Modeling and analysis on low energy consumption foot trajectory for hydraulic actuated quadruped robot

Abstract

According to the energy consumption characteristics of hydraulic actuator, the valuable foot trajectory characterized by using segmented cubic spline interpolation curve in the swing phase is proposed firstly to reduce the energy consumption of quadruped robots, which is implemented by using controlling parameters t_f to change the duration of leg raising and falling in one gait cycle, and then realized the directly control to the time ratio between the piston extension and retraction. Then, the total energy consumption of the hydraulic actuated quadruped robot SCalf-II is modeled. Meanwhile, the parameters of the foot trajectory that have a large impact on the energy consumption are determined. Finally, simulation analysis and verification experiments of the robot moving with constant speeds at the key parameters are performed. The results show that for the given foot trajectory, the optimization ranges of the gait cycle and duration of leg lifting from the lowest to highest are determined in which the energy required for the robot locomotion is at a relatively low level.

Keywords

Energy efficient hydraulic actuation quadruped robot optimal parameters of foot trajectory

Introduction

Energy efficiency is an important index that has attracted increasingly more attention in modern engineering, and the development of quadruped robots is no exception. As a legged robot that is relatively simple in terms of structure and control algorithms,^1,2 the quadruped robot has become an object of more focus. Currently, however, energy efficiency is still the main bottleneck restricting its development.³ By comparison with Cheetah, the electrical driven quadruped robot designed by MIT, the COT (Cost of transport) of BigDog, the hydraulically driven quadruped robot, is up to 15, which is near to 30 times of the former.

To improve the energy efficiency of a quadruped robot, many methods have been proposed by numerous researches. The most direct and effective way is the passive dynamic walking. The robot with this feature^4,5 does not depend on any energy input or active control and can realize walking on a shallow slope only with the movement inertia of components. Although with high energy efficiency, the passive dynamic walking do not have motion flexibility terrain adaptability, what greatly limits its application. Another efficient method for improving energy efficiency is using the elastic elements, such as the series elastic actuator^6,7 and variable stiffness actuator,^8,9 the driving devices that directly locates some elastic devices between the load and the output terminal of the actuator. Through direct force control or active impedance control, these actuators not only can reduce the consumed energy against angular inertia but also can improve contact compliance between the ground and robot feet and reduce energy consumption during contact impact.^10,11 However, the quadruped robot actuated by these actuators also has adaptability limitation to a certain extent because of their structure, mechanical properties of the material or the difficulty in component design, especially for the robot actuated by the actuator with straight movement.

The motion optimization, which do not rely on elastomers, is another effective method in improving the robot’s energy efficiency. Yang et al.¹² propound a practical foot trajectory planning method that uses the Fourier series to express the position, velocity, and acceleration of the joint, and parameters of mathematical functions are obtained by Pattern Search Method. Compared with other similar method, the energy consumption of the quadruped robot is dropped by 7.55% in experiment. The dephased gait generating method is proposed by Deng et al.,¹³ which can improve the walking speed simultaneously to reduce the energy consumption. Gao et al.¹⁴ use the fifth-order and six-order polynomials to express trajectories of the trunk body in forward and normal direction, respectively, and the foot trajectory is obtained by the kinematics of the robot. The results of the experiment confirm that this trajectory planning algorithm is better on reducing the impact force and improving the energy efficiency of the robot. Among the achievements by Kim et al.,¹⁵ the angle between robot’s feet and the ground is planned as a characteristic curve of trigonometric function. Compared with the traditional method characterized by fixed contact angle, this method can effectively reduce energy consumption of the robot by reducing swing amplitude of all rotary joints. In addition to the first method mentioned above, the rest can ensure the smooth transition of velocity and acceleration during phases switching, and the energy conservation of the last one is the best. However, due to the fact that the high energy efficiency depends mostly on the contact angle of robot’s feet with the ground, the adaptability of robot to the environment is weaken even with the last one.

Compared with quadruped robots driven by torque motors, the throttle loss, relief loss, pressure loss, and other issues make hydraulically driven quadruped robots more unsatisfactory in terms of energy efficiency. Although many hydraulic servo systems have excellent energy-saving performance, complex structures, lack of high dynamic performance, and strict requirements of matching load characteristics make them difficult to apply to legged robots field.¹⁶ So up to now, the valve-controlled system with relatively lower energy efficiency¹⁷ is still widely used in hydraulically driven robot design. Not only can the system exhibit high stability but also can improve the dynamic response of the robot. Although with high-performance hydraulic components that the energy efficiency of the system can improve effectively,¹⁸ this, however, comes at the expense of increased design cost.

In this work, for traditional foot trajectories with symmetrical property that are commonly used in quadruped robot, a foot trajectory with lower energy consumption characterized by using segmented cubic spline interpolation curve in the swing phase is studied. The time t_f of the foot which is measured from the ground to the highest point is always introduced into the foot trajectory equations by the way of the piecewise programming in the swing phase to change the time ratio between the piston extension and retraction. Then, on the basis of the energy consumption model of the hydraulic actuator, the complete energy consumption equation of the hydraulic actuated quadruped robot SCalf-II is obtained, and the sensitive parameters that have a great influence on the energy consumption are determined. Finally, the energy consumption of SCalf-II using the common foot trajectory is calculated. The influence of the gait cycle and the time of leg lifting in the swing phase on energy consumption is determined, which provides principle for the selection of controlled parameters for robot locomotion. In summary, contributions about this article are listed as follows:

In view of the fact that most of the current foot trajectory generation methods with low energy consumption may easily make the robot to lose the adaptation to the environment or trajectory smoothing, a highly energy-efficiency foot trajectory characterized by using segmented cubic spline interpolation curve in the swing phase is proposed, which is implemented by using controlling parameters t_f to change the duration of leg raising and falling in one gait cycle, and then realized the directly control to the time ratio between the piston extension and retraction.

The effect of t_f on energy consumption of quadruped robot is verified by the computer simulation and experiment, which offer the possibility for legged robot to be used in unknown terrain for stable and high energy-efficiency walking.

The remainder of this article is organized as follows. In the second section, the energy consumption model for the hydraulic actuator is described. In the third section, based on the above model, the energy consumption model for SCalf-II is built. In the fourth section, the gait trajectory is proposed and some gait parameters that affect energy consumption are studied by simulation. Finally, the comprehensive experiments are presented in the fifth section. The conclusions are drawn in the sixth section.

Energy consumption model for actuator

For the general description of the pressure characteristics in working chambers of the actuator, the following assumptions should be given:

Density and bulk modulus of oil can be considered constant.

The internal and external leakages of the hydraulic cylinder are ignored. Meanwhile, the internal leakage and dynamic characteristics of the servo valve are not considered.

The pressure loss in pipes and working chambers is ignored.

For the hydraulic actuator, the energy consumption can be classified into two parts: the useful work and the rest which has been converted into heat, and based on the schematic diagram shown in Figure 1, their mathematical formulations^19,20 can be described as follows

Figure 1.

Asymmetric hydraulic cylinder controlled by symmetrical servo valve.

\begin{array}{l} \{\begin{array}{l} W_{p} = & | (P_{S} - P_{p i s}) A_{p i s} {\dot{x}}_{p} + (A_{p i s} P_{p i s} - α A_{p i s} P_{r}) {\dot{x}}_{p} \\ + (P_{r} - P_{T}) α A_{p i s} {\dot{x}}_{p} | ({\dot{x}}_{v} \geq 0) \\ W_{n} = & | (P_{S} - P_{r}) α A_{p i s} {\dot{x}}_{p} + (α A_{p i s} P_{r} - A_{p i s} P_{p i s}) {\dot{x}}_{p} \\ + (P_{p i s} - P_{T}) A_{p i s} {\dot{x}}_{p} | ({\dot{x}}_{v} < 0) \end{array} \end{array}

where subscripts p and n represent the extension and retraction of the piston, respectively. P_S and P_T represent the oil supply pressure and returning oil pressure, respectively. $P_{p i s}$ and P_r are the pressures in the piston and ring chambers of the cylinder, respectively. The areas of both sides of the piston are denoted as $A_{p i s}$ and A_r , respectively, and their ratio is α. ${\dot{x}}_{p}$ is the velocity of the piston.

Taking into account that the part has been turned into heat is difficult to quantify and denote, the power consumed by the hydraulic actuator can be denoted as

W_{a c t}^{*} = | (A_{p i s} P_{p i s} - α A_{p i s} P_{r}) {\dot{x}}_{p} |

The force balance equation is given by

A_{p i s} P_{p i s} - α A_{p i s} P_{r} - f_{f} = f_{l}

where f_l is the external load, and f_f represents the friction and damping inside the hydraulic cylinder and can be denoted as²¹

f_{f} = (f_{s t} + K_{c} e^{\frac{| {\dot{x}}_{p} |}{v_{s t}}}) s i g n ({\dot{x}}_{p}) + C_{b} {\dot{x}}_{p}

where K_c and $v_{s t}$ are the parameters of static friction. C_b is the damping coefficient.

Taking into account equations (2) and (3), the simplified form of the power consumed by the hydraulic actuator can be deduced as

W_{a c t} = | (f_{l} + f_{f}) {\dot{x}}_{p} |

Through integrating equation (5), the energy consumption of the actuator can be expressed as

E_{a c t} = \int_{p} W_{a c t} d t

Energy consumption model for SCalf-II

According to equation (5), load forces ( $f_{l})$ acting on each hydraulic actuators must be calculated so as to obtain the energy consumption model of SCalf-II. To meet the demand of high dynamic performance, the constant pressure system is always adopted in hydraulically actuated quadruped robots, such as Scalf-II which was designed in 2012 for orientation in rough terrain,²² as shown in Figure 2. The topological structure of the robot’s legs is X style, the rear knee pointing to the front elbow, and each leg consists of three rotating joints for pitching movement of the assembly of the hip, thigh, and calf, respectively.

Figure 2.

Hydraulic actuated robot SCalf-II. (a) Robot prototype; (b) front-left leg of SCalf-II.

Load force acting on actuator in supporting phase

Considering the symmetries of leg structure geometry designing, the front-left leg can be only used to establish corresponding models. In general, the swaying of hip assembly is not a rhythmic movement and has light contributions to the foot trajectory implementation. Accordingly, the rotating joints of thigh assembly and calf assembly are only considered in leg kinematics modeling.¹² With variables of front-left leg defining in Figure 3 and the corresponding values shown in Table 1, the foot position (O_f ) in the reference coordinate system (O ₀) at the rotating joint of hip can be described as

Figure 3.

Definition of variables for front-left leg.

Table 1.

Leg specifications.

Symbol	Value	Units
$l_{u 1}$	0.236	m
$l_{d 3}$	0.41249	m
$l_{t 1}$	0.3395	m
r	0.035	m
$l_{u 2}$	0.04941	m
$l_{d 1}$	0.23402	m
$l_{d 2}$	0.04627	m
$l_{m 1}$	0.17574	m
$l_{m 2}$	0.21099	m
$θ_{1}$	16.1	$^{\circ}$
$θ_{2}$	12.46	$^{\circ}$
$θ_{3}$	20.58	$^{\circ}$
$θ_{4}$	1.41	$^{\circ}$
$θ_{5}$	3.68	$^{\circ}$

\{\begin{array}{l} x_{f} & = & l_{d 3} sin (θ_{r 1}) + l_{t 1} sin (θ_{r 1} + θ_{r 2}) \\ z_{f} & = & - l_{d 3} cos (θ_{r 1}) - l_{t 1} cos (θ_{r 1} + θ_{r 2}) - r \end{array}

where joint angles $θ_{r 1}$ and $θ_{r 2}$ can be calculated as

\{\begin{array}{l} θ_{r 1} & = & - arccos (\frac{x_{f}^{2} + z_{f}^{2} + l_{d 3}^{2} - l_{t 1}^{2}}{2 \sqrt{x_{f}^{2} + z_{f}^{2}} l_{d 3}}) - arctan (\frac{x_{f}}{z_{f}}) \\ θ_{r 2} & = & - arccos (\frac{l_{d 3}^{2} + l_{t 1}^{2} - x_{f}^{2} - z_{f}^{2}}{2 l_{d 3} l_{t 1}}) + π \end{array}

respectively.

Since the stability of the robot motion is not considered, the robot can be simplified to a planar model, as shown in Figure 4(a). In each gait cycle, the legs on two diagonals are either in the swing phase or supporting phase. Thus, the interaction force between the foot endpoint and the ground should be calculated during the supporting phase. Assuming no relative sliding and constant speed of the robot, the following relationships can be obtained by referring the two-dimensional (2D) planar model shown in Figure 4 (b)

Figure 4.

Simplified model of SCalf-II. (a): 3D translated into planar model; (b): 2D planar model.

\{\begin{array}{l} f_{r r, z} + f_{f l, z} = G \\ (b + Δ x_{f}) f_{f l, z} = (b - Δ x_{f}) f_{r r, z} \end{array}

where b represents the length from the center of mass of robot’s torso to the hip joint. G is the weight of the torso, and $f_{f l, z}$ and $f_{r r, z}$ are the interaction forces between feet and the ground. $Δ x_{f}$ is the distance from the hip F/E to the foot position along X axis.

Then, taking equations (7), (8), and (9) into account, the torques on the joints can be calculated as

\{\begin{array}{l} τ_{f l, hip} = \frac{(b - Δ x_{f}) G Δ x_{f}}{2 b} \\ τ_{f l, knee} = \frac{G Δ x_{p} l_{t 1} sin (θ_{r 1} + θ_{r 2}) (b - Δ x_{f})}{2 b} \\ τ_{r r, hip} = \frac{(b + Δ x_{f}) G Δ x_{f}}{2 b} \\ τ_{r r, knee} = - \frac{G Δ x_{p} l_{t 1} sin (θ_{r 1} + θ_{r 2}) (b + Δ x_{f})}{2 b} \end{array}

Taking the front-left leg as an example, the load forces on the piston can be calculated by equation (10)

[\begin{array}{l} f_{f l, hip} \\ f_{f l, knee} \end{array}] = [\begin{array}{l} \frac{τ_{f l, hip}}{l_{τ 1}} \\ \frac{τ_{f l, knee}}{l_{τ 2}} \end{array}]

where $l_{τ 1}$ and $l_{τ 2}$ are the effective arms of the actuator and can be obtained by equations (1 ) to (5) shown in Appendix 1.

Load force acting on actuator in swing phase

Again taking the front-left leg as an example, based on the Lagrange method,²³ the dynamics model of the leg can be written as follows

τ - τ_{f l} = M (Θ) \ddot{Θ} + V (Θ, \dot{Θ}) + G (Θ)

where the torque $τ_{f l}$ acted on the joints caused by external disturbing forces is zero. $τ$ is the 2 $\times$ 1 matrix, [ $τ_{f l, h i p}; τ_{f l, k n e e}$ ]. The coefficient matrix $M (Θ)$ , $V (Θ, \dot{Θ})$ and $G (Θ)$ can be obtained by equations (6) to (9) shown in Appendix 1.

And the load forces acting on actuators also can be calculated by equation (11).

Taking into account equation (14) and the robot structure shown in Figure 2, beside the gravity force, in the swing phase the load force acted on the actuators still includes the inertial force caused by rotation of leg components, and both of them manifest as a force that opposes motion during leg uplift stage. However, the gravity force manifest as a force that provides motive during leg falling.

Robot energy consumption modeling

Owing to the rhythm of the robot’s locomotion, the energy consumption of the legs on the diagonal in a gait cycle can be only considered and used to reflect the energy efficiency of the robot. Then, taking equations (6) to (12) into account, the energy consumption model of the robot can be deduced as follows

E = E_{hip} + E_{knee}

with the energy consumption of hip and knee actuators being expressed as follows

\{\begin{array}{l} E_{hip} = & \int_{hip} | (f_{f l} + f_{f}) \times {\dot{x}}_{p} | d t \\ E_{knee} = & \int_{knee} | (f_{f l} + f_{f}) \times {\dot{x}}_{p} | d t \end{array}

where the velocity of the piston ${\dot{x}}_{p}$ can be obtained by equation (1) shown in Appendix 1.

Gait parameters analysis

For the hydraulic servo actuator used on the quadruped robot, the piston diameter and oil supply pressure (P_S ) are mainly determined by the weight and load capacity of the robot. The choice of the diameter of the piston rod also must consider the robustness of the structure. Once the structure is made certain, the parameters described above are also determined. Thus, from the energy conservation standpoint, the most effective way to reduce the energy consumption of the robot is to adopt the appropriate gait and foot trajectory with low energy consumption.

Foot trajectory of trotting gait

With larger adaptation range of the motion velocity and better energy efficiency, the dynamic trotting gait²⁴ is always an optimal choice for the motion of the quadruped robot. A gait cycle is composed of the swing phase and supporting phase, and the major controlling parameters in gait expressing equation are the step height, step length, and gait cycle.

A foot trajectory is planned with the goal of maintaining stabilization and continuation of the travelling speed of the robot, so it need to be representative and versatile. To assure the stability of torso’s traveling speed, the velocity of the foot trajectory along the direction travel should be formulated as constant during the entire gait period. Meanwhile, to acquire a better coordinated movement, the velocity and acceleration at the switching points of phase switching should be set to zero and of course include the highest point of the swing phase. Regarding the research on trajectory planning, many scholars have designed numerous functions such as curve-fitting,²⁵ cycloid functions,²⁶ and Bezier-curve functions.²⁷ Because of its better controllability of parameters and simple calculation, the curve-fitting method is the most commonly used.

For the foot trajectory planned by the curve-fitting method, the main controlled parameters include the step height, step length, and gait cycle, and step height has no contribution to the speed of the robot. On the basis of the above analysis and the study on the motion law of each actuators²⁸ in prior work, it can be concluded that the workable way to further change the energy consumption of SCalf-II is to control the velocity of leg uplifting and falling. To this end, the traditional curve-fitting method should be improved by adding another important controlled parameter. To meet this requirements, in the swing phase a piecewise continuous fitting curve is used in the z-axis and the equation can be written as follows

\{\begin{array}{l} f_{x, s w} = L (A_{1} t^{5} + B_{1} t^{4} + C_{1} t^{3} + D_{1} t^{2} + E_{1} t + F_{1}) \\ f_{x, s u} = A_{2} t + B_{2} \\ f_{z, s w 1} = H (A_{3} t^{5} + B_{3} t^{4} + C_{3} t^{3} + D_{3} t^{2} + E_{3} t + F_{3}) \\ f_{z, s w 2} = H (A_{4} t^{5} + B_{4} t^{4} + C_{4} t^{3} + D_{4} t^{2} + E_{4} t + F_{4}) \\ f_{z, s u} = 0 \end{array}

where H, L, and T represent step height, step length, and gait cycle, respectively. t_f is the time at which the step height is raised to the highest. $f_{x, s w}$ , $f_{x, s u}$ , $f_{z, s w 1}$ , $f_{z, s w 2}$ , and $f_{z, s u}$ are defined in $(0 < t \leq T / 2)$ , $(T / 2 < t \leq T)$ , $(0 < t \leq t_{f})$ , $(t_{f} < t \leq T / 2)$ , and $(T / 2 < t \leq T)$ , respectively. The letter A to F with different subscripts represent polynomial coefficients of the equations.

According to the description of stability and continuation of robot motion mentioned above, the constraint formulations are summarized as follows

\{\begin{array}{l} f_{x, s w} (0) = 0 & {\dot{f}}_{x, s w} (0) = A_{2} & {\ddot{f}}_{x, s w} (0) = 0 \\ f_{x, s w} (\frac{T}{2}) = 0 & {\dot{f}}_{x, s w} (\frac{T}{2}) = A_{2} & {\ddot{f}}_{x, s w} (\frac{T}{2}) = 0 \\ f_{x, s u} (\frac{T}{2}) = 0 & f_{x, s u} (T) = - \frac{L}{2} \\ f_{z, s w 1} (0) = 0 & {\dot{f}}_{z, s w 1} (0) = 0 & {\ddot{f}}_{z, s w 1} (0) = 0 \\ f_{z, s w 1} (t_{f}) = H & {\dot{f}}_{z, s w 1} (t_{f}) = 0 & {\ddot{f}}_{z, s w 1} (t_{f}) = 0 \\ f_{z, s w 2} (t_{f}) = H & {\dot{f}}_{z, s w 2} (t_{f}) = 0 & {\ddot{f}}_{z, s w 2} (t_{f}) = 0 \\ f_{z, s w 2} (\frac{T}{2}) = 0 & {\dot{f}}_{z, s w 2} (\frac{T}{2}) = 0 & {\ddot{f}}_{z, s w 2} (\frac{T}{2}) = 0 \end{array}

where formulas in the first and second lines are used to define the displacement, velocity, and acceleration of the foot endpoint along x axis in the swing phase with the time of 0 and $T / 2$ in one cycle gait. Formulas in the third line are used to define the displacement of the foot endpoint along x axis in the supporting phase with the time of $T / 2$ and T in one cycle gait. Similarly, formulas in the fourth to seventh lines are used to define corresponding variables along the z axis.

Simulation and analysis

Simulation setup

To obtain the variety of the energy consumption with different parameter combinations and find more ideal value range of the controlled parameters, two different simulation experiments were performed. The simulation settings are shown in Table 2. The computation time of each group is 10 s.

Table 2.

Simulation parameters.

Group	t_f (s)	Speed (m/s)	L (m)
1	$T / 4$	0.15, 0.25,	0.1
1	$T / 4$	0.35 and 0.5	0.1
2	$2 T / 20$ to $8 T / 20$ , interval of $T / 20$	0.25 and 0.35	0.1

Group 1 simulation results

Considering that two diagonal legs perform the same trajectory, the energy consumption of front-left and rear-right legs is only considered. The energy consumption of actuators at different parts of the legs is shown in Figure 5, parts of which are enlarged. The curves are drawn with the gait cycle T as the abscissa and the range of T between 0.1 s and 1 s is calculated. It can be seen from the results that as the gait cycle increases, the energy consumption of the actuators at the corresponding position of the front and rear legs is monotonically decreasing. Increase in gait cycle is always accompanied by decrease in the step length. Thus, the above data show that for the same speed, the model of low gait cycle and large step length consumes less power.

Figure 5.

Energy consumption of actuators in group 1.

Figure 6 shows the total energy consumed by the robot. It can be seen from the results that the energy consumption decreases sharply with increasing the gait cycle when T is less than 0.6 s, while the change rate remains roughly constantly from 0.6 to 1 s. Thus, for the given foot trajectory, the ideal gait cycle should be greater than 0.6 s and the specific parameter can be determined according to the movement requirement of the robot.

Figure 6.

Energy consumption of robot in group 1.

Taking the speed of 0.15 m $/$ s as an example, Figure 7 shows the velocity of actuators mounted on the front-left leg in a gait cycle. In the swing phase, the average speed of the piston has decreased with the increasing of the gait cycle, which means that the inertia force of the leg also keeps decreasing. Combined with the results shown in Figures 5 and 6, the method of decreasing energy consumption by reducing the movement inertia has good effect.

Figure 7.

Velocity of actuators in group 1.

Simulation results of group 2

In group 2, the foot trajectories with different t_f values are shown in Figure 8. The gait cycle is set to 0.8 s, and the simulation results are drawn with the step length as the abscissa. As can be seen from Figures 9 and 11, with the exception of the hip actuator mounted on the rear-right leg, the energy consumption of the remaining actuators keeps monotonically decreasing. This phenomenon can also be explained by the velocity curves of the actuators.

Figure 8.

Foot trajectories in group 2. A and B represent separation point and contact point between the ground and foot.

Figure 9.

Energy consumption of actuators in group 2.

Figure 10.

Velocity of actuators in group 2.

Figure 11.

Energy consumption of robot in group 2.

Figure 10 shows the velocity of the actuators with different t_f values when the speed of the robot is 0.25 m $/$ s, and the absolute values of the corresponding velocities (the dashed-dotted line) are also shown in the figure. As can be seen from the results, with the exception of the hip actuator mounted on the rear-right leg, the average velocity of the rest of actuators declines with the increasing of t_f , which means that the movement inertia of the leg also reduces accordingly. Although the increasing of t_f also leads to the time of the leg uplifting growing, this has a less significant on the energy consumption compared with the former. However, for the actuator mounted on the rear-right leg, the increasing of t_f has an apparent affect on the average velocity of leg uplifting, but results in the increasing of piston speed in the stage of leg falling. Thus for the hip actuators installed on different legs, the energy consumption curves exhibit different trend. As can be seen from Figure 11, the total energy consumption curves of the robot first rise and then fall with increasing t_f , and their maximum are 14910 and 17200 J at a t_f value of 0.12 s ( $3 T / 20$ ), respectively.

From the results of discussion, it is shown that when the gait cycle and the step length are determined, to further reduce the energy required during the locomotion of the robot, the process of leg lifting in the swing phase can be extended for as long as possible. Then a shorter time of leg lowering means that the foot may be in contact with the ground at a greater speed (especially in unstructured terrain), thereby increasing the impact and affecting the stability of the robot. Thus, the actual parameters of the foot trajectory still should be determined according to the movement requirement of the robot.

Experimental verification

Experimental setup

The power of SCalf-II is provided by a single cylinder petrol engine with a constant rotational speed of 10000 rpm. The flow supplied by the combination of the petrol engine and a variable displacement piston pump with a displacement of 4 ml $/$ r is up to 40 L $/$ min. The instantaneous flow supplied by the accumulator mounted on the high-pressure circuit is higher than 20 L $/$ min. Thus, SCalf-II can reach speeds of up to 1.3 m $/$ s.

The load forces acted on the actuators can be obtained by the force sensor mounted on the piston, and the velocity of the piston can be calculated by differential of the data collected by the displacement sensor.

The experiment was also divided into two groups and carried out sequentially with reference to the simulation setup process. Considering the stability of SCalf-II in the movement process, the experimental settings are detailed in Table 3. In group 2, five gait cycles are adopted: 0.4, 0.5, 0.6, 0.7, and 0.8 s. To ensure a high trajectory tracing accuracy, position control is adopted.

Table 3.

Simulation parameters.

Group	t_f (s)	Speed (m/s)	L(m)
1	$T / 4$	0.15 and 0.25	0.1
2	$2 T / 20$ to $8 T / 20$ , interval of $T / 20$	0.25	0.1

In the experiment, each test was repeated three times for a duration of 10 s each time. The average value of the three experiments is used for comparison and analysis.

Experimental results

Figures 12 and 13 show the experimental results in group 1. In each images, the simulation results are also drawn for comparison.

Figure 12.

Experimental results of actuators in group 1. Solid line and dotted line represent the experiment results (ex) and simulation results (si), respectively.

Figure 13.

Experimental results of robot in group 1. Solid line and dotted line represent the experiment results (ex) and simulation results (si), respectively.

As can be seen from the results, the energy consumption of SCalf-II obtained using the energy calculation model expressed in equation (14) is very close to that of the experimental results, which indicates that the model has high reliability for predicting the energy consumption of a hydraulic actuated quadruped robot. Owing to the limitation of the stability of SCalf-II, the tests under the remaining gait cycles were not carried out. However, from the comparison with simulation results, the law of energy consumption concluded in the section of the group 1 simulation results is still met. The experimental results are smaller than the simulated values with the same conditions. The reason for this is that the data of the displacement and velocity applied in the calculation are provided by the linear displacement sensor, which has a static error (less than 1 mm) with the theoretical value. Moreover, the mathematical modeling is an approximation of the realistic robot system, so the error is inevitable.

The experimental results of group 2 are shown in Figures 14 and 15. The predicted results of the mathematical model accord well with the observed data, and the law concluded in the section of group 2 simulation results is also met.

Figure 14.

Experimental results of actuators in group 2. ex: experiment; si: simulation.

Figure 15.

Experimental results of robot in group 2. ex: experiment; si: simulation.

The above experiments show once again that, for hydraulic actuated quadruped robots with a similar structure to SCalf-II, the larger the gait cycle adopted, the less energy is required by the robot with the trotting gait at a constant speed. When the proper gait cycle is determined, energy consumption can be further reduced by extending the time of leg lifting in each cycle.

Conclusions

In this work, based on energy consumption model of the hydraulic actuator, the energy consumption model of the quadruped robot, SCalf-II, is established firstly. Then, combining the analysis on load characteristics and of the actuators and the previous research, the foot trajectory characterized by using segmented cubic spline interpolation curve in the swing phase is proposed, which is to reduce the energy consumption of the robot through introducing the controlled parameters t_f to change the duration of leg raising and falling in one gait cycle. Eventually, the variation of energy consumption affected by the controlled parameters of gait cycle and t_f is obtained by simulation and the corresponding experiments for verification are also carried out. The results provide principle for the selection of controlled parameters for robot locomotion. For instance, the theoretical value of the gait cycle should be greater than 0.6 s, and for t_f the theoretical value should be greater than $T / 4$ . To obtain the further information to select the parameters, the stability of the robot must also be taken into account.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed the receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Nature Science Research Project of Anhui province (NO. KJ2020A0287), the National Natural Science Foundation of China [Grant No. 91748211, 61973135, 62072191, 91948201], the Opening Project of Shanghai Robot R & D and Transformation Functional Platform [Grant No. k2020468].

ORCID iD

Zisen Hua

Appendix 1

The velocity of piston can be obtained by differentiating equation (7), which yields

where $θ$ represents $(θ_{r 1} + θ_{r 2})$ .

The actuator’s length can be expressed as the function of the joint angle, and the expressions are defined as follow

where auxiliary angles $θ_{u}$ and $θ_{d}$ can be expressed as

respectively.

Then, the effective lever arm of the actuator is deduced as

where auxiliary angles $θ_{τ 1}$ and $θ_{τ 2}$ can be expressed as

respectively.

The 2 $\times$ 2 mass matrix $M (Θ)$ can be written as

where m_u and m_d represent the mass of thigh an calf of the robot, respectively. I_u and I_d are the moment of inertia with respect to its center of mass, respectively.

The 2 $\times$ 1 vector $τ_{f l}$ is expressed as

$V_{Θ, \dot{Θ}}$ is derived as

and $G (Θ)$ is written as the following equation

where g is the gravity acceleration.

References

Fahmi

Focchi

Radulescu

, et al. Stance: locomotion adaptation over soft terrain. IEEE Trans Robot 2020; 36(2): 443–457.

Rong

. Mechanism design and kinematics analysis of a hydraulically actuated quadruped robot SCalf. Jinan: Shandong University, 2013, p. 10.

Seok

Wang

Chuah

, et al. Design principles for highly efficient quadrupeds and implementation on the MIT Cheetah robot. In: 2013 IEEE international conference on robotics and automation, Manhattan: IEEE, 2013, pp. 3307–3312.

McGeer

. Passive dynamic walking. I J Robotic Res 1990; 9(2): 62–82.

Kuo

. Stabilization of lateral motion in passive dynamic walking. I J Robotic Res 1999; 18(9): 917–930.

Liu

. Iterative learning impedance control for rehabilitation robots driven by series elastic actuators. Automatica 2018; 90: 1–7.

Rampeltshammer

Keemink

Van Der Kooij

. An improved force controller with low and passive apparent impedance for series elastic actuators. IEEE ASME Trans Mechatron 2020; 25(3): 1220–1230.

Liu

Yuan

, et al. Design and analysis of spring parallel variable stiffness actuator based on antagonistic principle. Mech Mach Theory 2019; 140: 44–58.

Grosu

Rodriguez-Guerrero

Grosu

, et al. Design of smart modular variable stiffness actuators for robotic-assistive devices. IEEE ASME Trans Mechatron 2017; 22(4): 1777–1785.

10.

Hutter

Gehring

Lauber

, et al. ANYmal-toward legged robots for harsh environments. Adv Robot 2017; 31(17): 918–931.

11.

Hutter

Gehring

Jud

, et al. ANYmal-a highly mobile and dynamic quadrupedal robot. In: 2016 IEEE/RSJ international conference on intelligent robots and systems (IROS), Manhattan: IEEE, 2016, pp. 38–44.

12.

Yang

Zhou

, et al. Energy efficient foot trajectory of trot motion for hydraulic quadruped robot. Energies 2019; 12(13): 2514.

13.

Deng

Liu

Ding

, et al. Motion planning and simulation verification of a hydraulic hexapod robot based on reducing energy/flow consumption. J Mech Sci Technol 2015; 29(10): 4427–4436.

14.

Gao

Liu

Ding

, et al. Low impact force and energy consumption motion planning for hexapod robot with passive compliant ankles. J Intell Robot Syst 2019; 94(2): 349–370.

15.

Kim

Kwon

, et al. The energy minimization algorithm using foot rotation for hydraulic actuated quadruped walking robot with redundancy. In: ISR 2010 (41st International Symposium on Robotics) and ROBOTIK 2010 (6th German Conference on Robotics), VDE, 2010, pp. 1–6.

16.

Abuowda

Okhotnikov

Noroozi

, et al. A review of electrohydraulic independent metering technology. ISA Trans 2020; 98: 364–381.

17.

Shi

Zha

, et al. Force-controlled compensation scheme for P-Q valve-controlled asymmetric cylinder used on hydraulic quadruped robots. J Bionic Eng 2020; 17(6): 1139–1151.

18.

Barasuol

Villarreal-Magana

Sangiah

, et al. Highly-integrated hydraulic smart actuators and smart manifolds for high-bandwidth force control. Front Robot AI 2018; 5: 51.

19.

Ren

. System modeling and experiment research on asymmetrical cylinder controlled by asymmetrical valve. Mach Tool Hydra 2004; 8: 73–75.

20.

Manring

Fales

. Hydraulic control systems. Hoboken: John Wiley & Sons, 2019.

21.

Boaventura

Hydraulic compliance control of the quadruped robot HyQ, Doctor of Philosophy. PhD Thesis, Advanced Robotics Department, University of Genova, Genova, 2013.

22.

Hua

Rong

, et al. Active compliance control on the hydraulic quadruped robot with passive compliant servo actuator. IEEE Access 2019; 7: 163449–163460.

23.

Hendzel

Rykala

. Modelling of dynamics of a wheeled mobile robot with mecanum wheels with the use of Lagrange equations of the second kind. Int J Appl Mech Eng 2017; 22(1): 81–99.

24.

Chen

Gao

. Energy expenditure of trotting gait under different gait parameters. Chin J Mech Eng 2017; 30(4): 943–950.

25.

Yuan

Zhou

Luo

. Crawling gait planning based on foot trajectory optimization for quadruped robot. In: 2019 IEEE international conference on mechatronics and automation (ICMA), Manhattan: IEEE, 2019, pp. 1490–1495.

26.

Rahme

Abraham

Elwin

, et al. Dynamics and domain randomized gait modulation with Bezier curves for sim-to-real legged locomotion. arXiv preprint arXiv:2010.12070, 2020.

27.

Choi

Curry

Elkaim

. Path planning based on Bézier curve for autonomous ground vehicles. In: Advances in Electrical and Electronics Engineering-IAENG Special Edition of the World Congress on Engineering and Computer Science 2008, Manhattan: IEEE, 2008, pp. 158–166.

28.

Hua

Rong

, et al. Analysis and verification on energy consumption of the quadruped robot with passive compliant hydraulic servo actuator. Appl Sci 2020; 10(1): 340.