With the rapid development in the aerospace industry field, space robots in free floating state attracts the attention of many countries, and the attitude transformation is an important research direction. Its realization methods mainly include reaction jet devices and reaction wheels at present. The reaction jet works by carrying the propellant. The propellant in the space is not renewable and has limited carrying capacity, so the life of the space robot will be obviously shorten by heavy use of spray. The inertia parameters of the reaction wheel are much smaller than those of the space robot, momentum saturation easily occurred in the attitude transformation, which leads to the delay of the target task. The reaction wheel has the quality that can not be ignored, that will greatly increase the cost of launching and maintaining the space equipment[1-4]. The phenomenon that a free-falling cat can rotate in the air has been studied since the end of the 19th century[5-6], which has attracted widespread attention of academics for exploring the rotation movement of a space robot under the condition of weightlessness. There has been many relevant analysis and research results since then. The bending theory brought up by Kane et al. is the most persuasive one[7]. Symmetric double rigid bodies analysis model was established based on the bending theory[8-10], the analysis results of the model explain the phenomenon very well.

In the joint space,the adjustment of the joint angle of the manipulator can affect the attitude of the base, and requirements of saving energy and reducing cost are satisfied. However, when the base attitude in this method satisfies the target, the pose of each link is not consistent with the initial state of the body[11-13]. Closed motion can ensure the consistency of attitude, but the efficiency of closed motion for attitude adjustment is relatively low. Even if the overall attitude transformation range is small, multiple closed motions are needed to achieve the target pose. The combination joint created by the bionic joint and closed motion can solve the problem in the overall attitude transformation, and deal with the situation that the link’s attitude is not consistent with the initial state. When there is an asymmetric case in the bionic joint, a certain disturbance is generated for the attitude transformation, so the correlation analysis is done in the attitude transformation, and the method of reducing interference is proposed.

1 Modeling

1.1 Bionic mechanism

The cat in the falling process bends at first, then the upper body does a similar cone movement relatively to the lower body (but the bodies do not twist relatively), so as to achieve the rotation. For the theory of bending, the double rigid bodies model proposed by Yanzhu Liu and others can well explain this phenomenon[14-15]. The modeling process is shown in Fig.1.The detailed analysis process can be checked in our previous articles referring to Refs. [16-17].

Fig.1 Modeling process

1.2 FFSR modeling

The free falling cat model is established in weightlessness conditions, and the overall attitude transformation of mechanism can be realized by a model with double rigid bodies. Therefore, the model can be applied to the attitude transformation of a free floating space robot (FFSR).

Suppose the space robot is composed of n+1 rigid bodies connected in series by n rotating joints. The number of the robot base (body) is set as link 0, and then other links are numbered in turn. Joint i is between link i-1 and between link i. The general model of the space robot is shown in Fig. 2, in which the coordinate system of link n consolidated at the end of the link n-1[18-21].

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Fig.2 General model of the space robot

For the convenience of calculation and analysis, the origin of the inertial coordinate system is located at the center of mass system. It is easy to get linear velocity and angular velocity at mass center of each link as follows:

First of all, two movements need to be defined:

(1)

(2)

Put Eqs.(1),(2) into a matrix form, we have

(3)

The movement of the link i is related to the position, attitude and speed of the base and to the rotation of the manipulator joints. The kinematics of link i can be derived from the kinematics of link 0 and joints angle movement.

Ignore the gravity and external force (torque), and assuming that the initial linear and angular momentum of the system are O, v0 and ω0 can be calculated according to the law of linear and angular momentum conservation.

(4)

The studied specific attitude transformation belongs to the overall attitude transformation, and the base attitude is represented as the expression of the overall attitude transformation. Eq.(4) can be expressed as follows:

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(5)

Put Eq.(5) into Eq.(3), the linear and angular velocities of other links can be obtained:

(6)

Theoretically, numerical integration of Eq.(6) leads to a position level solution. And then the numerical solution of the speed level of each link is obtained by an iterative method.

1.3 Closed motion

Firstly, the attitude transformation in the plane is analyzed, Supposing there is a planar mechanism with two degrees of freedom (TDF), as shown in Fig. 3.

Fig.3 Planar mechanism with TDF

The overall attitude changes can be found given a certain law of motion between joint 1 and joint 2. The joint 1 rotates α angle first, then the joint 2 rotates β angle, then the joint 1 recovers, finally the joint 2 recovers[22-23]. Repeat these movements for convenient observation, and then get the body’s posture as shown in Fig.4.

Fig.4 Attitude change of planar mechanism with TDF

③ The X3 rotation is achieved again by joints 2 and 4.

The simulation is represented by the body axis X-Z-X Euler angles, as shown in Fig.11.

2 Combination Model and It’s Closed Motion

Can the double rigid bodies joint model be combined with closed motion? It is found that any two joints of the joint model cannot be parallel to each other or vertical to the robot plane formed by the axis of links 1 and 4. Therefore, it is necessary to try to increase a rotational joint on the joint model. The joint 2 is selected to cooperate with the new freedom in order to achieve the closed motion in that joints 1 and 3 can never be perpendicular to the robot plane. A 3D model is designed as shown in Fig.5.

The rotational joint 4 is added below the link 4 so that it is parallel to the joint 2 and perpendicular to the robot plane. Applying the closed motion of the space robot, the joint drive (joints 1 and 3 are not driven) is shown in Fig.6.

Fig.5 Combination model

② Joints motion(joints 1,2 and 3)causes the attitude to rotate about a central axis by a certain angle, which is called the Z rotation.

Fig.6 Joints 2 and 4 drive settings

Fig.7 Attitude parameter X1 of link 1

Fig.8 Joints drive settings

The analysis results show that the overall attitude rotates a certain angle, but the angle is small, which is determined by the structure parameters of the space robot , the motion order and size of the joint angles.

The rotation range of single closed motion is limited due to the limitation of physical properties. Even if the motion is planned to have the maximum rotation angle, the desired rotation angle may not be achieved. So it needs several times before arriving at the target. For instance, if the rotation angle requires 105°, and each closed motion up to 5°, to achieve the target position needs 21 times of closed movements. In fact, the rotation angle of the closed motion is small but highly controllable, so it is of great practical significance.

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3 Two DOF Attitude Transformation

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① Joint motions of the two rotating joints (joints 1 and 4) in the same plane causes the overall attitude of the space robot to rotate about an angle in the vertical direction of the plane, which is called X rotation.

Fig.9 Attitude parameters of link1

The closed motion resultis obtained by simulation, as shown in Fig.7.The body axis X-Z-X is adopted to represent the attitude, and the Euler angle parameters are named X1, Z2 and X3 respectively in order to distinguish their order. As a result of the closed motion, only the attitude angle X1 has changed and the other results are 0. The overall attitude of the space robot rotates about -3.23° around the X axis.

The driving law of the joints in the ADAMS simulation model is shown in Fig.8.

The simulation is represented by the body axis X-Z-X Euler angles, as shown in Fig.9.

Ignoring the perturbation of the Z rotation in the second stage, the overall attitude of the space robot can be completely determined by the two parameters which consist of attitude change angle φ1 generated by closing movement X in the first stage and φ2 generated around the center axis of rotation Z in the second stage. Therefore, the TDF attitude transformation scheme is verified by the simulation.

4 Three DOF Attitude Transformation

where Rot is the rotation X; Rotation is the composite movement，rotation Z. So the overall attitude transformation is:

4.1 Attitude transformation

There are 12 ways to represent Euler angles of three DOF attitude transformation. In the TDF scheme, the motion of the attitude transformation consists of the X rotation and the Z rotation. So there are two schemes for applying two kinds of rotation to three DOF, that is, the body axis X-Z-X and Z-X-Z which are consistent with the Euler angle representation. Therefore, the two schemes can realize arbitrary transformation of attitude. The three DOF attitude transformation based on the bionic mechanism is analyzed by taking the body axis X-Z-X as an example.

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The value of the target attitude has been given in the form of body axis X-Z-X Euler angle, and its implementation will be divided into three stages as follows:

① The rotation X1 is achieved by joints 2 and 4 closed motion method;

② The rotation Z2 is achieved by the bionic mechanism joints 1, 2, and 3;

Fig.4 shows a certain angle between the final and initial attitude which is different from a fixed base robot. This series of motion can be called closed motion. Closed motion requires at least two or more directional joints to achieve, and it cannot went exactly back with the application of the original inverse motion. Otherwise, the overall attitude changes will not be occurred.

In the third phase, the X3 rotation can be achieved, the second stage is about the central axis of rotation, in the calculation of the best rotation angle and bending angle, the rotation angle is calculated up to a multiple of 180 °, and let the bending angle slightly reduced small to reach the second stage of the target corner and keep axes of the joints 2 and 4 parallel.

Through the three stages of the rotation, not considering the second stage possible attitude disturbance, the space robot can realize the attitude transformation of body axis X-Z-X which is a kind of generalized Euler angle. Any attitude in space can be represented by Euler angles X-Z-X, consequently, the scheme is feasible.

Adopt the same simulation model as shown in Fig.3. The new closed motion still implemented by joints 2 and 4 is added to the combination model based on two rotation stages. And the rotation angles of the joints 1 and 3 are a multiple of 180°, so that the three DOF attitude transformation of the space robot can be realized. Set the drive for each joint as shown in Fig.10.

So then, the overall transformation of planar robot can be realized by closed motion, and its rotation direction is the same as the joint pointing and perpendicular to the plane of the mechanism.

Fig.10 Drive settings

Fig.11 Attitude parameter of link 1

Ignoring the perturbation in the second stage, it can be observed that in the first stage only the value of X1 changes, and the second stage only see changes in the value of Z2, and only the value of X3 changes in the third stage. Therefore, it can be seen that the three DOF scheme is feasible and coincides with the form of the body axis X-Z-X Euler angles.

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4.2 Disturbance canceling

4.2.1 Method

The related terms are explained firstly.

① The transform Z refers to the transformation of the Euler angle and does not produce any disturbance, while the rotation Z refers to the rotation of the double rigid bodies model joints, and there will be additional attitude perturbations.

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② The transform X is the transformation in the sense of Euler angles, which is the same as the rotation of the X produced by the closed motion.

In the three DOF attitude transformation scheme, the perturbation of the body axis X-Z-X is only generated in the second stage. In the motion planning, rotation X occurs in the first and third stages in the three DOF attitude transformation scheme. On the basis of satisfying the moving target, make rotation X cooperate with the rotation Z, that is, the rotation X is superimposed with a certain offset to balance the disturbance of the second stage. There will be no disturbance to produce finally in theory. The deduction is listed as follows:

Rotation X1 of the third stage:

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(7)

Rotation X3 of the third stage:

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(8)

The rotation Z2 of the second stage is expressed as the form of body axis X-Z-X transformation：

(9)

The two DOF attitude transformation based on a bionic spatial mechanism is still very narrow for space robot applications. We will further analyze the implementation scheme of three DOF.

AR=Rot(X1,α1)Rotation(Z2,γ2)Rot(X3,α3)=

(10)

Combine the first and two term matrices on the right of Eq.(10):

Rot(X1,(α1+α21))=

(11)

Then combine the fourth and five matrices on the right of Eq.(10):

Rot(X3,(α3+α23))=

(12)

So, it can be obtained according to Eq.(10):

AR=Rot(X1,(α1+α21))Rot(Z2,γ2)Rot· (X3,(α3+α23))=

(13)

It can be seen from Eq.(13) that the disturbance can be counteracted by the rotation X in thefirst and third stages respectively. Specific applications are as follows.

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It is assumed that corresponding Euler angles are X1, Z2 and X3 respectively in the attitude transformation by the body axis X-Z-X. Firstly, planning the second stage of motion, so that the motion can reach the angle Z2, and calculate disturbance angles α21 and α23 in the second stage. Then，before the start of the first and three stages of the motion planning, the first phase and the third phase have initial values of α21 and α23, Therefore, the value after planning should subtract the initial value, that is, the first stage planning value is α1=X1-α21, and the third stage planning value is α3=X3-α23 .

Through the above analysis,it is found that the disturbance generated by the body axis X-Z-X mode can be completely counteracted by motion planning. Theoretically, the attitude transformation without disturbance can be realized. This is of great significance for practical applications.

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4.2.2 Simulation

The purpose of combination joint is to complete the attitude transformation of the body axis X-Z-X, and the corresponding Eulerian angles are X1, Z2 and X3 respectively. The implementation process belongs to the motion planning. The key simulation results are only limited to the presentation and validation of the new structure. Motion planning research belongs to the following problems, and the related research results will be displayed soon.

Firstly, the second stage motion of the plan is set to reach the Z2 angle which is displayed in the 5.1 solution. Then the perturbation angles corresponding to the X1 and X3 are separated, and they are superimposed on the first and third stages. Finally, the angle changes of X1 and X3 can be observed as shown in Fig. 12 after the simulation. It can be found that when the Z2 changes, the attitudes of the X1, X2 at the corresponding motion are stable, and the useless disturbance has been eliminated. Therefore, the method of eliminating disturbance proposed is verified.

Fig.12 Attitude parameter of link 1 under disturbance cancellation processing

5 Conclusion

Space robot has a unique closed motion phenomenon, and the motion can only realize one DOF attitude transformation. The model with double rigid bodies based on bending theory can eventually achieve a rotation of the overall attitude. A new combination model is established by interweaving their characteristics, which can realize attitude transformation in planar, two DOF and three DOF space by simulation. However the rotation of the joint results in an additional perturbation. Therefore, a scheme to eliminate attitude perturbation is proposed, and resulting effects are derived in theory. The concrete realization of the scheme may need further study, and thus the application of the spatial attitude transformation of a bionic mechanism will may be a research direction in the near future.

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小学生处于一个对许多未知事物都充满了好奇心但是贪玩好动缺乏耐性的时期，所以想要长期培养小学生的阅读习惯就需要充分激发起他们对阅读的积极性和兴趣，让他们感受到阅读不是一件枯燥的事情，而是一件非常生动有趣的事情，那么如何展现出阅读的生动趣味性呢？