The structural block diagram of the semi-physical test is shown in Figure 1.

The principle of the semi-physical test is to calculate the motion of the spacecraft under the action of an external force according to the spacecraft dynamic model and then execute the motion through the real mechanism. When the docking mechanism moves according to the results given by the dynamic model, an actual collision appears, and the sensor detects the real collision force and torque. The force and torque data are sent to the dynamic model, and the model calculates the motion of the spacecraft in the next cycle. The manipulator continues to drive the docking mechanism according to the model’s results, and the docking mechanism processes a real collision again. In this way, the iteration constitutes a semi-physical test. Theoretically, we need the manipulator to act corresponding movement as soon as the impact force and torque are generated. However, in fact, the measurement of the impact force and torque, the calculation of the mathematical model, and the execution of the manipulator’s movement all take time, which leads to the lagging effect and significantly affects the semi-physical test. The traditional method alleviates the lagging effect by improving the response speed of the actuator. In this paper, under the condition that the existence of the lagging effect lasts, a method of actuation delay compensation is proposed, and it will be discussed in the next chapter.

The semi-physical model in this paper is as follows. The model describes the relationship between the force and torque of the spacecraft and the acceleration and angular acceleration during parking docking with manipulator assistance. The definition of parking docking is that two spacecraft are connected by a manipulator or other components, and the initial conditions of docking are almost static.

where F_K,Q and T_K,Q are the measurement results of the docking force and torque, F_{s−manipulator} and T_{s−manipulator} are the equivalent resistances of the space manipulator in parking docking, and the mass m_K,Q and inertia tensor I_K,Q of the spacecraft are the parameters set by the operator. Angular velocity ω_K,Q is a variable involved in iterative calculation. Acceleration a_K,Q and angular acceleration α_K,Q are variables to be solved.

As shown in Figure 2, we imagine that there is spacecraft behind the docking mechanisms. In fact, they do not exist, and only the docking mechanisms are real. The coordinate systems of spacecraft centroids and the coordinate systems of docking mechanisms are also drawn in the figure, where A is the ground coordinate; S the F/T sensor coordinate system; K the passive spacecraft centroid, which is moveable during the test and is also named M, meaning that all the moving commands describe the movement between M and A; J the passive docking mechanism docking surface, whose positional relationship with K is invariant and is also named N, an additional observation coordinate system set by the operator; P the active docking mechanism docking surface; and Q the active spacecraft centroid. P and Q are static during the test. In the real docking process in the universe, K and Q are both moving. In the experiment, Q is stationary, and K’s motion reproduces K’s relative motion with respect to Q. The F/T sensor can measure the force and torque in the S coordinate system in real time. In the test, we need to conduct gravity compensation, zero compensation, and dynamic compensation for the measurement results and convert the compensated measurement results to the M coordinate system for the model. The compensation and conversion are shown in the following equations.

where R(n)GS is the rotation matrix between the S coordinate system and the gravity direction coordinate system and R(n)SM is the rotation matrix between the M coordinate system and the S coordinate system. R(n)GS⋅GG and R(n)GS⋅TGG+pSGS×GS are the gravity compensation items; F0S and T0S are the zero compensation items; ma^S and I^Sα^S + ω^S×I^Sω^S are the dynamic compensation items. The parameters G^G, pSGS, F0S, and T0S used in gravity compensation and zero compensation can be obtained by the least square method under the condition that R(n)GS, F(de-gravity)⁢(n)S, and T(de-gravity)⁢(n)S are known. The parameters m and I^S can also be obtained by the least square method under the condition that a^S, α^S, and ω^S are known. In this paper, dynamic compensation is ignored; according to the large mass ratio between the docking mechanism and the spacecraft, only gravity compensation and zero compensation are programmed.

The F/T in the K coordinate system is directly measured, and the force in the Q coordinate system needs to be calculated according to the following equation. C is the reference coordinate system set by the operator.

The acceleration, angular velocity and angular acceleration can be solved according to the method given in the following equation.

In this way, we can calculate the angular velocity and angular acceleration of the active spacecraft and passive spacecraft in the model according to the measured force and torque in each motion period. The initial position, initial velocity and initial angle in the dynamic model are set by the operator. In each motion period, we calculate the attitude, velocity and angular velocity at the end of this period according to the following formula, which is used as the initial condition for the motion calculation of the next cycle.

The acceleration a_ARM(n) and angular acceleration α_ARM(n) generated by the force of the space manipulator on the spacecraft are involved in the model calculation here. They are two accelerations set by the operator. Thus far, we have completed the dynamic model of the docking of two spacecraft, both of which are moving. Provided that the test platform in this paper only allows the passive part of the docking mechanism to move and the active part of the docking mechanism to be fixed on the base, we also need to calculate the relative motion between the two docking mechanisms. Then, the manipulator performs the relative motion to complete the equivalent test. The motion of the K coordinate system relative to the Q coordinate system is calculated according to the following equation.

Finally, we obtain the movement command of the manipulator in the following equation:

The initial position and velocity of the K coordinate system relative to the Q coordinate system are calculated according to the initial position and velocity of the docking dynamic model, and the manipulator is driven by the operator to reach that position. When the manipulator reaches the initial conditions, we can start the semi-physical test by switching the control mode of the manipulator to the semi-physical simulation mode.

Thus far, we have completed the dynamic model used in the semi-physical test. The model can output the motion velocity command for coordinate M.

After obtaining the completed semi-physical test model, we designed a small experiment on the test equipment to verify the implementation of the semi-physical test. The designed spring rod test is shown in Figure 3.

In the spring rod test, the two spacecraft theoretically did a pure elastic collision, which means the impact force of each adjacent two times should be equal in size and opposite in direction. In fact, however, we obtain the results in Figure 4.

The results demonstrate that the impact force increases with impact times. It can be inferred that in the process of impact, the movement of the manipulator, F/T measurement, computer simulation calculation, communication, and other factors lead to the lagging effect in the semi-physical test system, resulting in a longer contact time of the elastic rod and a larger rebound velocity.

After many attempts, we have found a way to reduce the actuation delay of this system. We add two items to the output velocity of the model and then drive the manipulator with the final velocity as the command. To describe the way we do, we define four velocities: manipulator velocity command v_cmd(n) and ω_cmd(n); dynamic model output velocity v_model(n) and ω_model(n); compensation velocity v_comp(n) and ω_comp(n); back velocity v_back(n) and ω_back(n).

The relationship among these four velocities is:

This means that without compensation, we should use the model output velocity as the command velocity; however, in fact, we added something else to the model output velocity as the command velocity.

The definition of the dynamic model output velocity is:

To describe the compensation velocity, the following variables are defined. The velocity difference dv_(n), dω_(n) lies in the difference between the current cycle model velocity and the previous cycle model velocity.

Remark: It should be noted that signals v_model(n) and ω_model(n) have gone through another data preprocessing, which is a combination of a low-pass filter and saturation.

The definitions of velocity difference symbols s_dv(n) and s_dω(n) are judging each component of the velocity vector and angular velocity vector. When the absolute value of the component is less than the set threshold v_min⁡dv, ω_min⁡dωv, the component becomes 0. When the absolute value of the component is greater than the set threshold, the symbol of the component is defined as the component. That is, if the component is positive, the value is 1, and if the component is negative, or the value is −1.

The velocity symbols s_v(n) and s_ω(n) are:

The compensation of velocities is defined as the following equations:

This is a scalar equation, which should be calculated separately for each component of all vectors. Where b₀ is the coefficient of velocity difference symbol, b₁ the proportionality factor of velocity difference and velocity symbol, c_m the integral coefficient of velocity difference polynomial, and the right of the equation are polynomial with degree m up to four. Likewise, h₀ the coefficient of the rotating velocity difference symbol, h₁ the proportionality factor of the rotating velocity difference and rotating velocity symbol, and k_m the integral coefficient of the rotating velocity difference polynomial. The composition of the compensation function is similar to the addition of the velocity difference polynomial and velocity difference polynomial integral. This compensation reduces the collision duration.

Then, the regression function is defined as the following equations:

This equation is also a scalar equation, which should be calculated separately for each component of all vectors. k_pv, k_iv, k_pω, and k_iω are the proportionality factor and integral coefficients. The equations are PI-like controllers. With a proper combination of compensation and regression, one can allow the command velocity to overcome the actuation delay in the semi-physical test in this paper.

After compensation, the semi-physical simulation motion mode expression of the manipulator is:

We use a single-degree-of-freedom collision experiment to observe the effect of actuation compensation. The comparison between the before and after compensation effects can be seen in Figure 5. The blue curve in the figure represents the motion speed curve calculated by the model according to the actual measured force. As a motion command, this speed can be directly output to the robot for execution, or output to the robot for execution after the compensation algorithm. We can see that before compensation, the output speed of the model changes from 1 mm/s before collision to nearly −3 mm/s after the collision, and the absolute value of the speed increases. This phenomenon does not conform to the theoretical elastic collision. In order to observe the cause of this phenomenon, we employ the red curve to draw the measured execution speed of the robot after receiving the command. It can be seen from the figure before compensation that the actual execution speed of the robot lags behind the blue movement speed command curve, and there is a short pause when the speed reaches 0. So we designed a compensation algorithm. The speed command output from the model is delayed by the compensation algorithm, and becomes a green speed command curve and output to the robot. The fact is that the compensated green speed command curve and the model output speed curve have obvious deviations at the time of the collision. After the collision, the compensated speed curve and the model output speed curve gradually coincide. The actual motion curve of the robot represented by the red curve still lags behind the speed command after compensation, which is determined by the hardware characteristics of the system. However, the red robot execution speed curve is closer to the blue model output speed curve than before compensation, indicating that the compensation algorithm makes up for the implementation lag caused by the hardware performance. At the same time, the blue model output speed curve becomes–1 mm/s after the collision, which is basically consistent with the absolute value of the speed before the collision. It can be concluded that our compensation algorithm corrects the motion distortion in the semi physical collision process, making the motion closer to the real physical characteristics.

This is consistent with our originally noticed designing functions.

To study how the compensation algorithm compensates for the actuation delay of the semi-physical test, we carry out a simulation of a single-degree-of-freedom collision experiment. In the simulation, the execution velocity of the manipulator lags behind the command velocity by 32 milliseconds, and a short pause occurs when the execution velocity passes zero. The comparison of before and after compensation in the simulation of different initial velocities is demonstrated in Figures 6–8.

In the simulation, the values of the coefficients b₀ = 0.1, b₁ = 2, c₁ = 2.57, c₂ = −16.81, c₃ = 146, c₄ = 1337, k_pv = 0.005, and k_iv = 0.000005.

Observing the simulation results of different velocities, without compensation, the absolute value of the rebound velocity is greater than that of the incident velocity. With compensation, the absolute value of the rebound velocity is almost the same as that of the incident velocity. The actuation delay compensation performs well.

To observe the actions of each component of compensation, we draw 7 components of compensation velocity on the basis of different incident velocity conditions in Figure 9.

Where the blue line is c1⋅∑0n(sv⁢(n)⁢|dv(n)|1), the red line is c2⋅∑0n(sv⁢(n)⁢|dv(n)|2), the green line is c3⋅∑0n(sv⁢(n)|dv_(n)^|3), the yellow line is c4⋅∑0n(sv⁢(n)⁢|dv(n)|4), the pink line is b₀s_dv, the black line is b₁s_v(n)|dv_(n)|, and the purple line is v_back(n). It displays that at low velocity, the role of the blue lines c1⋅∑0n(sv⁢(n)⁢|dv(n)|1) is obvious. With velocity increasing, the roles of the red, green and yellow lines, c2⋅∑0n(sv⁢(n)⁢|dv(n)|2), c3⋅∑0n(sv⁢(n)⁢|dv(n)|3), and c4⋅∑0n(sv⁢(n)⁢|dv(n)|4) matter more. The simulation results demonstrate that within the velocity range required by the test, the actuation delay compensation method provided by us is of outstanding practice.