In this study, a hyper-redundant manipulator was designed for detection and searching in narrow spaces for aerospace and earthquake rescue applications. A forward kinematics equation for the hyper-redundant manipulator was derived using the homogeneous coordinate transformation method. Based on the modal function backbone curve method and the known path, an improved modal method for the backbone curves was proposed. First, the configuration of the backbone curve for the hyper-redundant manipulator was divided into two parts: a mode function curve segment of the mode function and a known path segment. By changing the discrete points along the known path, the backbone curve for the manipulator when it reached a specified path point was dynamically obtained, and then the joint positions of the manipulator were fitted to the main curve by dichotomy. Combined with engineering examples, simulation experiments were performed using the new algorithm to extract mathematical models for external narrow space environments. The experimental results showed that when using the new algorithm, the hyper-redundant manipulator could complete the tasks of passing through curved pipes and moving into narrow workspaces. The effectiveness of the algorithm was also proven by these experiments.

Hyper-redundant manipulators have tremendous application potential for complex working environments and narrow spaces, and currently play important roles in aerospace engineering [

Path-following motion [

In this study, a hyper-redundant manipulator was designed, and its positive kinematics equation was derived by a homogeneous matrix coordinate transformation. Focusing on the problems that path-following motion is only suitable for a hyper-redundant manipulator with a moving base and that the mode function backbone curve is not suitable for moving along the tangent direction of a known path, a cost-effective method is proposed to achieve motion planning for a hyper-redundant manipulator based on a combination of the backbone curve and the path-following concept. The algorithm caused the position of the end effector to move along the tangent direction of a known path, and the joint position of the entering path was always close to the known path. This algorithm allows hyper-redundant manipulators to complete the motion in pipelines or other narrow workspaces.

A hyper-redundant manipulator is shown in

The structural and joint angle parameters of the hyper-redundant manipulator are shown in _{i}_{i}_{0} direction in the coordinate system of the base was defined as the direction from the center of the first universal joint to the position of the end effector when all the joints had angles of _{0} direction in the coordinate system of the base was defined as the vertical upward direction, and the _{0} direction in the coordinate system of the base was determined by the right-hand rule. A homogeneous coordinate transformation was used to obtain the positive kinematics model [

Joint number ( |
Yaw | Pitch | _{i} |
---|---|---|---|

1 | _{1} |
_{1} |
179 |

2 | _{2} |
_{2} |
179 |

3 | _{3} |
_{3} |
179 |

4 | _{4} |
_{4} |
179 |

5 | _{5} |
_{5} |
179 |

6 | _{6} |
_{6} |
179 |

7 | _{7} |
_{7} |
179 |

8 | _{8} |
_{8} |
179 |

9 | _{9} |
_{9} |
179 |

10 | _{10} |
_{10} |
130 |

11 | _{11} |
_{11} |
130 |

12 | _{12} |
_{12} |
136 |

The homogeneous transformation matrix used during this study is presented as

The coordinate transformation process from the basic

The homogeneous transformation matrix from the basic

The transformation matrix of the ^{th} joint is expressed by

The kinematics equation for the hyper-redundant manipulator can then be expressed by

When the joint angles are inserted into

The geometric method, based on the backbone curve, was used to solve the inverse kinematics of the hyper-redundant manipulator in this study. The backbone curve of the hyper-redundant manipulator had to be determined first. Then the positions of the hyper-redundant manipulator joints must be fitted to the backbone curve. Finally, the geometric method was used to solve for the joint angles.

The modal method uses differential geometry to solve the inverse kinematics of hyper-redundant manipulators. The backbone curve of the mode function is shown in

The mode function of the backbone curve [

In

In

When solving for the direction vector,

In _{m}^{th} iteration, and _{m}^{th} iteration into

The motion-planning algorithm for the hyper-redundant manipulator based on path-following and the backbone curve is primarily useful for passing the end effector through curved pipes and narrow spaces.

As shown in

In

Five primary steps were used to solve for the set of backbone curves in the motion process.

^{th} step, the backbone curve was composed of the new mode function curve segment,

In

^{th} step.

As shown in

^{th} point were found, and this point was defined as point

_{1} as the radius. Circle

When a hyper-redundant manipulator is fitted, the length of the backbone curve must be appropriately greater than the length of the manipulator if the backbone curvature is large.

The universal joint angles for the hyper-redundant manipulator were solved using the closed vector method. In

The process of solving for the manipulator's joint angles was divided into three primary steps [

_{2}, could be expressed by

In

As shown in

The origin of the mode function curve was at ^{th} step is shown in

When solving for the backbone curve at each step, the hyper-redundant manipulator and the backbone curve were fitted by the algorithm described in Section 3.3, and the results are shown in ^{th} and 12^{th} universal joints were located on the known path and moved along the direction tangent to the path. ^{th}–12^{th} universal joints were close to the known path to achieve the expected motion. The inverse solutions are shown in

The red and blue curves in

As shown in

Using the new method proposed in this study, a series of results were generated. A backbone curve was obtained and is shown in

^{th} and 12^{th} universal joints were located in one cabin, and the 9^{th} and 10^{th} universal joints were located in another cabin.

Using the homogeneous coordinate transformation method to derive the positive kinematics equation for a hyper-redundant manipulator can reduce the number of coordinate systems, simplify the derivation process for the transformation matrix, and save calculation time. An improved modal backbone curve method was proposed in this study. First, with changes in the discrete points along the known path, the backbone curves that the hyper-redundant manipulator used to reach these points were dynamically obtained. Then, the joints of the hyper-redundant manipulator were fitted to the modal backbone curves. Finally, the inverse kinematics of the hyper-redundant manipulator were solved based on the spatial geometry method. This method solved the motion-planning problem of an industrial hyper-redundant manipulator entering a known narrow environment.

Engineering application experiments verified the hyper-redundant manipulator's ability to move through curved pipes and narrow workspace areas. The effectiveness of the new algorithm was also proven by these experiments.