Design a Hybrid Energy-Supply for the Electrically Driven Heavy-Duty Hexapod Vehicle

Increasing the power density and overload capability of the energy-supply units (ESUs) is always one of the most challenging tasks in developing and deploying legged vehicles, especially for heavy-duty legged vehicles, in which significant power fluctuations in energy supply exist with peak power several times surpassing the average value. Applying ESUs with high power density and high overload can compactly ensure fluctuating power source supply on demand. It can avoid the ultra-high configuration issue, which usually exists in the conventional lithium-ion battery-based or engine-generator-based ESUs. Moreover, it dramatically reduces weight and significantly increases the loading and endurance capabilities of the legged vehicles. In this paper, we present a hybrid energy-supply unit for a heavy-duty legged vehicle combining the discharge characteristics of lithium-ion batteries and peak energy release/absorption characteristics of supercapacitors to adapt the ESU to high overload power fluctuations. The parameters of the lithium-ion battery pack and supercapacitor pack inside the ESU are optimally matched using the genetic algorithm based on the energy consumption model of the heavy-duty legged vehicle. The experiment results exhibit that the legged vehicle with a weight of 4.2 tons can walk at the speed of 5 km/h in a tripod gait under a reduction of 35.39% in weight of the ESU compared to the conventional lithium-ion battery-based solution.


Introduction
Legged vehicles can pass through the rugged ground since they move via discrete support footholds [1][2][3]. Many endeavors were made in constructing legged vehicle systems [4][5][6][7][8]. However, many problems still restrict the application of legged vehicles [9,10]. The energy supply represents one of the most challenging fields in developing and deploying legged vehicles, especially for heavy-duty vehicles [11], in which significant power fluctuations in energy supply exist, with peak power frequently surpassing the average value.
Energy consumption of the legged vehicle significantly affects its driving range and independent working ability [12]. Various energy consumption models were developed for the legged vehicle systems to evaluate energy consumption effectively. Kar et al. carried out the energy analysis for a simplified hexapod robot model, considering the effects of structural parameters, friction coefficient, and duty cycle on energy consumption under its waveform gait but ignoring the considerable thermal energy generated in the support leg joints [13]. Furthermore, Nishii established the actuator output energy model for the specific actuator to determine the joint mechanical power and the thermal energy losses in the motor to evaluate the energy of the whole robot system [14]. Researchers have also conducted many studies on the energy optimization of legged robots [15][16][17][18]. Erden proposed an improved gradient descent method based on an optimal control algorithm to optimize the energy consumption during the swinging phase of the leg [19]. Roy analyzed the effect of gait parameters on the system's energy consumption under the fixed axis turning gait and the crab walking gait [20,21].
During high-speed walking, the energy-supply unit of the heavy-duty legged vehicle needs to provide positive driving power and absorb negative driving power (electric braking power). The violent alternating load of the joint leads to frequent positive and negative power switching and overload [22,23]. The energy-supply and joint driving systems must have a high response and good following to improve the motion performance of the vehicle [24,25]. Besides, the energy supply unit's weight should be carefully considered concerning the legged vehicle's loading capacity and transportation needs. The legged vehicle's motion capability and walking mileage will improve as its weight is reduced.
The applied energy-supply units for vehicles mainly have the following primary forms: (1) The generator energy-supply unit is primarily used in large industrial and mining vehicles [26,27]; (2) The battery energy-supply unit is primarily used in electric buses and electric cars [28]; (3) the supercapacitor energy-supply unit is mainly used in urban light rail, which needs the timing and distance charging conditions [29]; (4) There are also some hybrid energy-supply units, which commonly consist of the battery and generator [30]. The generator energy-supply unit cannot realize negative power absorption and meet the energy demand of the highfrequency operation of the actuator. Increasing the highpower discharge resistor is necessary to ensure the reliability and safety of such a system. As for the battery energy-supply unit, although it has better continuous discharge characteristics [28,31], it needs a larger capacity battery to meet the energy supply of the legged vehicle. It is difficult to achieve robust power and energy system lightweight simultaneously with conventional power battery configurations because of their heavy weight. As for the pure capacitor energy-supply unit, although stronger shock resistance and better response characteristics due to the energy density, it does not have the conditions for individual use [32]. Moreover, the applied energy-supply forms for the vehicles are mainly used for relatively smooth operating conditions of the system and may not adapt to severe power fluctuation.
Based on the above analysis, a robust energy-supply unit with high power density for the heavy-duty legged vehicle still needs to be developed to improve its independent working ability. This paper focuses on the energy-supply unit of the heavy-duty legged vehicle. A hybrid energy-supply unit combining lithium-ion batteries and supercapacitors is proposed. It gives full play to the advantages of the discharge characteristics of lithium-ion batteries and the peak-energy release/absorption characteristics of supercapacitors, which can improve instantaneous high-power output and have lasting dynamic performance. The parameters of the lithium-ion battery pack and supercapacitor pack are optimally matched using the genetic algorithm based on the energy consumption model of the heavy-duty-legged vehicle. The proposed hybrid energy supply scheme can adapt to significant power fluctuation with less weight. The work is motivated by a real need to increase the energy autonomy of a heavy-duty hexapod vehicle.
The rest of this paper is organized as follows. In Sect. 2, the energy consumption model of the heavy-duty hexapod vehicle is built to obtain the optimized gait parameters and the gait power consumption characteristics. In Sect. 3, a hybrid energy-supply unit scheme is proposed combining the discharge characteristics of lithium-ion batteries and supercapacitors. The experiments to validate the performance of the hybrid energy-supply unit are conducted in Sect. 4. Finally, conclusions and possible future research are provided in Sect. 5.

Energy Analysis of the Heavy-Duty
Hexapod Vehicle

Set-Up of the Heavy-Duty Hexapod Vehicle
The schematic figure of the designed hexapod vehicle is shown in Fig. 1. It weighs 4200 kg and consists of a body and six legs rectangularly configured with three spaced evenly on each side. The leg is tri-segmented and equipped with four actively actuated rotating joints inspired by large terrestrial mammals [33]. Each joint is actuated by an electric cylinder which contains a DC brushless motor and a planetary roller screw. The axis of the side joint is perpendicular to the coronal plane of the leg. The axes of the hip, knee, and ankle joints are perpendicular to the sagittal plane of the leg. The hip and knee joint actuators adopt two motors and two parallel electric cylinders. The side and ankle joint actuators adopt a single motor and a single cylinder. The planetary roller screw transforms the rotational motor motion into linear motion, which provides linear speed and thrust.

Energy Consumption Model of the Vehicle
The energy consumed during the hexapod vehicle walking process can be obtained as the sum of the energy consumed by each joint drive system. The energy from the energysupply unit of the vehicle is supplied to the rotational motor through the motor driver. The scheme of hexapod vehicle energy consumption is shown in Fig. 2. As each joint of the heavy-duty hexapod vehicle is actuated by an electric cylinder, the system energy consumption should consider both the output shaft loads and motor characteristics. It can help to analyze the power fluctuations during heavy-duty hexapod vehicle walking. The motor will generate the inductive electric potential by converting electrical energy to its mechanical counterpart. The electric model of the DC motor is shown in Fig. 3.
Hence, the voltage balance equation can be expressed as follows: , R a is the electric resistance of the motor, L a is the rotor inductance, k e is the back electromotive constant. Generally, the voltage generated by L a is much smaller than that by R a and e b (t) . Thus, it is not considered in this study.
The torque generated by the electromagnetic conversion of the motor is required to drive the motor rotor and the  where (t) = k m i(t) , J m includes the motor rotor inertia and the equivalent inertia of the electric cylinder screw nut, k m is the torque constant. And L (t) = J L̈m (t) = n cl F L (t) , where J L is the equivalent load inertia of the motor, F L is the thrust of the electric cylinder, and n cl is the transmission ratio between the rotational motor shaft motion and the linear motion of the electric cylinder end. The energy consumption of the electrical cylinder can be written as: According to Eq. (2), Eq. (3) can be expressed as: where v cl is the electric cylinder velocity and a cl is the electric cylinder acceleration.
The value of the torque constant k m is the same as the back electromotive constant k e when the international system of units is used. Thus, the energy consumption of the electrical cylinder is: Based on Eq. (5), the electrical cylinder power is composed of the heat consumed by the resistance P R , the mechanical energy consumed by the rotor P rotor , and the mechanical energy of the load P load .
(2) (t) = J m̈m (t) + L (t) The energy consumption of the hexapod vehicle is given as the sum of:

Energy Consumption Analysis
The heavy-duty hexapod vehicle uses the tripod gait to realize high-speed walking. The demand for energy consumption when the vehicle walks at the highest expected speed determines the energy-supply unit design. Improving the walking energy utilization efficiency of the vehicle and reducing its walking energy consumption and peak power can help to reduce the weight of the energy-supply unit. Furthermore, the optimal gait parameters can reduce the vehicle's energy consumption. The gait cycle and duty factor are important parameters among the gait parameters. The optimal gait parameters are obtained based on the established energy consumption model. The energy consumption demand when the vehicle walks at the highest expected speed is calculated based on the optimal gait parameters, which is the design input of the vehicle energy-supply unit.
The expected vehicle walking speed is 1.4 m/s when walking with a tripod gait. The relationship between the walking speed V , gait cycle T , stride length L , and duty factor is Both sets of legs can leave the ground simultaneously at the switching time, possibly causing instability or impact, especially under heavy load conditions. For this reason, the tripod gait is improved by increasing the period during which all six legs land simultaneously. Figure 4 shows the sequence of the tripod gait. The duty factor of gait is greater than 0.5. The dark strip indicates that one leg is in the support phase. The blank denotes that one leg is in the swing phase.
The influence of cycle time and duty factor on energy consumption is mainly revealed in the allocation of motion time between the swing and support phases. Under the The sequence of the tripod gait specific speed requirement, the gait cycle change will cause the stride length change. On the other hand, the duty factor change will directly redistribute the total gait cycle time to the swing and support phases to optimize energy consumption. After analyzing the energy consumption model using various duty factors and gait cycles, their combined effect on the energy consumption is shown in Fig. 5. The vertical axis indicates that the minimum total drive power of the hexapod vehicle is obtained when the gait cycle is 1 s and the duty factor is 0.54. The power of the heavy-duty hexapod vehicle can be calculated according to the obtained gait parameters. When the hexapod vehicle walks at 1.4 m/s using a tripod gait, the motor characteristics of the middle leg are shown in Fig. 6. As the hexapod vehicle walks straight in the simulation, the side joint of the leg stays. Thus, Fig. 6 shows the hip, knee, and ankle motor characteristics. Time from 0.00 to 0.54 s is the stance phase and time from 0.54 to 1.00 s is the swing phase. The solid green lines in Fig. 6 are data carves of the hip joint motors (the data of motor 1a and motor 1b are the same in this walk simulation). The red dash lines in Fig. 6 are data carves of the knee joint motors (the data of motor 2a and motor 2b are the same in this walk simulation). The blue dash-dotted line in Fig. 6 are data carves of the ankle joint motor. Figure 6a shows the motor rotation speed according to the gait time. Figure 6b shows the motor torque according to the gait time. Figure 6c shows the motor power according to the gait time. Figure 6d shows the motor torque according to the motor rotation speed. Figure 6e shows the motor power according to the motor rotation speed. Figure 6e shows the motor rotation speed in the frequency domain.

Energy Supply Composition Scheme
The energy-supply unit of the heavy-duty legged vehicle is required to release the positive power frequently and absorb the negative power to adapt to the power fluctuations. The lithium-ion battery has the advantages of high energy density and low cost. However, its disadvantage is the low discharge rate. A high discharge rate can significantly shorten its service life. Suppose the lithium-ion battery is selected as a single energy source to meet the requirements of power fluctuations and significant overload coefficients. In that case, the capacity of the lithium-ion battery pack is bound to be too large and heavy. The equivalent circuit model of the lithium-ion battery is shown in Fig. 7.
The output voltage of the lithium-ion battery is expressed as follows: where V batt is the output voltage of the lithium-ion battery, V bat_oc is the open-circuit voltage of the lithium-ion battery, I batt is the load current of the lithium-ion battery, R series is the internal resistance of the lithium-ion battery, V D is the electrochemical polarization voltage, V K is the concentration polarization voltage, D and K are the equivalent time constant, R transient_L and R transient_s are the analog polarization resistance. In this model, two RC circuits simulate the instantaneous response of the battery.
where C transient_L and C transient_s are simulated polarization capacitors.
The charged state of the lithium-ion battery can be expressed as: where SOC bat0 is the state of charge at the initial moment of the lithium-ion battery, and C bat is the actual capacity of lithium-ion battery.
The supercapacitors have the advantages of fast charging and discharging speed, many cycles, high rate charging and discharging, and small internal resistance in the charging and discharging process. Their disadvantages are low energy density, which forms an excellent complement to lithium batteries. The equivalent circuit of the supercapacitor is shown in Fig. 8, ignoring the self-discharge factor of the supercapacitor.
The charging and discharging power of the supercapacitor can be expressed as where V cap (t) is the voltage of the capacitor, I cap (t) is the discharge current of the supercapacitor, and R cap is the internal resistance of the supercapacitor. It proves that the charge-discharge power of the supercapacitor is proportional to the charge-discharge current. The working voltage of the supercapacitor can be expressed as: where C cap is the capacity of the supercapacitor, Δt is the duration of discharge. The charged state of the supercapacitor can be expressed as: where V cap_min represents the battery discharge cut-off voltage and V cap_max represents the maximum allowable voltage of the capacitor. In this paper, the hybrid energy supply scheme is adopted considering the advantages of the supercapacitor and the lithium-ion battery. It combines the supercapacitor and lithium-ion battery as the energy supply for the motor to drive the legged vehicle. The hybrid energy supply scheme gives full play to the advantages of the lithium-ion battery's high energy density and the supercapacitor's high power density. It can improve the instantaneous high power output and have lasting dynamic performance. The hybrid system can adapt to more significant power fluctuations with less weight than the lithium-ion battery system. Besides, the hybrid energy supply scheme has apparent advantages in resisting power fluctuations, which can prolong the battery module service life.
The hybrid power configuration scheme is shown in Fig. 9. By detecting the potential change at the port of the supercapacitor, the lithium-ion battery output power is qualitatively changed by changing the input and output characteristics of DC/DC cooperating with the supercapacitor. Additionally, the DC/DC only controls the lithium-ion battery discharge current, which is much smaller than the DC bus discharge current. Therefore, the capacity of the required DC/DC converter is significantly reduced, which is favorable when aiming to reduce the cost and improve reliability. Further, since the supercapacitor withstands the short-time high rate discharge current, it also effectively reduces the fast type requirements and the DC/DC converter performance index [34,35].

Optimal Matching of Composite Power System
Three requirements must be met simultaneously to maximize mileage for an electric hexapod vehicle at high speed with a tripod gait. Firstly, sufficient energy must be provided to ensure the motion mileage. Secondly, power must be available at any moment during the moving process. Finally, the electrical power regime of the whole machine must be known.

Power Constraints
To reach the required mileage, the depth of lithium-ion battery discharge is DOD bat . The hybrid power system still needs to provide the maximum power, which should not be below the maximum demanded output power: where P bat_max (DOD bat ) denotes the maximum power that can be output when the battery discharge depth is DOD bat , P cap_max is the maximum power that the supercapacitor can provide, P need_max is the peak vehicle power demand, b 1 represents the number of batteries in a series, b 2 is the number of batteries connected in parallel, c 1 is the number of supercapacitors connected in series, c 2 is the number of supercapacitors connected in parallel, and finally, and DC denotes the DC/DC discharge efficiency (value of DC = 0.95 is taken in this paper).

Energy Constraints
Since the supercapacitor energy density is much lower than that of lithium batteries, its initial energy is not considered in this paper. Thus, to satisfy the motion mileage requirement, the lithium-ion battery should provide no less than the energy required for vehicle motion where E bat DOD bat denotes the energy released from the battery when the single-cell discharge depth is DOD bat , S is the driving range, V represents the tripod gait speed, P need_avr is the average power required for the tripod gait at speed V, and bat is the battery discharge efficiency ( bat = 0.94 was taken in this paper).

Voltage Constraints
The minimum supercapacitor working voltage should be above the minimum allowable operating voltage of the electric equipment and not greater than the rated voltage. The minimum operating voltage of the lithium-ion battery should not be below the minimum allowable DC/DC converter voltage and not greater than the rated DC/DC converter voltage: where V Dmin and V Dnom represent the minimum allowable input and rated voltages of the DC converter, respectively, V min and V nom represent the minimum allowable operating and rated voltage of the electric equipment, respectively, V bmin and V cmin are minimum voltages of the lithium-ion battery and supercapacitor unit, respectively, Ceil represents the upward rounding, and Floor represents downward rounding.

Optimization Model
This paper selects the cost and weight indexes to construct the objective function considering the economic cost and energy-supply unit weight. The objective function is used to optimize the number of serial and parallel connections. The objective function can be written as where c and m denote the weight coefficients of economic and weight costs, respectively, m b and m c are the weight of the lithium-ion battery and supercapacitor units, while c b and c c denote the prices of lithium-ion battery and supercapacitor units, respectively. b 1 represents the number of batteries in a series, b 2 is the number of batteries connected in parallel, c 1 is the number of supercapacitors connected in series, and c 2 is the number of supercapacitors connected in parallel.

Genetic Algorithm-Based Optimal Matching
The genetic algorithm is an algorithm for solving both unconstrained and constrained nonlinear optimization problems. It is based on a natural selection process and mimics biological evolution. The algorithm iteratively modifies a population composed of individual solutions (units). The genetic algorithm randomly selects individuals from the current population during each iteration. It uses them as parents to generate the next generation of children. The optimal solution is obtained once the stop criteria are met. The optimization problem in this paper is a constraintbased nonlinear integer programming problem. The genetic algorithm is used to ensure the optimal matching of the composite power system so that the objective function F will have the minimum value. The genetic algorithm-based optimization is realized with the MATLAB genetic algorithm toolbox. The algorithm parameters are shown in Table 1.
The lithium-ion battery and the supercapacitor unit parameters are given in Tables 2 and 3. The objective function's economical cost and weight cost are selected as 0.2 and 0.8, and parameters such as driving range, power requirement, and system voltage operating range are included in the constraints. Figure 10 shows the population iteration curve. The mean population value reaches the optimal value after 65 iterations when the weight function reaches the value of 31.5101. According to the optimization function F, the   2.85 V Battery weight (g) 68 ± 1 g computation results achieve better economic cost and weight cost of the energy-supply unit under power constraints, energy constants, and voltage constants. The remaining optimization results are shown in Table 4.

Energy-Supply Unit for the Heavy-Duty Legged Vehicle
The heavy-duty hexapod vehicle should be able to walk for a long time to carry out regular transport tasks. Although the hybrid energy supply scheme can provide a higher quality energy supply for the vehicle to meet the lightweight requirement, it is more restrictive in the driving range. To increase the endurance of the vehicle, a generator module is added based on the hybrid power system configuration scheme shown in Fig. 9. The battery can be continuously charged by a generator, which further enhances the endurance of the hexapod vehicle. The hybrid energy-supply unit scheme is shown in Fig. 11.
The energy supply for the heavy-duty hexapod vehicle is designed according to the optimal parameters of the battery pack and capacitor pack and the hybrid power system configuration scheme. The circuit frame diagram and detailed design of the energy-supply unit are shown in Fig. 12. The battery pack and capacitor pack parameters in the energysupply unit are shown in Tables 5 and 6. The maximum discharge current of the energy-supply unit can reach 75 A. The weight that the battery pack and supercapacitor pack occupied is 34.992 kg.   Fig. 11 Hybrid energy supply configuration scheme for the heavyduty legged vehicle Fig. 12 Hybrid energy-supply unit for the heavy-duty legged vehicle

Experiments
A single-leg prototype experimental platform is constructed to obtain the drive power characteristics of the joint motors and validate the performance of the designed energy-supply unit. The experimental platform is shown in Fig. 13, which includes the prototype of the single mechanical leg and the load-lifting device. The total mass of the leg is 200 kg. The hip joint of the leg prototype connects to the hinge joint on the load-lifting device.

Power Characteristics Experiments of the Leg Motors
To obtain the power characteristics of each joint motor, the electric cylinders on each joint are set to move sinusoidally with a 70 mm amplitude and a frequency of 1 Hz. The leg is powered by a 540 V constant voltage DC energy supply. The experimental motor data obtained for each joint are shown in Figs. 14, 15, 16. According to the experimental data, the rapid cyclic reciprocating joint motion causes drastic changes by inducing the electric potential of the motor, which in turn causes fluctuations in the bus current and voltage. The maximum bus current of the hip joint motor can reach 17 A, the maximum bus current of the knee joint motor can reach 13 A, and the maximum bus current of the ankle joint motor can reach 13 A. Those current and voltage fluctuations also cause motor power fluctuation, which poses high requirements for the vehicle energy-supply unit. The leg prototype is set to track a sinuous foot point trajectory to obtain the joint motors' power characteristics. As shown in Fig. 17, the leg prototype can realize horizontal swing sinusoidally with a 400 mm amplitude and a frequency of 0.8 Hz.
The detailed joint motor rotation speeds are shown in Fig. 18. The energy supply data of the leg is shown in Fig. 19. According to the experimental data, there are large fluctuations in the current and voltage, which increase the requirements of the energy-supply unit. The maximum total current of the leg can reach 32 A. The designed energy-supply unit can meet the current requirements of the leg's locomotion.

Comparison Experiment of the Energy-Supply Unit
The purpose of the experiment is to test the performance of the designed energy-supply unit when it powers the whole heavy-duty vehicle walking. The hybrid energy-supply unit is set to work in the power requirements according to the simulation data in Fig. 6. Another energy-supply unit is set to work with the same power requirements for experimental comparison. The energy-supply unit for experimental comparison uses a lithium-ion battery pack to supply the current and voltage. The maximum discharge current of the lithiumion battery energy-supply unit is the same as the designed hybrid energy-supply unit to ensure its performance can   Table 7.
The output current and voltage of the designed hybrid energy-supply unit are shown in Fig. 20. The output current and voltage comparisons of battery current and voltage comparison between the hybrid energy-supply unit and the lithium energy-supply unit are shown in Fig. 21. According to the experiment data, the hybrid energy-supply unit's battery current fluctuations are smoother than the lithium-ion battery energy-supply unit's battery current. Although the lithium-ion battery energy-supply unit has continuous discharge characteristics, the response characteristic of the battery system has a large gap compared with the hybrid energy-supply unit. The supercapacitor module absorbs the most current fluctuations of the hybrid energy-supply unit, which helps extend the battery module's service life. That can also enhance the reliability of the hybrid energy-supply unit.
The lithium-ion battery energy-supply unit uses much more battery units to make the maximum discharge current reach 75 A, dramatically increasing the weight of the battery pack. Its battery pack weighs 54.144 kg. However, the weight of the battery pack and supercapacitor pack in the designed hybrid energy-supply unit is just 34.992 kg. The hybrid energy supply configuration helps to obtain a 35.39% weight reduction. Considering the characteristics of supercapacitors and lithium batteries and analyzing the experimental data comprehensively, the hybrid energy supply scheme is the better choice, which effectively combines the shock resistance and high response characteristics of supercapacitors and the continuous discharge capability of lithium batteries.

Conclusion
This paper proposes a hybrid energy-supply unit for the heavy-duty legged vehicle, which can release positive power frequently and absorb the negative power to adapt to power fluctuations. It comprises a lithium-ion battery pack and supercapacitor pack, which can provide instantaneous high power output and lasting power. The energy-supply unit is  developed based on analyzing the heavy-duty legged vehicle's typical working conditions and the discharge characteristics of lithium batteries and the supercapacitor. Additionally, the energy-supply unit's battery pack and capacitor pack parameters are optimized via a genetic algorithm based on the power constraints, energy constraints, voltage constraints, and weight and mass cost objective function. The hybrid system can adapt to more significant power fluctuations with less weight than the lithium-ion battery system. The designed energy-supply unit has high power density and can release positive work and absorb negative power, which meets the requirements of the high-frequency response and high overload multiple of legged locomotion. The experiment shows that the hybrid energy-supply unit reduces 35.39% of the weight of the power module compared to the lithium-ion battery energy-supply unit. Future research of the authors will focus on the energy control strategies, aiming to develop a smoother energy supply strategy for the heavy-duty legged vehicle. More experiments with different energy control strategies will be designed and implemented for the heavy-duty legged vehicle.