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The paper focuses on finding a dual number solution to position and attitude coordinated control for a multi-rigid-body system. First, a relative motion coupling model of a multi-rigid-body system is established under the framework of dual number and dual quaternion theory. Then, a coordinated control strategy that uses graph theory based on a derived new type of dual quaternion is proposed to simultaneously control the position and attitude of a multi-rigid-body system. Finally, the resulting Lyapunov function is proved to be almost globally asymptotically stable. The simulation results show that the proposed algorithm not only achieves unified control of position and attitude but also exhibits better tracking control performance.

Multiagent systems have become an important topic of research in recent years, with an increasing number of scholars researching them. The object of this research is a set of self-governing agent groups in which multiple agents exchange information such as goals, strategies, and plans. Compared with a single subsystem, a multiagent system is more robust. A multibody system is a special type of multiagent system that not only has the advantages of a multiagent system but is also widely used in the fields of unmanned aerial vehicle (UAV) cooperation tracking, satellite formation flights, intelligent transportation, distributed sensor networks, and swarm robot systems, which are all of increasing interest to researchers. Formation control, as an important research direction of the coordinated control of multi-rigid-body systems, has become a pressing issue in the field of control research in recent years [

As a basic problem of the coordinated control of a multiagent system, the problem of consistency had been a matter of concern for an increasing number of researchers [

To offer a solution for the outlined problems, this paper proposes a distributed coordinated control algorithm of multi-rigid-body systems based on dual number theory and graph theory. The rest of the paper is organized as follows. Section

The concept of dual number was first proposed by Clifford as follows [

A dual vector is a special type of dual number that has real and dual parts that are vectors. For example, the dual vector

While the torque acting on a rigid body depends on the choice of a reference point, the force does not depend on a reference point.

If the angular velocity of a rigid body at a certain time is

A dual quaternion can be described as a dual number whose real and dual parts are both unit quaternions, as defined by

In (

Quaternions and dual quaternions have similar properties:

From the basic properties of the dual quaternions, the following conclusion can be drawn:

If

Multi-rigid-body systems are essentially a class of multiagent systems, and each rigid-body is considered an individual component in a multiagent system. Each individual component acquires its own control laws through interaction with its neighbours and thereby realizes overall control. In this paper, the topology of the multi-rigid-body system is modelled by graph theory.

Directed and undirected graphs are the most suitable mathematical objects for modelling the information topology of communications and perceptions between multiple rigid bodies. The directed graph is denoted by

The interaction between individual agents is realized through the following protocol [

The rigid-body kinematics are given by

The relative kinematics equation of the multi-rigid-body system, written in terms of dual numbers, is

For the

The dual velocity can be written as

After taking the derivatives of formula (

The position and attitude coupling dynamic model of the

where the dual matrix of the inertia of the rigid body is given in [

In this formula,

The following expression can be obtained from formula (

Substituting formula (

Formula (

Hence, we complete the proof.

According to the Euler theorem, the attitude of one coordinate system relative to the other can be obtained by rotating the angle

The logarithmic mapping of a unit dual quaternion can be written as

Using the quaternion logarithm and the dual quaternion Lie group properties [

When

The real and dual parts of the logarithmic dual quaternion correspond exactly to position and attitude. Therefore, we define a new dual quaternion as follows:

The design of a multi-rigid-body system coordination control law mainly solves the pose coordination problem. As long as all state variables are bounded, it is ensured that each rigid body tracks the desired pose. Suppose that two rigid bodies are represented by

For the relative motion coupling model based on the dual quaternion (

The ideal position and attitude coordinated control law is designed as follows:

We present the stability conclusions of the proposed multi-rigid-body system coordinated control algorithm in the form of a theorem.

Assume that the control law is given by formula (

Consider a Lyapunov function candidate:

Since

Then

Then

Substituting

Hence, we complete the proof.

Similarly, by taking the derivative of formula (

Then, formula (

Substituting the value of

To make

Formula (

According to [

Hence, we complete the proof.

Using the presented theoretical analysis, we apply the proposed algorithm to the coordinated control of formation rigid bodies. Taking the pose-coordinated control of six rigid bodies as an example, we verify the effectiveness of the control law. Each rigid body is regarded as a node, and the undirected graph is used to model the information flow between the rigid bodies. The interactions between the rigid bodies are shown in Figure

Undirected graph of rigid bodies.

The initial attitude and position of the rigid bodies are listed in Table

Initial attitude and position of the rigid bodies.

Rigid body | Initial attitude ( | Initial position (m) |
---|---|---|

1 | | |

2 | | |

3 | | |

4 | | |

5 | | |

6 | | |

Figures

Attitude tracking curve.

Angular velocity tracking curve.

Position tracking curve.

Velocity tracking curve.

To further demonstrate the effectiveness of the algorithm, we simulate the attitude, angular velocity, position, and velocity of each rigid body. The simulation results are shown in Figures

Attitude curves of rigid bodies 1, 3, and 5.

Angular velocity curves of rigid bodies 1, 3, and 5.

Position curves of rigid bodies 1, 3, and 5.

Velocity curves of rigid bodies 1, 3, and 5.

Figures

A new type of dual quaternion is investigated in this study. From the derived form of dual quaternion together with graph theory, we proposed a novel coordinated control law to simultaneously control position and attitude. Then, the Lyapunov function is used to prove that the model is almost globally asymptotically stable in terms of the control law. The simulation results indicate that the controller that considers the coupling effect performs better than the traditional controller that considers the relative position and attitude separately. Additionally, the proposed algorithm in this paper is simpler and can simultaneously control both relative position and attitude.

The data used to support the findings of this study are included within the article.

The author declares no conflicts of interest.

This work was supported by the Supporting Fund for Teachers’ Research of Jining Medical University (No. JY2017KJ052).