**2. Robot Modelling**

The quadruped robot SCalf was build by the Center for Robotics of Shandong University, as shown in Figure 1. SCalf is a hydraulic actuated robot. The length and width of the robot are 1.4 m and 0.75 m respectively, and the overall weight is 200 kg. The robot consists of a trunk and four legs. The structure of the legs must be designed considering the fact that an improper design can limit the reachable space of the robot legs when operating in the normal quadruped gaits [11]. Therefore, the leg of SCalf is designed as a 3 DOFs (degree of freedom) structure to make the foot endpoint move in a three-dimension workspace with the hip joint as its origin. The three joints of the leg are defined as the rolling hip joint, pitching hip joint, and pitching knee joint. To make the robot easy to control, the SCalf is built symmetrically. The backward/forward configuration is used to improve the gradeability and load capacity.

**Figure 1.** The overall structure of the SCalf robot.

A variable displacement piston pump is connect to the gasoline engine to provide power for the whole robot system. The onboard hydraulic system and the control system are integrated inside the trunk of the robot. The onboard hydraulic system can distribute a hydraulic flow to the joint actuators and can control the motions of the robot. The robot can be used to transport loads on rough terrains like jungle or mountain, and its load capacity is 200 kg.

To calculate the energy consumption of the robot, the following preconditions are made:

(a) Only the motions in the sagittal plane are studied. The robot is walking on a flat surface with the trot gait.

(b) When walking on a flat surface, the height change of the robot trunk's center of mass (COM) is ignored.

(c) The trunk of the robot can be regarded as a cuboid, and its COM is assumed to be at the geometric center of the body

## *2.1. The Kinematics Modelling*

The legs of the robot have the same structure and mechanical parameters. Here, the right front (RF) leg is used as an example. The single leg model is shown in Figure 2. According to precondition (a), the motions of the rolling hip joint are not considered and the rolling hip joint position is set as 0.

**Figure 2.** A simplified model of the right front (RF) leg.

In Figure 2, the RF leg can be divided into the hip, thigh, and shank parts. Since the motions of the rolling hip joint are neglected, the hip part can be regarded as the base. The origin of the coordinate frame {*Oh*} is set at the rolling hip joint. The *z*-axis direction is vertical to the ground facing upward. The *x*-axis is pointing to the motion direction of the robot.

The mechanical parameters in Figure 2 are introduced as follows. The lengths of the hip, thigh, and shank parts are *l*0, *l*1, and *l*<sup>2</sup> respectively, where *l*<sup>0</sup> = 45 mm. *m*<sup>1</sup> and *m*<sup>2</sup> are the masses of the thigh and the shank parts, respectively. The joint positions of the pitching hip and pitching knee joints are denoted as *θ*<sup>1</sup> and *θ*2, respectively. The COMs of the thigh and shank parts are defined by *lm*1, *lm*2, *ε*1, and *ε*2. When the thigh and shank parts rotate around their own joint shafts, the moments of inertia are *I*<sup>1</sup> and *I*2, respectively. The values of these mechanical parameters are listed in Table 1 [12].


**Table 1.** The parameters of the thigh and shank parts.

Based on the mechanical parameters, the location of the foot endpoint in coordinate frame{*Oh*} can be written as follows, where *θ*<sup>12</sup> = *θ*<sup>1</sup> + *θ*<sup>2</sup>

$$\begin{cases} x = -l\_2 \sin \theta\_{12} - l\_1 \sin \theta\_1 \\ z = -l\_2 \cos \theta\_{12} - l\_1 \cos \theta\_1 - l\_0 \end{cases} \tag{1}$$

By solving Equation (1), the joint positions of the RF leg can be obtained as

$$\begin{cases} \theta\_1 = -\phi - \arctan(\frac{\mathbf{x}}{-z - l\_0}) \\\\ \theta\_2 = \pi - \arccos(\frac{l\_1^2 + l\_2^2 - \xi^2}{2l\_1l\_2}) \end{cases} \tag{2}$$

where

$$\begin{cases} \xi = \sqrt{(-z - l\_0)^2 + \mathbf{x}^2} \\ \phi = \arccos(\frac{l\_1^2 + \xi^2 - l\_2^2}{2l\_1 \xi}) \end{cases} \tag{3}$$

The Jacobian matrix of the single leg model can also be calculated by Equation (1).

$$J = \begin{bmatrix} -l\_1 \cos \theta\_1 - l\_2 \cos \theta\_{12} & -l\_2 \cos \theta\_{12} \\ -l\_1 \sin \theta\_1 + l\_2 \sin \theta\_{12} & l\_2 \sin \theta\_{12} \end{bmatrix} \tag{4}$$
