Dynamics modeling and control problem of a two-link manipulator mounted on a spacecraft (so-called carrier) freely flying around a space target on earth’s circular orbit is studied in the paper. The influence of the carrier’s relative movement on its manipulator is considered in dynamics modeling; nevertheless, that of the manipulator on its carrier is neglected with the assumption that the mass and inertia moment of the manipulator is far less than that of the carrier. Meanwhile, we suppose that the attitude control system of the carrier guarantees its side on which the manipulator is mounted points accurately always the space target during approaching operation. The ideal constraint forces can be out of consideration in dynamics modeling as Kane method is used. The path functions of the manipulator’s end-effector approaching the space target as well as the manipulator’s joints control torque functions are programmed to meet the soft touch requirement that the end-effector’s relative velocity to the space target is zero at touch moment. Numerical simulation validation is conducted finally.

Tethered space robot and space manipulator are two means for on-orbit operation such as satellite maintenance, refueling, retrieval, or space debris removal. The main advantage of the former over the latter is its long operation distance (from tens of meters to dozens of kilometers), while the disadvantage of the former is obvious such as inherent instability in retrieval procedure and huge difficulties in control design for the procedure [

Generally, the system dynamics of space manipulator is more complex than that on the ground [

Trajectory optimization of manipulator with redundant degree of freedom for capturing space target is researched in [

The closed loop control law is designed for manipulator when approaching and capturing space target in [

For tethered space robot dynamics and control problem, the involved orbit dynamics is taken into account adequately [

A space target is supposed to fly on a circular orbit around the earth. An operation spacecraft (called carrier uniformly afterwards in the paper) flies freely around the target governed by the well-known homogenous C-W equations [

Illustration of space manipulator capturing an on-orbit target around the earth.

Some coordinate systems and the corresponding unit orthogonal basis are defined as follows:

Earth-centered inertial coordinate system

Space target fixed coordinate system

Carrier body-fixed coordinate system

Link I fixed coordinate system

Link II fixed coordinate system

In accordance with the above definition, the vector groups

The following equations can be easily obtained accordingly:

What should be pointed out is that a vector keeps its value when being moved in parallel. So unit orthogonal basis of vector groups

In Figure

The following equations can be obtained easily:

We know the origin of coordinate system

The position vector of the origin of coordinate system

The time derivative of (

In the same way, the time derivative of (

The rotation angular velocity and acceleration of coordinate system

The position vector of the center of mass of link I relative to the origin of coordinate system

Substituting (

Taking the first and second order time relative derivative of (

The rotation angular velocity and acceleration of link I relative to coordinate system

The all forces upon link I (including the inertia forces) except for the constraint ones will be analyzed below.

The inertia force upon link I induced by linear acceleration

The centrifugal force induced by rotation angular velocity

The inertia force induced by rotation angular acceleration

The Coriolis force induced commonly by

The relative inertia force induced by relative acceleration

All the moments of force (including the inertia forces) with respect to link I’s center of mass will be analyzed below.

It can be proved that the moments of

The moment of inertia force induced by relative angular acceleration

Finally, the torque exerted on link I by the motors mounted on joints I and II is

The resultant force and moment exerted on link I are, respectively,

Now we study link II.

The position vector of the center of mass of link II relative to the origin of coordinate system

Taking the first and second order time relative derivative of (

The relative rotation angular velocity and acceleration of link II to coordinate system

The inertia force upon link II induced by linear acceleration

The centrifugal inertia force induced by rotation angular velocity

The inertia force induced by rotation angular acceleration

The Coriolis force induced commonly by

The relative inertia force induced by relative acceleration

All the moments of force (including the inertia forces) with respect to link II’s center of mass will be analyzed below.

For the same reason as for link I aforementioned, among the moments of the inertia forces induced by the translational motion or rotation of coordinate system

In addition, the moment of inertia force induced by relative angular acceleration

Finally, the torque exerted on link II by the motor mounted on joint II is

The resultant force and moment exerted on link II are, respectively,

The essence of Kane method is to project all the active and inertia forces on the directions of the generalized coordinate curves, converting them into generalized forces, and then to write the equilibrium equations of the generalized forces. So it is also called generalized D’Alembert’s principle. For the multibody system of two-link manipulator researched in the paper, rotation angles

The equilibrium equations of the generalized forces are

Substituting (

The related information of space target such as its position relative to the carrier or the end-effector can be obtained easily by the camera mounted on the carrier-manipulator system [

So

The parameters

The command of the position of the end-effector under coordinate system

The actual position coordinates of the end-effector are denoted by

The two-order time derivative of (

The position vector of the end-effector relative to the space target is

Substituting (

Taking the second order derivative of (

Substituting (

It can be noticed that (

The inlet parameters for numerical simulation are shown in Table

Inlet parameters for numerical simulation.

Inlet parameters | Values |
---|---|

| |

| |

| |

| |

^{2}) | |

^{2}) | |

| |

| |

| |

| |

| |

| |

| |

The results of numerical simulation are shown in Figures

The relative trajectory of the carrier to the space target.

The relative trajectory of the end-effector to the space target and the corresponding configuration variation of the manipulator.

The distance and phase angle of the carrier relative to the space target.

The command versus the real value of the end-effector position coordinate.

The distance and relative velocity of the end-effector to the space target.

The rotation angles, angular velocities of the links I and II, and the corresponding driving torques of the motors.

The results of the numerical simulation indicate that the actual position of the end-effector closely tracks the command; in fact, the response curves almost overlap the command curves exactly. At the final time

The simulation neglecting the inertia forces exerted on the manipulators induced by the carrier body-fixed coordinate system has been conducted as well. The deviations of the simulation results neglecting the inertia forces and moments from that considering the inertia forces and moments mentioned above have been shown in Figures

The deviation of driving torques neglecting the inertia forces and moments from that considering them.

The deviation of link’s rotation angles neglecting the inertia forces and moments from that considering them.

The deviation of end-effector’s position coordinates and its distance relative to the space target neglecting the inertia forces and moments from that considering them.

In Figures

The two-link manipulator’s dynamics model during its end-effector approaching of a space target is derived using Kane method under the situation that the carrier is flying freely around the space target, governed by the C-W equations. The relative orbit dynamics of the carrier to the space target is considered in system dynamics modeling. The path functions of the end-effector of the manipulator as well as the corresponding driving torques functions of the joints are programmed in accordance with the requirement of soft touch between the end-effector and the target for the purpose of avoiding the damage to them by touch impact. The results of numerical simulation show the effectiveness of the control law proposed in the paper. The issues left for the future study are as follows: (

As shown in Figure

Illustration of a rigid body in noninertial coordinate system.

The inertia force induced by acceleration

The moment of the infinitesimal force with respect to point

The corresponding total moment with respect to point

Substituting (

That is,

The definition of the center of mass of rigid body is used in the above derivation.

The inertia centrifugal force induced by angular velocity

The moment of the infinitesimal force with respect to point

The corresponding total moment with respect to point

Substituting (

That is,

The Coriolis force induced commonly by angular velocity

The moment of the infinitesimal force with respect to point

The corresponding total moment with respect to point

Substituting (

That is,

In summary, we conclude that in two-dimensional situations, namely, planar situations, the inertia force moments exerted on a rigid body with respect to its center of mass induced by the coordinate system’s linear acceleration, centripetal acceleration, and the rigid body’s Coriolis acceleration are all zero.

The author declares that there is no conflict of interests regarding the publication of this paper.

This work is supported by the Fundamental Research Funds for the Central Universities (Grant no. NS2016082).