kim, K.; four-wheeled Mecanum robot platform in our lab. Experimental results show that the accuracy of Zhang, L. Practical Model for Energy the model reached 95%. The results of energy consumption modeling can help robots save energy by Consumption Analysis of helping them to perform rational path planning and task planning. Omnidirectional Mobile Robot. Sensors 2021, 21, 1800. https:// Keywords: Mecanum mobile robots; minimum-energy control; energy modeling; energy measure- doi.org/10.3390/S21051800 ments; complex environment

Academic Editor: Md Nazmul Huda

Received: 31 December 2020 1. Introduction Accepted: 21 February 2021 The Mecanum wheel is the most common omnidirectional wheel structure used in Published: 5 March 2021 practical applications. Compared with other omnidirectional wheel structures, it has high complexity capabilities [1]. Mecanum wheel robots have the advantage when it comes to Publisher’s Note: MDPI stays neutral stability, high load capacity, simple structure, and flexible movement. Therefore, they are with regard to jurisdictional claims in published maps and institutional affil- widely used in industrial production [2]. While Mecanum wheel robots have the advantage iations. of movement flexibility, they also consume more energy than ordinary wheeled robots [3]. In recent years, Mecanum wheel robots have been widely used in warehousing and logistics. With the increasing degree of automation of mobile robots, mobile robot energy models are becoming more complex. However, the energy modeling of Mecanum wheel mobile robots in complex environments has not been given enough attention. In this Copyright: © 2021 by the authors. paper, we mainly study the power consumption modeling of the Mecanum wheel robot Licensee MDPI, Basel, Switzerland. in upslope and downslope environments. A separate analysis of the effects of various This article is an open access article factors on power consumption is presented, and a power consumption model in a complex distributed under the terms and conditions of the Creative Commons environment of Mecanum wheel robots is proposed. Attribution (CC BY) license (https:// The special structure of the four-wheeled Mecanum wheel robot (FMWR) (see Figure1 ) creativecommons.org/licenses/by/ gives it the ability to move omnidirectionally and with high loads [4]. The particular struc- 4.0/). ture of the Mecanum wheel provides the robot with better flexibility than ordinary wheeled

Sensors 2021, 21, 1800. https://doi.org/10.3390/s21051800 https://www.mdpi.com/journal/sensors Sensors 2021, 21, x FOR PEER REVIEW 2 of 19

Sensors 2021, 21, 1800 The special structure of the four-wheeled Mecanum wheel robot (FMWR) (see Figure2 of 18 1) gives it the ability to move omnidirectionally and with high loads [4]. The particular structure of the Mecanum wheel provides the robot with better flexibility than ordinary wheeledrobots. Therobots. omnidirectional The omnidirectional movement movement performance performance enables mobileenables robots mobile to robots carryout to carrybetter out transportation better transportation tasks in crowded, tasks in narrow, crowded, or highly narrow, dynamic or highly environments. dynamic However,environments.the special However, structure the and special movement structure of theand Mecanum movement wheel of the also Mecanum increases thewheel energy also con-in- creasessumption the of energy the robot, consumption so, compared of the to ordinary robot, so, wheeled compared robots, to ordinary the Mecanum wheeled wheel robots, robot thehas Mecanum more energy wheel consumption. robot has more [5]. Almostenergy consum all robotsption. need [5]. the Almost energy all from robots the need batteries the energythey carry from [6 the]. The batteries energy they situation carry of[6]. the The battery energy dramatically situation of limits the battery the task dramatically completion limitsof the the robots task [completion7]. Energy-efficient of the robots strategies [7]. Energy-efficient for mobile robots strategies can expand for mobile the range robots of canuses, expand perform the morerangemissions, of uses, perform and accomplish more missions, more complexand accomplish operations more [8 –complex10]. Energy op- erationsmodeling [8–10]. has significance Energy modeling for battery has energy significance management for battery and rangeenergy estimation management of mobile and rangerobots estimation [11–13]. Accuracy of mobile of robots energy [11–13]. models A canccuracy help of robots energy with models task planning, can help energyrobots withprediction task planning, [14–18], sensorenergy setupprediction [19,20 [14–18],], and energy sensor optimal setup [19,20], path planning and energy [21– optimal23], and pathhave planning very significant [21–23], applications and have very in autonomous significant planning,applications autonomous in autonomous exploration planning, [24], autonomousand even robot exploration design. [24], and even robot design.

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FigureFigure 1. 1. SchematicSchematic of of the the four-wheeled four-wheeled Mecanum Mecanum wheel wheel robot robot (FMWR). (FMWR). At present, the main research on the energy consumption of Mecanum wheel robots At present, the main research on the energy consumption of Mecanum wheel robots focuses on the energy consumption of the smooth-running phase of the Mecanum wheel focuses on the energy consumption of the smooth-running phase of the Mecanum wheel robot during the motion on the plane. The energy modeling for this part has been relatively robot during the motion on the plane. The energy modeling for this part has been rela- well developed [25,26]. However, there is a lack of analysis of the difference in energy tively well developed [25,26]. However, there is a lack of analysis of the difference in en- consumption of the Mecanum wheel robot between different terrains, and there is no ergy consumption of the Mecanum wheel robot between different terrains, and there is modeling analysis of the factors affecting the energy consumption of the robot, the energy no modeling analysis of the factors affecting the energy consumption of the robot, the consumption fluctuations in the transition phase between the two stable operation phases energy consumption fluctuations in the transition phase between the two stable operation are also only briefly explained [25,26]. This results in incomplete modeling scenes, and the phases are also only briefly explained [25,26]. This results in incomplete modeling scenes, model cannot adapt to the various environments in which the robot operates. Moreover, and the model cannot adapt to the various environments in which the robot operates. the accuracy of the modeling needs to be improved. Moreover, the accuracy of the modeling needs to be improved. The main contribution of this paper is to analyze and model the energy consumption of a MecanumThe main contribution wheel robot inof complexthis paper environments, is to analyze theand analysis model the of theenergy factors consumption that have a ofsignificant a Mecanum impact wheel on robot the robot’s in complex energy environmen consumption,ts, whichthe analysis are represented of the factors as parameters that have ain significant the modeling. impact Finally, on the the robot’s difference energy between consumption, the power which consumption are represented of Mecanum as param- wheel etersrobots in and the ordinary modeling. wheeled Finally, robots the due diff toerence their special between structure the ispower emphasized. consumption According of Mecanumto the previous wheel study, robots we and divided ordinary the wheeled energy consumptionrobots due to oftheir the special Mecanum structure wheel is robot emphasized.into three According major systems to the (control previous system, study, motion we divided system, the and energy sensing consumption system), modeled of the the energy consumption of each of these three systems, and analyzed the interactions and connections between the energy consumption of the three systems. In this paper, we focus on the energy consumption fluctuations of robots, which are often neglected in previous

Sensors 2021, 21, 1800 3 of 18

energy consumption studies. We investigate the energy consumption of the Mecanum wheeled robot during uphill and downhill slopes, analyze the differences between the Mecanum wheel robot and ordinary wheeled robots during its motion due to its special structure, and study the ﬂuctuations of energy consumption during the transition phase of its motion on different surfaces and the reasons for them. The main objective of the study is to increase the applicability of the Mecanum wheel robot modeling as well as to improve the accuracy of the modeling to help the Mecanum wheel robot predict its energy consumption more accurately. By modeling the energy consumption of the robot in complex terrain and adding the analysis of the factors affecting the energy consumption of the robot, the scenarios for modeling the energy consumption of the Mecanum wheel can be further increased. The analysis of the factors affecting the energy consumption can help to improve the accuracy of the modeling effectively. The modeling results can help the robot ﬁnd the path with the lowest energy consumption during path planning and reduce the robot’s energy consumption, as well as help the robot predict the energy required for the task and perform reasonable task planning.

2. Related Works At present, there are many known studies on power consumption models of omnidirectional mobile robots, including omnidirectional wheels and Mecanum wheels robots, terrain and trajectory planning, and so on. Sedat Dogru and Lino Marques [1] modeled the energy consumption of the skid steer robot throughout its motion and analyzed each of the factors affecting the robot’s energy consumption individually. B. K. Kim et al. [8,27–29] fo- cused on the optimal energy consumption trajectory planning strategy for a three-wheeled omnidirectional wheel robot, proposed that the energy consumption of the robot could be reduced by reducing the speed change of the robot, and planned the lowest energy consumption path for the three-wheeled omnidirectional wheel robot by kinematic and dynamic modeling, combined with the Pontryagin minimax principle, which was verified to save about 30% energy consumption than the ordinary path planning algorithm. Xie [30] investigated the trajectory planning method for the minimum energy consumption of the Mecanum wheel robot. The method used was to apply the established energy consumption model of the Mecanum wheel to the trajectory planner based on the extended dynamic window method. According to the verification, it can meet the purpose of reducing energy consumption while achieving the online obstacle avoidance function. However, in the study of modeling energy consumption of the Mecanum wheel robot, only the energy consumption of the motion system was modeled, neglecting the energy consumption of the control system and sensing system. Researches are also evolving rapidly in numerous specific scenarios. Amir Sadrpour et al. [31] examined the problem of mission prediction and battery energy prediction for unmanned ground-operated vehicles using batteries as energy sources that undertook specific tasks. Jesús Morales et al. [32] modeled the energy consumption of a skid steer robot on the hard ground through the two perspectives of power generated by rigid terrain and power provided by the motor. Broderick et al. [19] found that, for small and lightweight robots, the energy consumption of the robot’s non-motor systems (e.g., sensing, communication, and computing) accounted for a large percentage of the robot’s energy consumption. Therefore, appropriate scheduling strategies, as well as advanced energy conservation techniques and energy-efficient materials, could have a major effect on reducing the energy consumption of robots [33]. Structures, such as wheeled robots with redundant brakes [34], limb robots [35], and snake robots [36] that can re- place the energy consumption of the motion system are also being investigated. 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 [27]. Brateman et al. [20] pointed out that energy consumption could be effectively reduced by coordinating the relationship between the robot motor speed and the processing frequency of the processor and the scanning frequency of the sensors. Power demanded by one motor Sensors 2021, 21, 1800 4 of 18

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 [37].

3. Energy Modeling In order to improve the accuracy of robot modeling in complex environments, we divide the power consumption of robots into three major systems, namely motion system, control system, and sensing system—modeling and analyzing the energy consumption of these three systems separately. Finally, the connection between the power consumption of the three systems is sought.

3.1. Motion System The ﬁrst is that the power consumption of the motion system conforms to the following formula [1]. Z Z Emotion = Pmotordt = IaEmdt (1)

Emotion is the energy consumed by the robot’s motion system, Pmotor is the total power of the robotic motion system, Ia is the total current ﬂowing through the robot’s motion system. Em is the electromotive force on the robot motor. The power consumption of the robot’s motion system is the power consumption of the robot’s DC motor, but it is complicated to determine the measurement of the current for the robot’s motor, because the currents around the motor are not the same, and the current of the robot motor is not the same at different times. Thus, it is very inaccurate to measure the power consumption of the robot motor in this way. Therefore, we decompose the power consumption of the motor and divide the power consumption of the motor into consumption at the place.

Pmotor = Pcolo + Plolo + Poutput (2)

Pmotor is the power of the robot’s motor, Pcolo is the power of the robot’s copper consumption, Plolo is the power consumption of the robot’s iron loss, Poutput is the mechanical output of the robot. 2 Pcolo = Ia Ra (3)

Pcolo is the power of the motor to lose energy when the robot is in motion, Ia is the current ﬂowing through the motor, Ra Is the internal resistance of the robot motor. For a DC motor, the torque T and the rotation speed of the shaft ωsha f t are proportional to the armature current Ia and the electromotive force E. At the same time, the torque and speed output of the motor are proportional to the force Fwheels and the angular velocity w of the wheel. τ is the conversion factor of force and energy consumption, which is related to the radius of the wheel and the connection of the wheel to the motor. k(T) is the speed energy factor of the robot, β(T) is the temperature energy consumption coefﬁcient of the robot; their values, as well as the correspondence with speed and temperature, are given in Equations (30) and (31) below.

2 Pcolo = τ·Fwheel + k(T) + β(T) ∗ t (4)

Plolo = Pedlo + Phylo + Pother (5)

The iron loss of the robot was divided into three parts, where Pedlo is eddy current loss, Phylo is hysteresis loss, Pother is the iron loss after removing eddy current loss and hysteresis loss of other parts of the loss. Because the eddy current loss and hysteresis loss of the robot cannot be measured, the iron loss of the robot is directly related to the speed, acceleration, and running time of the robot. 4 2 2 2 Plolo = λ1 ∑(vi ∗ a) + λ2 t ∗ ai ∗ v + λ3 t ∗ a ∗ v (6) i=1 Sensors 2021, 21, 1800 5 of 18

λ1, λ2, and λ3 are the energy consumption coefﬁcients of the robot motors, vi is the real-time speed of each wheel of the robot, v is the actual motion speed of the robot, ai is the real-time acceleration corresponding to each wheel of the robot. According to the characteristics of the Mecanum wheel robot, the mechanical output of the robot can be subdivided into the mechanical energy of the output and the power consumption due to the friction losses caused by the characteristic structure of the Mecanum wheels, which is:

Poutput = Pmechanical + Pf ica (7)

Pmechanical = IaEb (8) Through the principles of motor science, we can learn that the motor’s mechanical output is transformed into the torque T and rotational speed ωsha f t of the motor shaft, and they are proportional to the input current Ia and input voltage Eb of the motor. For the robot, the output force Fwheels of the wheels is proportional to the torque T of the motor, while the rotational speed of the wheel is equal to the product of the torque ωsha f t of the motor and the diameter of the wheel [2].

2 Pmechanical = k1Fwheels + k2ωFwheels (9)

where k1 and k2 are proportionality constants, which represent information about the motor’s torque, armature resistance, and other parameters [2].

4 Pf ica = µ1 ∑ (Ni·vi) cos θ+µ2N·v cos θ cos γ (10) i=1

where µ1 and µ2 are the coefﬁcients of friction of the robot wheels, their values and variation laws are given in Equations (25) and (26). Ni is the positive pressure acting on each drive wheel, θ is the angle between the direction of robot motion and the positive direction of the robot, γ is the angle between the inclined plane and the horizontal plane of the robot movement So:

4 2 Poutput = k1Fwheels + k2ωFwheels + µ1 ∑ (Ni·vi) cos θ+µ2N·v cos θ cos γ (11) i=1

So, the robotic motion system can be analyzed as: 4 4 4v 4vi 2 2 4v 2 2 Pmotion = λ1 ∑ vi ∗ 4t + λ2 ∑ 4t ∗ v ∗ t + λ3 t ∗ 4t ∗ v + k1Fwheels + k2ωFwheels i=1 i=1 4 (12) 2 +µ1 ∑ (Ni·vi) cos θ + µ2N·v cos θ cos γ + τ·Fwheels + k(T) + β(T) ∗ t i=1 The impact on the power consumption of the motion system in different environments will be analyzed below.

3.2. Control System The control system mainly sends commands to the sensing system to control the sensors and poll the sensor readings periodically and sends commands to the motion system to control the motion state of each motor. It was found that the speed of the robot, the scanning frequency of the sensors and the running time of the robot all have an impact on the energy consumption of the control system. According to the different states of the robot operation, we can divide the power consumption of the robot control system into three phases: the standby phase, the phase in which the robot speed changes drastically, and the ﬁnal smooth-running phase. The consumption of the control system of the robot in this test is mainly for processing the data of each sensor of the robot, analyzing the motion state of the robot, and controlling the motion state of the robot to Sensors 2021, 21, 1800 6 of 18

perform the motion, according to the controller’s setting. Finally, the control system has the basic operating power consumption during the running process and the heat energy consumption generated by the increase in time during the running process [24].

 E = R P dt  stanby stanby  n 4  R v ∆v 1  E = ρ · f s · i + ρ i + ρ ·t 10 + P spech ∑ i i vmax 1 ∑ ∆t 2 stanby Econtrol = i=1 i=1 dt (13)  n  R 1  Estable = ∑ ρi· f si + ρ2·t 10 + Pstanby dt i=1

Econtrol is the power consumed by the robot control system during the motion, Estanby is the power consumption consumed by the robot control system in the standby state, Espech is the power consumption of the robot’s control system when the motion state of the robot changes, Estable is the power consumption of the robot’s control system during smooth operation. Pstanby is the power consumption of the motion system of the robot in the standby state. This part constitutes the basic power consumption of the robot control system. This part of the power consumption exists in all phases of robot operation. ρ is a parameter that characterizes the amount of data of the sensor of the robot, which is proportional to the amount of data generated by the robot’s sensor for data acquisition. For the control system, the more data generated by the robot’s sensing system, the more computing resources that need to be used, and the more energy the control system consumes. f s is the sampling frequency set by the robot’s sensor standard. For the robot of this experiment, the standard sampling frequency of each sensor has been set before the experiment was performed. In order to save power during motion and minimize power consumption, we combine the real-time sampling frequency of the robot with the speed of the robot. The sampling frequency of the robot’s sensing system is proportional to the real-time speed of the robot. In addition, ρ1 is the acceleration parameter of the robot, which characterizes the relationship between the power consumption and the acceleration of the control system 4 ∆vi during the speed change of the robot. ∑ ∆t is the sum of the absolute values of the four i=1 drive wheel accelerations of the robot, because the control system can individually control each drive wheel of the robot during the movement. ρ2 is the time power consumption parameter of the control system. During the movement of the robot, the control system of the robot will dissipate part of the energy consumption in the form of heat energy due to the increase of the running time. This part of the power consumption can be expressed as a function of time.

3.3. Sensor System The main function of the robot’s sensing system is to detect the environment around the robot and send the detected data to the robot’s control system for processing. The power consumption of the robot sensing system satisﬁes the following formula:

n Esensor = ∑(Psensor) (14) i=1 i

Esensor represents the power consumption of the robot sensing system, (Psensor)i is the power of the ith sensor, the power consumption of the sensing system can be superimposed on the sum of the power consumption of each sensor.

Ps( fS) = CS0 + CS1· fS (15)

The power consumption of a sensor is determined by both the type of sensor and the acquisition frequency. The energy consumption of different types of sensors varies greatly. For example, for cameras and LIDAR, the energy consumption varies greatly, and Sensors 2021, 21, 1800 7 of 18

for the same sensor with different frequencies of data acquisition, the energy consumption also varies greatly, we can use a linear function to characterize the energy consumption of the sensor, where CS0 represents the basic energy consumption of the sensor, and CS1 represents the sensor constant coefﬁcient of the sensor, their values only depend on the type of sensor [36].

4. The Inﬂuence of Various Factors on the Power Consumption of the Robot Next, we will analyze in detail the various inﬂuencing factors affecting FMWR power consumption in a complex environment in detail:

4.1. Research on the Power Consumption of Robots Uphill and Downhill 4.1.1. Power Analysis of Robot Uphill Process The upslope of the robot mainly affects the speed and acceleration of the robot, which has a huge impact on the robot’s control system, motion system, and sensing system. The influence on the control system and the sensing system is mainly reflected in the change of the corresponding part of the robot’s power consumption due to the change of the speed; the speed change of the robot can be obtained by substituting it into the formula. The following is mainly to analyze the power consumption of the motion system of the robot motion system during the ascending process. After analyzing the robot’s driving force rise and speed change caused by the angle problem during the uphill process, the impact on the iron loss is reflected in the speed. We do not conduct detailed analysis, and analyze the changes in iron loss and mechanical output. Figure2 shows the analysis of the forces on the robot in the direction of the incline during the uphill movement. We simplified the force analysis of the robot during the motion, and only analyzed the force along the slope, which can help us simplify the force analysis and kinematic and dynamics analysis. Firstly, according to the force analysis (see Figure2), the robot is subjected to several forces in the direction of the slope during its uphill movement. In addition to the driving force provided by the robot itself, the forces on the robot in the direction of the incline mainly include the component of the robot’s gravity in the direction of the incline and the frictional force acting on the robot. Because the robot is moving uphill, the velocity with respect to the slope is upward along the slope, so the frictional force acting on the robot is downward along the slope. Therefore, the direction of Sensors 2021, 21, x FOR PEER REVIEWthe combined force formed by the component of gravity along the slope and the friction8 of 19 force is downward along the slope. The robot needs to provide the upward force along the slope to ensure the balance of forces if it wants to maintain a stable motion on the slope. The drive force Fup provided by the robot is upward along the incline, and the value of Fup the value of F is the sum of the value of the component of gravity F along the is the sum of theup value of the component of gravity Fgravity along the slope andgravity the value of =+ theslope friction and the force valueFf riction of the, i.e., frictionFup = forceFgravity F+frictionFf riction, i.e.,. BecauseFFup gravity of the F special friction . Because construction of the of the Mecanum wheel, it is subjected to frictional forces, including both rolling friction special construction of the Mecanum wheel, it is subjected to frictional forces, including between the wheel and the inclined surface and the sliding friction between the roller and both rolling friction between the wheel and the inclined surface and the sliding friction the slope. between the roller and the slope.

FRobot

VRobot θ

F Friction F Gravity γ

Figure 2. Schematic diagram of the force analysis along the slope direction during the uphill Figure 2. Schematic diagram of the force analysis along the slope direction during the uphill movement of the robot. movement of the robot. After stress analysis, it is found that the following equation exists during the ascending process: =+ FFup gravity F friction (16) = θ γ Fmggravity sin sin (17) = μγ Fmgfriction cos (18) μ where Ffriction is the magnitude of the total frictional force on the robot, is the friction factor of the robot, its value is related to both the magnitude of the static friction and the magnitude of the sliding friction of the robot, the specific values and variation laws are

given in Equation (24). m is the mass of the robot, Fgravity is the component of the robot’s

gravity along the slope direction, Fup is the force that the robot needs to provide upwards along the slope during the uphill climb. Because the robot is subjected to frictional forces down the slope and the downward component of gravity along the slope, the robot must provide a corresponding upward force along the slope to ensure smooth robot operation. Because it exists during the operation of the robot: = FFup wheels (19)

So, the power consumption of the robot can be extended to:

44 =+λλvvvi 2222222 + λ   + μγ Pvmotion12 i *****cos vttvkmg 3   1 ii==11tt   t  ++μθγωγγθγθω++ ()2mgk12 sin sin k mg cos mg sin sin () mgk 12 sin sin k (20) 4 ++μθμγτ()+ θβ++()2 () + 12 Nvi icos Nv cos cos  F gravity F friction kT ( T )*t i=1

Sensors 2021, 21, 1800 8 of 18

After stress analysis, it is found that the following equation exists during the ascending process: Fup = Fgravity + Ff riction (16)

Fgravity = mg sin θ sin γ (17)

Ff riction = µmg cos γ (18)

where Ff riction is the magnitude of the total frictional force on the robot, µ is the friction factor of the robot, its value is related to both the magnitude of the static friction and the magnitude of the sliding friction of the robot, the speciﬁc values and variation laws are given in Equation (24). m is the mass of the robot, Fgravity is the component of the robot’s gravity along the slope direction, Fup is the force that the robot needs to provide upwards along the slope during the uphill climb. Because the robot is subjected to frictional forces down the slope and the downward component of gravity along the slope, the robot must provide a corresponding upward force along the slope to ensure smooth robot operation. Because it exists during the operation of the robot:

Fup = Fwheels (19)

So, the power consumption of the robot can be extended to: 4 4 4v 4vi 2 2 4v 2 2 2 2 2 Pmotion = λ1 ∑ vi ∗ 4t + λ2 ∑ 4t ∗ v ∗ t + λ3 t ∗ 4t ∗ v + µ k1m g cos γ i=1 i=1 +µ(2mgk1 sin θ sin γ + ωk2)mg cos γ + mg sin γ sin θ(mgk1 sin γ sin θ + ωk2) (20) 4 2 +µ1 ∑ (Ni·vi) cos θ+µ2N·v cos θ cos γ + τ· Fgravity + Ff riction + k(T) + β(T) ∗ t i=1

4.1.2. Power Analysis of Robot Downhill Process During the descent, the frictional force of the robot is always upward in the direction of the slope; the component of gravity along the direction of the slope is smaller than the frictional force of the robot. In order to ensure that the robot can move smoothly downward, the robot needs to provide a drive force downward along the slope to ensure the balance of forces on the robot. The relationship between the forces at this point is:

Fdown = Ff riction − Fgravity (21)

Fdown is the upward force along the slope that the robot needs to provide during the descent, because the combined force formed by the component of gravity along the slope and the component of friction along the slope is downward along the slope. During the whole movement, the robot is force balanced, the robot’s wheels will roll down in the direction of the slope, and there will be no reverse rotation of the wheels. For Mecanum wheel robots, due to the special design of the Mecanum wheels (the rollers of the Mecanum wheels can rotate around themselves), when the component of gravity along the slope is bigger than the limit of sliding friction between the roller and the slope (i.e., the limit of rolling friction of the Mecanum wheels), the robot will lose the ability to control its own motion state (this out-of-control state we do not study). For ordinary vehicles, the limit of downhill is the sliding friction between the tire and the contact surface force limit. This limit is greater than the rolling friction limit of the drive wheel, Therefore, the process of Fdown increasing in the opposite direction will occur. For an omnidirectional wheel robot, such as the Mecanum wheel, there is a trend of Fdown decreasing to 0. So, Sensors 2021, 21, 1800 9 of 18

4 4 4v 4vi 2 2 4v 2 2 2 2 2 Pmotion = −λ1 ∑ vi ∗ 4t + λ2 ∑ 4t ∗ v ∗ t + λ3 t ∗ 4t ∗ v + µ k1m g cos γ i=1 i=1 −µ(2mgk1 sin θ sin γ + ωk2)mg cos γ + mg sin γ sin θ(mgk1 sin γ sin θ + ωk2) (22) 4 2 +µ1 ∑ (Ni·vi) cos θ+µ2N·v cos θ cos γ + τ· Fgravity + Ff riction + k(T) + β(T) ∗ t i=1

4.2. The Influence of the Center of Gravity on the Power Consumption of the Robot The offset of the robot’s center of gravity relative to the center position directly affects the iron consumption and mechanical output of the robot. The impact on iron consumption is mainly reflected in the impact on the friction force, i.e., on the positive pressure of each wheel, which is defined as F = µ × N, where N is the positive pressure on the contact surface. For a robot whose center of gravity deviates from the center of gravity, the positive pressure on the contact surface of each of its wheels is different, where the positive pressure on the contact surface can be expressed as:

1 δ 1 δ N = ± a ± b ·N (23) i 2 a 2 b

where N is given by Equation N = mg cos γ; m, g, γ represent the mass of the robot, the gravity constant, and the inclination angle of the terrain; a and b represent the horizontal and vertical distances between each wheel; ±δa and ±δb represent the distance between the center of gravity of the robot and the center of the robot [2]. Figure3 shows a diagram of the change in the robot’s center of gravity, with the black dots indicating the current position of the robot’s center of gravity. The δa and δb indicate the lateral and vertical shift of the robot’s center of gravity.

2 1 δb 2a 2 − a R0 + δa µ = ·µR + ·µS (24) R0 + Ly R0 + Lx

Lx + δb Ly + δa µ1 = ·µR + ·µS (25) R0 + b R0 + a

Lx − δa Ly − δb µ2 = ·µR + ·µS (26) R0 + Lx R0 + Ly

µR and µS correspond to the sliding friction and rolling friction of the robot, R0 corresponds to the turning radius of the robot, for the Mecanum wheel robot, because of its special construction, the friction factor is different when moving with different turning radius. Lx Sensors 2021, 21, x FOR PEER REVIEW 10 of 19 and Ly are the distances from the center of the driving wheel to the center of the vehicle, as shown in Figure2.

δ a Lx δ b b

a FigureFigure 3.3. SchematicSchematic diagramdiagram ofof thethe analysisanalysis ofof thethe robot’srobot’s centercenter ofof gravitygravity position.position.

δ 2 1 − b 2a  R + δ 2 a (24) μ= 0 a μμ+  ++R S RL00yx RL

L + δ L + δ μμ= xb+ ya μ 1 ++RS (25) Rb00 Ra

L − δ L − δ μμ= xa+ yb μ 2 ++R S (26) RL00xy RL μ μ R and S correspond to the sliding friction and rolling friction of the robot, R0 corresponds to the turning radius of the robot, for the Mecanum wheel robot, because of its special construction, the friction factor is different when moving with different turning

radius. Lx and Ly are the distances from the center of the driving wheel to the center of the vehicle, as shown in Figure 2.

input to the motor. Since both Ia and Ra in Formula (3) increase, the temperature of the robot will continue to rise. Therefore, if the heat cannot be dissipated reasonably, the robot will lose more and more energy. At the same time, the increased current flowing through the motor drive board will cause the energy consumed by the motor drive board to increase accordingly, and the resulting speed change will cause the control system to lose more energy. The change of resistance with temperature conforms to the formula: =+−α RT() R00()1 () T T (27)

Among them, R()T represents the resistance value of the robot when the temperature is T , α is the temperature coefficient, T represents the temperature at the current

time, R0 and T0 represent the resistance value at a certain time and the temperature at this time. Table 1 shows the parameters of the robot motors, which are only related to the type of motor used in the robot and are not related to the robot structure.

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4.3. The Effect of Temperature on the Power Consumption of the Robot The energy consumption of three systems of the robot is temperature related. As the robot moves, part of the energy of the motor will be converted into heat to be lost. As the temperature of the motor increases, the torque and speed constant of the robot will decrease [38]. To compensate for the decreased torque, it is necessary to increase the current input to the motor. Since both Ia and Ra in Formula (3) increase, the temperature of the robot will continue to rise. Therefore, if the heat cannot be dissipated reasonably, the robot will lose more and more energy. At the same time, the increased current ﬂowing through the motor drive board will cause the energy consumed by the motor drive board to increase accordingly, and the resulting speed change will cause the control system to lose more energy. The change of resistance with temperature conforms to the formula:

R(T) = R0(1 + α(T − T0)) (27)

Among them, R(T) represents the resistance value of the robot when the temperature is T, α is the temperature coefﬁcient, T represents the temperature at the current time, R0 and T0 represent the resistance value at a certain time and the temperature at this time. Table1 shows the parameters of the robot motors, which are only related to the type of motor used in the robot and are not related to the robot structure.

−2 −4 k1(T) = k10(1 + αR(T − T0)) (1 + αM(V − Vmax)) (28)

−2 k2(T) = k20(1 + αR(T − T0))(1 + αM(V + Vmax)) (29) −2 −4 k(T) = k10k20(1 + αR(T − T 0 )) (1 + αM(V + Vmax)) (30) −2 β(T) = k20(1 + αM(V + Vmax)) (31)

where k10 and k20 are the motor constants measured at the temperature of the motor, T0 is the temperature coefﬁcient of the motor, αR and αM is the speed coefﬁcient of the motor.

Table 1. Friction constants as measured through different wheels.

Front Back Mean M1 (Left motor) k10 0.0019 0.0021 0.0020 k20 1.5223 1.5325 1.5274 M2 (Left motor) k10 0.0022 0.0026 0.0024 k20 1.5445 1.5326 1.5385

Table2 shows the temperature parameters of the robot motors, which are only related to the type of motor used in the robot.

Table 2. Friction constants as measured through different wheels.

Front Back Mean M1 (Left motor) αR 1.25 1.56 1.405 αM 1.32 1.44 1.38 M2 (Left motor) αR 1.35 1.37 1.36 αM 1.38 1.35 1.365 Sensors 2021, 21, 1800 11 of 18

5. Simulation and Experimental Veriﬁcation The experimental setting is mainly to study the power consumption of the robot in various situations during uphill and downhill. Before the experiment, we set up a test bench that can freely adjust the uphill and downhill angles. In order to ensure that the power of the robot will be stable on the test bench, we set the slope of the test bench to 4 m during the test. The robot we used in the experiment is a four-wheeled Mecanum wheel robot called MiniBalance, in our laboratory. The main parameters of the robot are the mass of the robot: m = 12 kg, the overall dimensions of the robot are length: L1 = 0.86 m, width: L2 = 0.52 m, the motor of the robot is the planetary gear motor model MD36N, the design of the robot is independent suspension, which has good shock absorption function and can ensure that the robot can still be used normally in the process of shifting the center of gravity, the maximum load weight of the robot is 30 kg. The robot is designed with 24 v power supply and the control board of the bottom control system is stmf407. Table3 shows the parameters of the theoretical model established above. By substituting the parameters into the theoretical model established above and implementing the model mathematically in MATLAB, we can obtain the simulated power consumption values in this case.

Table 3. The value of parameter in the theoretical model.

Parameter Value

λ1 0.15 λ1 0.08 λ1 0.01 ρ1 0.2 ρ2 0.05 τ 0.3 T0 20

During the experiment, we increased the height of the test bed continuously, and used inclinometer (the accuracy of the inclinometer can reach 0.05◦) for accurate angle measurement, then control the FMWR to start from the flat bottom first, and then go uphill when the speed is stable. This process requires PID to intervene to ensure that the speed of the robot remains constant during the uphill. When the robot stabilizes in the uphill process, it immediately enters the horizontal movement stage to buffer, and then enters the downhill stage. During the downhill process, the robot control system needs to control the speed of the robot to travel at a preset speed to prevent out of control situations. In order to study the influence of various factors in the environment on the power consumption of the robot, we use different speeds and temperatures and the center of gravity to perform the same experimental process during the experiment. We verify the correctness of our model by comparing the differences in power consumption. Figure4 shows the experimental pictures of the robot in the real scene. Figure4a shows a schematic diagram of the robot’s center of gravity by changing the position of the weight loaded on the robot. The blue object in the figure is iron weighing 10 kg. Figure4b shows the robot during the uphill and downhill experiments. From the picture, we can see that we use a green rubber carpet to increase the friction, and the angle of the uphill and downhill can be adjusted freely. Sensors 2021, 21, x FOR PEER REVIEW 12 of 19

Table 3. The value of parameter in the theoretical model.

Parameter Value λ 1 0.15 λ 1 0.08 λ 1 0.01 ρ 1 0.2 ρ 2 0.05 τ 0.3

T0 20

During the experiment, we increased the height of the test bed continuously, and used inclinometer (the accuracy of the inclinometer can reach 0.05 ) for accurate angle measurement, then control the FMWR to start from the flat bottom first, and then go uphill when the speed is stable. This process requires PID to intervene to ensure that the speed of the robot remains constant during the uphill. When the robot stabilizes in the uphill process, it immediately enters the horizontal movement stage to buffer, and then enters the downhill stage. During the downhill process, the robot control system needs to control the speed of the robot to travel at a preset speed to prevent out of control situations. In order to study the influence of various factors in the environment on the power consumption of the robot, we use different speeds and temperatures and the center of gravity to perform the same experimental process during the experiment. We verify the correctness of our model by comparing the differences in power consumption. Figure 4 shows the experimental pictures of the robot in the real scene. Figure 4a shows a schematic diagram of the robot’s center of gravity by changing the position of the weight loaded on the robot. The blue object in the figure is iron weighing 10 kg. Figure 4b Sensors 2021, 21, 1800 shows the robot during the uphill and downhill experiments. From the picture, we12 ofcan 18 see that we use a green rubber carpet to increase the friction, and the angle of the uphill and downhill can be adjusted freely.

Figure 4.4. Robot experiment process diagram.diagram. The measurement method adopted by the robot is shown in Figure5. For the control The measurement method adopted by the robot is shown in Figure 5. For the control system and the sensor system, the power is less than 10 W and the sampling value of system and the sensor system, the power is less than 10 W and the sampling value of the the power consumption is relatively high. We use power monitor AAA10F to measure power consumption is relatively high. We use power monitor AAA10F to measure power SensorsSensors 2021 2021, 21, 21, x, xFOR FOR PEER PEER REVIEW REVIEWpower consumption. For the motion system, because of its high power, we use IT6952A1313 of of 19 19 consumption. For the motion system, because of its high power, we use IT6952A to meas- to measure. The actual measurements shown in the article are averaged after several ure. The actual measurements shown in the article are averaged after several experimental experimental measurements, and we operate in accordance with statistical principles. This measurements, and we operate in accordance with statistical principles. This ensures that ensuresthethe results results that of of thethe the resultsexperiments experiments of the are experiments are not not subject subject are to to chance not chance subject and and toguarantees guarantees chance and the the guaranteesaccuracy accuracy of of thethe the accuracyexperimentalexperimental of the validation validation experimental as as well well validation as as the th ereproducibility reproducibility as well as the reproducibilityof of the the experiments. experiments. of the experiments.

EnergyEnergy input input PowerPower Monitor Monitor SensorSensor AAA10FAAA10F DataData readback readback SystemSystem

EnergyEnergy input input MotorMotor BrushedBrushed DC DC IT6952AIT6952A WheelsWheels DataData readback readback ControllerController MotorMotor

Energy input PowerPower Monitor Monitor Energy input ControlControl AAA10FAAA10F SystemSystem DataData readback readback Figure 5. Power measurement system. FigureFigure 5. 5.Power Power measurement measurement system. system.

BeforeBefore thethe the experimentalexperimental experimental veriﬁcationverification verification ofof of eacheach each system,system, system, ﬁrst,first, first, thethe the speedspeed speed variationvariation variation duringduring during thethethe motionmotion motion shouldshould should bebe be analyzedanalyzed analyzed andand and modeled.modeled. modeled. FigureFigure Figure6 6 shows 6shows shows the the the speed speed speed variation variation variation of of of the the the robotrobotrobot throughoutthroughout throughout itsits its motionmotion motion whenwhen when wewe we setset set thethe the maximummaximum maximum speeds peedspeed ofof of thethe the robotrobot robot toto to 11 m/s.1m/s. m/s.

Speed-varying image over time 1.2

0.8 (m/s) 0.6

0.4

0.2 Speed

0 0 2 4 6 8 10 12 14 16 18 20 Time (t)

FigureFigure 6.6. 6. The The speed speed of of the the robot robot duringduring during thethe the movement.movement. movement. 5.1. Motion System 5.1.5.1. Motion Motion System System For the motion system, we need to verify that the power consumption of the robot ﬂuctuatesForFor the duethe motion motion to changes system, system, in slope we we need andneed centerto to verify verify of gravitythat that the the during power power the consumption consumption robot’s movement. of of the the robot Therobot fluctuatesfluctuates due due to to changes changes in in slope slope and and center center of of gravity gravity during during the the robot robot’s’s movement. movement. The The firstfirst is is the the change change in in the the power power consumption consumption of of the the robot robot’s’s control control system system due due to to changes changes inin the the angle angle of of the the slope slope and and the the speed speed of of the the robot robot itself itself during during the the uphill uphill process: process: FigureFigure 7 7shows shows the the comparison comparison between between the th emeasured measured and and si simulatedmulated values values of of the the robotrobot’s’s power power during during the the process process of of uphill uphill and and downhill. downhill. In In the the validation validation process, process, with with twotwo degrees degrees as as a asampling sampling interval, interval, we we performed performed experimental experimental validation validation and and MATLAB MATLAB simulation,simulation, and and compared compared the the solved solved simulated simulated values values with with the the experimental experimental measured measured values,values, and and it it can can be be seen seen through through Figure Figure 7 7that that the the simulated simulated values values are are very very close close to to the the actualactual measured measured values. values. The The accuracy accuracy of of our our modeling modeling is is proved proved to to be be relatively relatively high. high. FigureFigure 8 8shows shows the the real real-time-time power power consumption consumption of of the the r obotrobot during during its its entire entire movement. movement.

Comparison of measured and simulated values of robot power consumption 40

25 (w) (w) 20 Measured power consumption during robot uphill Simulation value of power consumption during robot uphill 15 Measured power consumption during robot downhill Power Power Simulation value of power consumption during robot downhill 10

0 0 2 4 6 8 10 12 14 16 18 20 Angle ( ° )

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the results of the experiments are not subject to chance and guarantees the accuracy of the experimental validation as well as the reproducibility of the experiments.

Power Monitor Energy input Sensor

AAA10F Data readback System

Energy input Motor Brushed DC IT6952A Wheels Data readback Controller Motor

Power Monitor Energy input Control AAA10F System Data readback Figure 5. Power measurement system.

Before the experimental verification of each system, first, the speed variation during the motion should be analyzed and modeled. Figure 6 shows the speed variation of the robot throughout its motion when we set the maximum speed of the robot to 1 m/s.

Speed-varying image over time 1.2

0.8 (m/s) 0.6

0.4

0.2 Speed

0 0 2 4 6 8 10 12 14 16 18 20 Time (t)

Figure 6. The speed of the robot during the movement.

Sensors 2021, 21, 1800 13 of 18 5.1. Motion System For the motion system, we need to verify that the power consumption of the robot fluctuates due to changes in slope and center of gravity during the robot’s movement. The ﬁrstfirstis isthe thechange changein inthe thepower powerconsumption consumption of of the the robot’s robot’s control control system system due due to to changes changes inin the the angle angle of of the the slope slope and and the the speed speed of of the the robot robot itself itself during during the the uphill uphill process: process: FigureFigure7 7shows shows the the comparison comparison between between the the measured measured and and simulated simulated values values of of the the robot’srobot’s powerpower duringduring thethe processprocess of of uphill uphill and and downhill. downhill. In In the the validation validation process, process, with with twotwo degrees degrees as as a a samplingsampling interval,interval, we we performedperformed experimentalexperimental validation validation and and MATLAB MATLAB simulation,simulation, andand comparedcompared thethe solvedsolved simulated simulated valuesvalues with with the the experimental experimental measured measured values,values, and and it it cancan bebe seenseen throughthrough Figure Figure7 7that that the the simulated simulated values values are are very very close close to to the the actualactual measuredmeasured values.values. TheThe accuracyaccuracy ofof ourour modelingmodeling isis provedproved toto bebe relativelyrelatively high.high. FigureFigure8 8shows shows the the real-time real-time power power consumption consumption of of the the robot robot during during its its entire entire movement. movement.

Comparison of measured and simulated values of robot power consumption 40

25 (w) 20 Measured power consumption during robot uphill Simulation value of power consumption during robot uphill 15 Measured power consumption during robot downhill Power Simulation value of power consumption during robot downhill 10

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0 0 2 4 6 8 10 12 14 16 18 20 ° Angle ( ) Figure 7. ComparisonComparison between between the the measured measured value value and and the thetheoretical theoretical value value when when the power the power con- consumptionsumption is stable is stable during during the therobot’s robot’s uphill uphill and and downhill. downhill.

Robot power consumption values at different angles 50

45 Angle is 0 degrees Angle is 6.5 degrees

40 Angle is 12 degrees

35 Angle is 20 degrees

30 (w) 25

Downhill process

Power 20

5 Uphill process

0 0 2 4 6 8 10 12 14 16 18 20 Time (t) Figure 8.8. Power consumption during robot movement.movement.

Through Figure 88,, we can see see that that the the robot robot’s’s power power has has a very a very significant significant rise rise in the in thestartup startup process, process, the process the process of starting of starting uphill, uphill, and andthe process the process of finally of finally entering entering the flat. the flat. This is because, in these several processes, the speed of the robot will decrease due to This is because, in these several processes, the speed of the robot will decrease due to the the power being unable to increase instantly. Because of the role of PID control, the robot power being unable to increase instantly. Because of the role of PID control, the robot will will increase its power to increase the speed to a predetermined value, so the process of increase its power to increase the speed to a predetermined value, so the process of in- increasing the power consumption of the robot corresponds to the process of increasing creasing the power consumption of the robot corresponds to the process of increasing the the speed of the robot. speed of the robot. As the robot’s uphill angle increases, the power difference between the robot’s uphill As the robot’s uphill angle increases, the power difference between the robot’s uphill and downhill will increase; according to our research, the Mecanum wheel robot reached and downhill will increase; according to our research, the Mecanum wheel robot reached the limit of the robot’s uphill and downhill at 20 degrees due to its special structure. At the limit of the robot’s uphill and downhill at 20 degrees due to its special structure. At 20 20 degrees, it is both the limit for the robot to go uphill and the limit for the robot to go degrees, it is both the limit for the robot to go uphill and the limit for the robot to go down. down. During the descent, the robot has a long-term power of 0. DuringFigure the 9descent, shows thethe effectrobot ofhas the a long robot’s-term center power of gravityof 0. offset on the position of the robot’sFigure center 9 shows of gravity the effect on the of powerthe robot consumption’s center of gr ofavity the robot.offset on The the data position in Figure of the9 arerobot the’s actualcenter measurementsof gravity on the of power the effect consumption of the change of the in therobot. robot’s The centerdata in of Figure gravity 9 are on energythe actual consumption. measurements In order of the to showeffect theof the results change more in concisely, the robot we’s onlycenter show of gravity the actual on measuredenergy consumption. values here, In andorder the to errorshow between the results the more simulated concisely, and we actual only valuesshow the does actual not exceedmeasured 0.5 values w after here, the actual and the comparison. error between the simulated and actual values does not exceed 0.5 w after the actual comparison.

Figure 9. Influence of the shift of the center of gravity on the power consumption of the robot.

For the shift of the center of gravity, the method shown in Figure 4a is adopted; the position of the center of gravity of the robot is shifted by placing heavy objects at different positions of the robot. Figure 9 shows the position of the center of gravity of the robot gradually shifted outward from the position of the center of the robot; through Figure 9, it can be seen very clearly that the power consumption of the robot increases significantly

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Figure 7. Comparison between the measured value and the theoretical value when the power consumption is stable during the robot’s uphill and downhill.

Figure 8. Power consumption during robot movement.

Through Figure 8, we can see that the robot’s power has a very significant rise in the startup process, the process of starting uphill, and the process of finally entering the flat. This is because, in these several processes, the speed of the robot will decrease due to the power being unable to increase instantly. Because of the role of PID control, the robot will increase its power to increase the speed to a predetermined value, so the process of increasing the power consumption of the robot corresponds to the process of increasing the speed of the robot. As the robot’s uphill angle increases, the power difference between the robot’s uphill and downhill will increase; according to our research, the Mecanum wheel robot reached the limit of the robot’s uphill and downhill at 20 degrees due to its special structure. At 20 degrees, it is both the limit for the robot to go uphill and the limit for the robot to go down. During the descent, the robot has a long-term power of 0. Figure 9 shows the effect of the robot’s center of gravity offset on the position of the robot’s center of gravity on the power consumption of the robot. The data in Figure 9 are the actual measurements of the effect of the change in the robot’s center of gravity on energy consumption. In order to show the results more concisely, we only show the actual Sensors 2021, 21, 1800 14 of 18 measured values here, and the error between the simulated and actual values does not exceed 0.5 w after the actual comparison.

FigureFigure 9. 9.Influence Influence of of the the shift shift of of the the center center of of gravity gravity on on the the power power consumption consumption of of the the robot. robot. For the shift of the center of gravity, the method shown in Figure4a is adopted; the Sensors 2021, 21, x FOR PEER REVIEW For the shift of the center of gravity, the method shown in Figure 4a is adopted;15 of the 19 position of the center of gravity of the robot is shifted by placing heavy objects at different position of the center of gravity of the robot is shifted by placing heavy objects at different positions of the robot. Figure9 shows the position of the center of gravity of the robot positions of the robot. Figure 9 shows the position of the center of gravity of the robot gradually shifted outward from the position of the center of the robot; through Figure9, it gradually shifted outward from the position of the center of the robot; through Figure 9, canas the be seencenter very of gravity clearly thatof the the robot power shifts consumption outward. As of the center robot increasesof gravity significantly of the robot it can be seen very clearly that the power consumption of the robot increases significantly asshifts the centeroutward, of gravitythe load of on the one robot or shiftsmore outward.wheels of Asthe the robot center will of increase gravity significantly, of the robot shiftswhich outward, will lead theto the load increase on one of or power more consumption wheels of the of robot the corresponding will increase significantly, motor of the whichrobot. willMoreover, lead to the the increase increase of of power power consumption consumption is of not the a corresponding linear relationship motor with of the the

robot.movement Moreover, of the the robot increase’s center of powerof gravity, consumption but a feature is not of aexponential linear relationship increase with—that the is, movementthe farther ofthe the robot robot’s’s center center of ofgravity gravity, is from but a the feature center, of exponentialthe more energ increase—thaty consumption is, theis increased farther the for robot’s the same center outward of gravity movement is from theof the center, robot the’s center more energyof gravity. consumption This kind isof increasedshift of the for center the same of gravity outward is very movement likely to ofcause the robot’sthe overload center situation of gravity. of the This robot kind mo- of shifttor. of the center of gravity is very likely to cause the overload situation of the robot motor. FigureFigure 10 10 shows shows the the variation variation of of the the energy energy consumption consumptio ofn theof the robot’s robot motion’s motion system sys- withtem thewith change the change of temperature, of temperature, which haswhich a very has obvious a very obvious effect on effect the energy on the consumption energy con- ofsumption the motion of the system. motion It issystem. important It is toimportant note that to the note data that shown the data in Figures shown9 inand Figure 10 ares 9 theand instantaneous 10 are the instantaneous power values power collected values after collected the robot afte changesr the robot temperature changes ortemperature the center ofor gravity the center and of reaches gravity a steadyand reaches state ofa steady operation, state and of operation, each of the and data each here of is the obtained data here by averagingis obtained the by data averaging from multiple the data experiments, from multiple so thatexperiments, the effects so of that accidental the effects factors of acci- can bedental excluded. factors The can higher be excluded. the temperature The higher of the the robot, temperature the higher of the energyrobot, the consumption higher the ofenergy the motion consumption system. of the motion system.

Effect of temperature on robot power consumption 15

14.5

13.5

13 (w) 12.5

12 Power

11.5

11 Robot power measurement Robot power consumption simulation value 10.5

10 0 5 10 15 20 25 30 o Angle( C) FigureFigure 10. 10.Influence Influence of of temperature temperature on on the the power power consumption consumption of of robot robot motion motion system. system. 5.2. Control System 5.2. Control System The verification of the robot control system’s power consumption is mainly to verify The verification of the robot control system’s power consumption is mainly to verify the fluctuation of the power consumption of the robot’s control system during the process the fluctuation of the power consumption of the robot’s control system during the process of uphill, downhill, and final stabilization of the robot. Test the accuracy of our power of uphill, downhill, and final stabilization of the robot. Test the accuracy of our power consumption modeling of the robot control system. consumption modeling of the robot control system. Figure 11 shows the power consumption of the control system at different stages of the robot’s movement. Through Figure 8, we can clearly see that the basic consumption of the robot control system is basically the same. The difference is the amount of data processed by the robot at different stages and the acceleration pulses issued by the robot due to different speeds. According to our theory, the robot has the largest number of acceleration pulses during the uphill process, followed by the startup process, and finally the process of stable operation. The above figure is very good to confirm our idea, so it can explain our modeling is accurate.

(a) Start process power (b) Smooth running process power (c) Uphill process power

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as the center of gravity of the robot shifts outward. As the center of gravity of the robot shifts outward, the load on one or more wheels of the robot will increase significantly, which will lead to the increase of power consumption of the corresponding motor of the robot. Moreover, the increase of power consumption is not a linear relationship with the movement of the robot’s center of gravity, but a feature of exponential increase—that is, the farther the robot’s center of gravity is from the center, the more energy consumption is increased for the same outward movement of the robot’s center of gravity. This kind of shift of the center of gravity is very likely to cause the overload situation of the robot motor. Figure 10 shows the variation of the energy consumption of the robot’s motion system with the change of temperature, which has a very obvious effect on the energy consumption of the motion system. It is important to note that the data shown in Figures 9 and 10 are the instantaneous power values collected after the robot changes temperature or the center of gravity and reaches a steady state of operation, and each of the data here is obtained by averaging the data from multiple experiments, so that the effects of accidental factors can be excluded. The higher the temperature of the robot, the higher the energy consumption of the motion system.

Figure 10. Influence of temperature on the power consumption of robot motion system.

5.2. Control System Sensors 2021, 21, 1800 15 of 18 The verification of the robot control system’s power consumption is mainly to verify the fluctuation of the power consumption of the robot’s control system during the process of uphill, downhill, and final stabilization of the robot. Test the accuracy of our power consumptionFigure 11 showsmodeling the of power the robot consumption control system. of the control system at different stages of the robot’sFigure movement. 11 shows Through the power Figure consumption8, we can clearly of the seecontrol that system the basic at different consumption stages of of the robotthe robot’s control movement. system is basicallyThrough Figure the same. 8, we The can difference clearly see is that the the amount basic consumption of data processed of bythe the robot robot control at different system is stages basically and the the same acceleration. The difference pulses is issued the amount by the of robot data pro- due to differentcessed by speeds. the robot According at different to our stages theory, and the acceleration robot has the pulses largest issued number by the of robot acceleration due pulsesto different during speeds. the uphill According process, to followed our theory, by the robot startup has process, the largest and number finally the of acceler- process of stableation operation.pulses during The the above uphill figure process, is very foll goodowed to by confirm the startup our idea,process, so itand can finally explain the our modelingprocess of is stable accurate. operation. The above figure is very good to confirm our idea, so it can explain our modeling is accurate.

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Figure 11. Power of the control system in different states of motion. Figure 11. Power of the control system in different states of motion.

5.3. Sensor System5.3. Sensor System The power consumptionThe power consumptionof the sensor ofsystem the sensor is closely system related is closely to the relatedspeed of to the speed of the robot. When therobot. speed When of the speedrobot ofis thenot robotvery fast, is not the very sensor fast, thecan sensor operate can in operate a non-full- in a non-full-scale way. When the speed of the robot increases gradually, the frequency of the sensor gets scale way. When the speed of the robot increases gradually, the frequency of the sensor higher and higher, so the power generated will be higher and higher (see Figure 12). gets higher and higher, so the power generated will be higher and higher (see Figure 12).

Figure 12. Power of the sensorFigure system. 12. Power of the sensor system.

Figure 12 showsFigure the robot12 shows sensing the robotsystem’s sensing power system’s consumption power consumptionat different sampling at different sampling frequencies. Figurefrequencies. 12a shows Figure the 12 energya shows co thensumption energy consumptionof the sensing of system the sensing when systemthe when the robot samplingrobot frequency sampling is 0.04 frequency s. Figure is 12b 0.04 shows s. Figure the energy 12b shows consumption the energy of the consumption sens- of the ing system whensensing the robot system sampling when the frequency robot sampling is 0.02 frequencys. According is 0.02 to the s. Accordinggraph, it can to thebe graph, it can concluded thatbe the concluded higher the that sampling the higher frequency the sampling of the frequency sensor, the of the higher sensor, the the energy higher the energy consumption consumptionof the sensing of system. the sensing According system. to Accordingthe setting, to the the faster setting, the thespeed faster of the the speed of the robot, the higherrobot, the the sampling higher the frequency sampling will frequency be, and willby generalizing be, and by generalizing it, we can get it, we that can get that the the faster the speedfaster theof the speed robot, of the robot,higher the the higher energy the consumption energy consumption of the sensing of the system. sensing system. We We demonstrateddemonstrated the energy the consumption energy consumption characteristics characteristics of the sensor of the system sensor in system the pre- in the previous vious researchresearch article [24,25], article [so24 ,we25], will so wenot will elaborate not elaborate on them on here. them here.

6. Discussion and Conclusions Before comparing the robot energy consumption modeling with the actual energy consumption measurements, we first measure the energy consumption of the robot during motion several times and find out the actual power value of the robot during motion, according to the statistical processing method. Figure 13 shows the comparison of multiple experimental measurements of the robot’s energy consumption throughout the motion of the robot. The comparison shows that the results of each experimental measurement are very close to each other and are consistent with the results of the robot’s energy consumption measurements shown in Figure 13, which shows that our experimental results are not influenced by chance. The data in Figure 14 are averaged over several experimental measurements, which ensure the accuracy of the robot modeling verification.

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6. Discussion and Conclusions Before comparing the robot energy consumption modeling with the actual energy consumption measurements, we first measure the energy consumption of the robot during motion several times and find out the actual power value of the robot during motion, according to the statistical processing method. Figure 13 shows the comparison of multiple experimental measurements of the robot’s energy consumption throughout the motion of the robot. The comparison shows that the results of each experimental measurement are very close to each other and are consistent with the results of the robot’s energy consumption measurements shown in Figure 13, which shows that our experimental results are not influenced by chance. The data in Figure 14 Sensors 2021, 21, x FOR PEER REVIEW 17 of 19 Sensors 2021, 21, x FOR PEER REVIEWare averaged over several experimental measurements, which ensure the accuracy of17 theof 19 robot modeling verification.

Comparison between measured values of multiple experiments 50 Comparison between measured values of multiple experiments 50 Measured value of the first test Measured value of the first test 45 Measured value of the second test Measured value of the second test 45 Measured value of the third test 40 Measured value of the third test 40

35 35

30 30 (w)

(w) 25 25

Power 20

15 15

10 10

5 5

0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 12 14 16 18 20 Time (t) Time (t) FigureFigure 13.13. ComparisonComparison ofof powerpower consumptionconsumption valuesvalues fromfrom multiplemultiple experimentalexperimental measurements.measurements. Figure 13. Comparison of power consumption values from multiple experimental measurements.

Comparison of theoretical and actual values of robot power 50 Comparison of theoretical and actual values of robot power 50 Theoretical value of robot power 45 Theoretical value of robot power 45 Measurement of robot power Measurement of robot power 40 40

35 35

30 30 (w)

(w) 25 25

Power 20 Power 15 15

10 10

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0 0 0 2 4 6 8 10 12 14 16 18 20 0 2 4 6 8 Time10 (s) 12 14 16 18 20 Time (s) Figure 14. Comparison of the measured value of the robot’s real-time power with the actual value FigureFigure 14.14.Comparison Comparison ofof thethe measuredmeasured valuevalue ofof thethe robot’srobot’s real-timereal-time power with the actual value during the movement. duringduring thethe movement.movement.

FigureFigure 1414 shows the the relationship relationship between between the the actual actual and and measured measured values values of ofthe the ro- robot’sbot’’s power power during during the the entire entire movement movement of of the the robot. robot. The error betweenbetween thethe theoreticaltheoretical andand actualactual values of of the the calculated calculated power power of of the the robot robot is within is within 5%, 5%, and and the theaccuracy accuracy rate rateisis aboutabout is about 95%.95%. 95%. ItIt cancan It be canbe provedproved be proved thatthat thatthethe modelingmodeling the modeling ofof thethe of threethree the three majormajor major robotrobot robot systemssystems systems isis veryvery is verysuccessful. successful. ByBy comparing comparing the the modeling modeling with with actual actual values, values, it is it found is found that thethat accuracy the accuracy of the of robot the modelingrobot modeling is about is 95%, about and 95%, the a modelingnd the modeling has reached has reached the expected the expected modeling modeling accuracy. accu- The differenceracy. The betweendifference the between modeling the and modeling the actual and value the isactual mainly value because is mainly the consumption because the ofconsumption the robot’s otherof the parts robot of’’s itsother motion parts system of its motion during system the modeling during processthe modeling has not process been thoroughlyhas not been searched. thoroughly searched. This paper proposes a power consumption model of a robot in a complex environment. The influences of the angle of the upslope and downslope of the robot during the movement, the temperature of the robot, and the change of the center of gravity of the robot on the power consumption of the robot are fully studied. Compared to traditional modeling, we pay more attention to the fluctuation of the power consumption of the robot during the transition of the different motion states of the robot. By incorporatinging thethe fluc-fluc- tuationstuations ofof thethe robotrobot intointo thethe modeling,modeling, wewe improvedimproved thethe accuracyaccuracy ofof thethe robotrobot modelingmodeling toto aboutabout 95%,95%, whichwhich greatlygreatly improvedimproved thethe accuracyaccuracy ofof thethe robotrobot modeling.modeling. AtAt thethe samesame time,time, wewe analyzedanalyzed thethe statestate ofof MecanumMecanum wheelswheels duringduring actualactual operationoperation aand found that Mecanum wheel robots and ordinary wheeled robots are different in the process of uphill and downhill. The angle limit of the Mecanum wheel robot on uphill and downhill is much smaller than that of the ordinary wheeled robot. This is due to the particular struc- tureture ofof MecanumMecanum wheelswheels—thatthat is,is, thethe MecanumMecanum wheelwheel robotrobot sacrificessacrifices thethe robotrobot’’s ability toto adaptadapt toto complexcomplex environments,environments, toto increaseincrease thethe flexibilityflexibility ofof thethe robot.robot.

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This paper proposes a power consumption model of a robot in a complex environment. The influences of the angle of the upslope and downslope of the robot during the movement, the temperature of the robot, and the change of the center of gravity of the robot on the power consumption of the robot are fully studied. Compared to traditional modeling, we pay more attention to the fluctuation of the power consumption of the robot during the transition of the different motion states of the robot. By incorporating the fluctuations of the robot into the modeling, we improved the accuracy of the robot modeling to about 95%, which greatly improved the accuracy of the robot modeling. At the same time, we analyzed the state of Mecanum wheels during actual operation and found that Mecanum wheel robots and ordinary wheeled robots are different in the process of uphill and downhill. The angle limit of the Mecanum wheel robot on uphill and downhill is much smaller than that of the ordinary wheeled robot. This is due to the particular structure of Mecanum wheels—that is, the Mecanum wheel robot sacrifices the robot’s ability to adapt to complex environments, to increase the flexibility of the robot.

Author Contributions: Conceptualization, L.Z. and K.K.; methodology, L.Z.; software, L.H.; validation, L.H, L.Z.; formal analysis, L.H. and F.Z.; investigation, L.H.; resources, L.H.; data curation, L.H.; writing—original draft preparation, L.H.; writing—review and editing, K.K.; visualization, K.K.; supervision, L.Z.; project administration, L.Z.; funding acquisition, F.Z. All authors have read and agreed to the published version of the manuscript. Funding: This project was supported by the National Key R&D Program of China (Grant No. 2017YFB1302400), National Natural Science Foundation of China (Grant No. 61773242 and Grant No. 61473179), Key Program of Natural Science Foundation of Shandong Province, China (Grant No. ZR2015QZ08), Key Program of Scientific and Technological Innovation of Shandong Province, China (Grand No. 2017CXGC0926), Key Research and Development Program of Shandong Province, China (Grant No. 2017GGX30133), Natural Science Foundation of Shandong Province, China (Grant No. ZR2020MF073). Conflicts of Interest: The authors declare no conflict of interest

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