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Quadruped robot has great potential to walk on rough terrain, in which static gait is preferred. A hierarchical structure based controlling algorithm is proposed in this paper, in which trajectory of robot center is searched, and then static gaits are generated along such trajectory. Firstly, cost map is constructed by computing terrain features under robot body and cost of selecting footholds at default positions, and then the trajectory of robot center in 2D space is searched using heuristic A

Legged robot offers great potentials in some complex and dangerous environment, because it has more flexibility and adaptability than that of wheeled counterparts. This potential has sparked a great deal of research on legged robot in recent years, especially on quadruped robot [

This paper is focused on designing hierarchical controlling system for quadruped. Research on controlling algorithm for quadruped robot has changed gradually to design static gait used on complex and dangerous terrain, such as excellent works [

The remainder of the paper is organized as follows. Section

The cost map of terrain under robot body is computed from the following 3 terrain features: terrain slope, step height, and terrain roughness. In order to estimate those features, as shown in Figure

2D projection of grid based elevation map.

Let the grid

Plane function is defined as (

For optimizing the parameters of the plane handily, the triangle

Let the intermediate variables be computed from

Slope of the terrain centered at the grid

The elevation of the grid

For each grid

The processing diagram for computing the step height.

In (

There are 4 features of the terrain beneath foothold:

Both the normal and curvature are estimated from 3D points inside the sphere with a radius, so the terrain features are related to the radius. In order to get terrain information with different scales, two radiuses are employed,

Now the cost function is defined as the weighted sum of feature values, which is represented formally as

Default foothold is defined as the projection point of shoulder of the quadruped robot; a triangle is made around the default foothold which lies at center of it. There are 4 triangles related to default footholds, respectively. Average cost for each default foothold is computed from all the grids in the triangle. Equation (

Similarly, let the center of the robot body lie in the grid

Differing from the case of using traditional A

The movement direction

Designing heuristic function is a key step to search optimal path using A

According to the theory of triangle inequality, the heuristic function here satisfies the constraint. The searched 2D path of robot body center includes both coordinates of the points and movement direction on the path.

In terms of energy savings and robot stability, standard order of switching flying foot is left front foot(LF)→ right back foot(RB)→ right front foot(RF)→ left back foot(LB). Figure

Nominal static gait pattern of one circle.

The star in Figure

In addition to selecting proper foothold, adjusting pose of the robot body is an important step to drive the quadruped robot to walk. The pose is generally determined by the parameters: height of robot body center, angle of yaw, angle of roll, and angle of pitch. The following part will briefly introduce how to get such parameters.

Yaw angle computation.

To sum up, there are 5 steps to compute the pose of robot body, given 3 supporting feet and the target of the flying foot.

3D point

Rotational transformation

Another rotational transformation

Rotational transformation of robot body is computed by

There are two cases that may produce collision between robot (body or leg) and terrain: one is flying foot from original point to target point, the other is adjusting pose of robot body.

Foot is planned to avoid the collision between leg and terrain; a simple method is employed to generate foot trajectory; foot is moving along rectangular box. Take right front foot as an example, which is shown in Figure

Foot Trajectory.

The robot is projected into 2D plane

Key points in the foot trajectory.

1 | 2 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

| 0.100 | 0.100 | 0.100 | 0.135 | 0.170 | 0.170 | 0.170 |

| -0.270 | -0.235 | -0.200 | -0.200 | -0.200 | -0.235 | -0.270 |

Before flying foot, gravity of robot body should be adjusted into the supporting triangle, which may produce collision between leg and terrain. Figure

Moving trajectory of the robot body.

There are 5 key poses of robot; if none of these poses produce collision, then it is safe to adjust pose of robot body when fixing standing feet. Similarly, after projecting the robot onto 2D plane

Key poses in the body adjusting trajectory.

1 | 2 | 3 | 4 | 5 | |
---|---|---|---|---|---|

| 0 | 0.025 | 0.05 | 0.05 | 0.05 |

| 0 | 0 | 0 | −7.5° | −15° |

State after taking next step of robot is defined from 4 elements: foothold, height of center of robot body, pitch angle, and roll angle. It is formally represented as

Now the problem is changed into how to select such 4 elements.

In (

Let the number of states for next step be

Set of

According to the order in set

Terrain complexity has a great influence on the stability of robot, so foothold should be selected with the lowest cost as much as possible. Comparing to translation, the rotation has greater impact on the stability of robot, so the height parameter is firstly adjusted. According to the model of quadruped robot, it is relatively safe to adjust pitch angle firstly.

When the quadruped robot explores one step forwardly, there are

Let

Robot state

No collision will occur when the robot is moving from state

If

The searching process with recursive backtracking is represented by a recursive tree shown in Figure

Tree-style structure for recursively searching series of static gaits.

None of the searching paths on left and below of green line can reach the 4th step, because some switches of robot states would make collision. States on right and above green line have not been searched yet. In this searching process, only 5 states in the green line are restored. All of the states below the green line may be constructed in the worst case, but if there is no state transformation from

The following heuristic rules can be used to prune some unnecessary branches.

If the kinematic constraint can not be satisfied for current height value, then the larger height value needs not be considered.

If collision happens between rear leg of robot and terrain for current pitch angle, then the smaller pitch angle needs not be considered.

If collision happens between left leg of robot and terrain for current roll angle, then the larger roll angle needs not be considered.

The more steps the robot searches forwardly, the more optimized the searched series of static gaits.

The model of the quadruped robot in this paper is simplified as long as there are three degrees in one leg. A simple model with DH parameters is shown in Figure

DH parameters based model of the quadruped robot.

In Figure

Variable DH parameters of the model above are shown in Tables

The DH parameters for the left legs.

LF | LB | |||||||
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

1 | | | | | | | | |

2 | | | | | | | | |

3 | | | | | | | | |

4 | | | | | | | | |

The DH parameters for the right legs.

RF | RB | |||||||
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

1 | | | | | | | | |

2 | | | | | | | | |

3 | | | | | | | | |

4 | | | | | | | | |

Range of the joint angle.

| | | |

| |||

| | | |

Constant parameters are shown in Table

Constant parameters of the model.

| | | | |
---|---|---|---|---|

0.15m | 0.07m | 0.02m | 0.09m | 0.1m |

In order to validate the effectiveness of the algorithm for planning standard static gait, a simple simulation is carried out. The quadruped robot is driven along some straight line, and the footholds are divided into two categories: valid foothold at flat terrain and invalid foothold at terrain with edge.

Structured terrain model is generated in OpenRAVE [

Parameters for generating candidate body pose.

| | | | | | | | |
---|---|---|---|---|---|---|---|---|

| | | | | | | | |

Standard static gaits for two circles are shown in Figure

Series of standard static gaits for two circles on structured terrain.

The strategy with recursive backtracking is used to search gait series for the quadruped robot, recursive depth which is the gait circles for searching forwardly once and has great impact to the searching result, and the depth is 2 for the searching process shown in Figure

Two cases are shown in Figure

Different searching results by looking one and four steps ahead, respectively.

Looking one step ahead (recursive depth=1)

Looking four steps ahead (recursive depth=4)

Simulation in Section

Another terrain model is generated in simulation environment, whose size is

Geometric features of 3D terrain under the robot body.

2.5D elevation map based on grid

Roughness

Slope

Step height

The radius for computing terrain features (roughness, slope, and step height) is

Cost map is constructed from three feature maps using (

Parameters for computing cost value.

| | | | | |
---|---|---|---|---|---|

| | | | | |

The larger some feature weight is, the greater its contribution to the cost function is. Taking slope feature as an example, if

The computed cost map is shown in Figure

Cost map of the 3D terrain.

The cost map in Figure

Trajectory of 2D body center.

Coordinate of the start point is

The 2D path of body center with parameter

Motion trajectory of the quadruped.

In this paper, a hierarchical algorithm for planning quadruped robot walking on rough terrain is proposed. There are two steps to execute the algorithm; the first step is searching the trajectory of robot center using heuristic algorithm A

The datasets generated during and/or analysed during the current study are available from the corresponding author upon reasonable request.

The authors declare that they have no conflicts of interest.

This work has been supported by Fundamental Research Funds for the Central Universities (Grant no. 2572017BB12); the State Key Laboratory of Robotics and System (HIT) Open Fund (Grant no. SKLRS-2017-KF-06); the National Natural Science Foundation of China (Grant no. 16051575097); the National Science Foundation for Postdoctoral Scientists of China (Grant nos. 2015M571381 and 2017T100217); and the Natural Science Foundation of Heilongjiang Province (Grant no. JJ2019LH2071).