## 3 Energy-Saving Principle of HPCA  
### Motion Description of SCalf II  
SCalf II is the hydraulically actuated quadruped robot designed by Shandong University in 2012 for orientation in rough terrains [25]. As can be seen from Figure 2A, 12 degree-of-freedom (DOF) topology is adopted because it cannot only provide a sufficient range of motion but can reduce the complexity of mechanical structures and control algorithms. Each leg (Figure 2B) includes three rotating joints: two pitch joints used for flexion and extension (hip F/E and Knee F/E) at the knee and hip, and a roll joint used for adduction and abduction at the hip (hip A/A). Under normal conditions, hip F/E and knee F/E are the main contributors to foot trajectory implementation, and hip A/A only participates in movement during attitude adjustment of the robot torso. With better energy efficiency and stability, trotting [26] has become the most basic and most commonly used gait for quadruped robots, and a gait cycle can be decomposed to supporting and swing phases. The torso moves forward in the supporting phase, while the leg swings forward in the swing phase. To have a smooth transition from the supporting phase to the swing phase and eliminate the fluctuation of the torso, the velocity of the robot along the x-axis should be constant during the whole gait period, and the velocity along the z-axis should be set to zero at the starting point, the ending point, and the highest point of the swing phase. Abrupt change points should not appear on the velocity and acceleration curves throughout the cyclic gait. In this paper, the foot trajectory with duty  
Figure 1: Hydraulic servo actuator. (**A**) Actuator with original structure and corresponding hydraulic principle; (**B**) hydraulic servo actuator with passive compliance (HPCA) and corresponding hydraulic principle. factor \(\beta=0.5\) (the fraction of a cycle time during which the leg is in the supporting phase) described in previous work [1; 27] is used and can be expressed as:  
\[\left\{ \begin{array}{l}x_{fsw}=L(\frac{192t^{5}}{T^{5}}-\frac{240t^{4}}{T^{4}} +\frac{80t^{3}}{T^{3}}-\frac{t}{T})\\ x_{fsu}=-\frac{Lt}{T}+L\\ z_{fsw}=H(\frac{256t^{4}}{T^{4}}-\frac{256t^{3}}{T^{3}}+\frac{64t^{2}}{T^{2}}) \\ z_{fsu}=0,\end{array} \right. \tag{1}\]  
where subscripts \(fsw\) and \(fsu\) indicate the swing phase and supporting phase, respectively, and \(L\), \(H\), and \(T\) are, respectively, the length, height and cycle of the gait. Taking the left-front and right-rear legs as examples, Figure 3 shows the foot trajectory and the motion curves of the actuators (hip F/E and knee F/E) calculated by leg kinematics [25]. The three displacement curves shown in Figure 3C use the initial points (the starting points of each gait cycle) shown in Figure 3B as starting points. The distances from points \(O_{1}\) to \(O\) and \(O_{2}\) to \(O\) are both 0.15 m. As can be seen from the curves, each actuator performs the same motion process in the swing phase, retracting first and then extending, while in the supporting phase, there is a big difference between the actuator trajectories at different positions, monotonous retraction or monotonous elongation. Since the mass of the leg components is much lighter than that of the torso, the loads applied to the actuators in the swing phase are obviously smaller than that in the supporting phase. Unfortunately, the output force of the hydraulic actuator is generated by the pressure difference between the piston chamber and ring chamber, which is affected by the supply pressure. Thus, the inessential energy-loss is predominantly in the swing phase. Moreover, it can be seen from Figure 3B that, for the knee F/E, the direction of the torque exerted by the interaction force between the ground and foot on the rotating joint is not affected by the position of the initial points but not for the hip F/E. The same principle has always held by the direction of the actuator output forces.