Abstract
For space robots, it is difficult to track continuous time-varying manifolds on SE(3) by using traditional closed-loop control strategies, which are designed to track the position and the attitude separately. Therefore, the dynamics model should be rebuilt, and the corresponding control strategy should be redesigned. Firstly, the dynamics equations for a space robot in the joint space and workspace are established separately in the framework of Lie group SE(3) and screw theory based on the Lagrange principle. Secondly, based on the proposed feedback form, a PD (proportional derivative) control law of output force on the end-effector is designed, and a closed-loop continuous tracking control strategy is proposed using the force Jacobian matrix and the kinematic model. The simulation results show that the control scheme has good performance when the system state changes gently. Furthermore, a robust sliding mode tracking control scheme is designed. The simulation results show that the proposed robust control law has better accuracy than the PD control law because the system state changes wildly. Finally, a robust fuzzy sliding mode tracking control scheme is designed to deal with the chattering phenomenon. The simulation results show that the proposed robust fuzzy control law can eliminate the chattering well and decrease the joint control torque significantly. The robustness of the proposed robust fuzzy control law is also verified by numerical simulation.
1. Introduction
The emergence of on-orbit assembly stems from the growing need to build large space structures [1]. Related technology has been greatly promoted through the development of on-orbit service technology, especially the on-orbit demonstration of space robot technology. However, the process of assembling large space structures is so elaborate that it requires an accurate and reliable operation ability for space robots. Accordingly, a proper motion planning method and corresponding tracking control strategy should be introduced. Due to the advantages beyond traditional theory, Lie group SE(3) and screw theory have been applied to research in space robotics, such as kinematic modelling [2] and trajectory planning. However, compatible dynamics and tracking control strategy are not considered in detail.
The dynamics of space robots belong to the category of multibody system dynamics. Thus, the modelling is mainly based on the Newton-Euler method and the Lagrange method. The former analyzes the forces of each rigid body based on classical Newtonian mechanics and then obtains dynamic equations in iterative form according to the relationship of internal forces [3]. The advantages of such a method are the clear physical meaning and the relatively small calculation [4]. However, the equations will become more complicated as the number of rigid bodies increases [5]. The Lagrange method is based on the Hamiltonian principle, and the dynamic relationship between the system state variables and the generalized forces is built based on the conservation of energy [6]. The advantages of such a method are that modelling process is relatively easy, and the dynamic equation is in the analytic form [7], but the physical meaning is not clear, and the partial differential calculation is more complex. A detailed comparison of these two methods is presented in [8].
In addition, the Kane method is also introduced in robotic dynamics research [9]. This method subtly uses the partial velocity as the generalized coordinate, which is combined with the concept of generalized force to establish the dynamic equations according to the Darenbel-Lagrange principle [10]. The Kane method combines the advantages of the Newton-Euler and Lagrange methods, which have high calculation efficiency and concise form. Some improvement methods are proposed [11, 12], which further enhance the performance of the Kane method. Xu and Huang [13] compare classical methods with the Kane method and deduce the dynamic equations of a five-DOF robot using both methods. Yin and Ge [14] derive the dynamic equations of a dual-arm space robot using the Kane-Huston method and verify the feasibility with simple simulation.
Traditional dynamics modelling, which is based on geometric relationships, makes the derivation process relatively complicated. It brings a lot of convenience by introducing the Lie group and Lie algebra, which have a concise and unified form and decreases the difficulty of theoretical derivation. The theory of the Lie group has been applied to the dynamics of fixed-base robots [15, 16]. Besides, Liu et al. [17] combine the screw theory and the Kane method and propose a new concept of space robot dynamics.
Trajectory tracking control is crucial to accomplishing the task for the space robots, and the performance of the tracking controller determines the accuracy and reliability of the task. Many trajectory-tracking control strategies are designed based on the dynamic model. And some control theories for fixed-base ground robots can be introduced to tracking control of space robots, such as decomposition rate control [18], calculating torque control [19], robustness control [20], and reinforcement learning control [21]. Traditional dynamics modelling, which is based on geometric relationships, makes the derivation process relatively complicated.
Trajectory tracking of space robots can be performed either in joint space or in cartesian space. According to tracking targets, it can be further classified into point-to-point and continuous trajectory tracking. The latter means that the end-effector is required to move along the desired continuous trajectory, such as peg-in-hole insertion [22], which makes tracking control relatively difficult. Li and Liang [23] establish the kinematic equation of the space robot system by using the D-H (Denavit-Hartenberg) method and design the continuous trajectory control algorithm according to the momentum conservation law. Such an algorithm could be extended to a space robot with an arbitrary tree structure. Galicki [24] defines a nonsingular terminal sliding model in the workspace and proposes a robustness controller based on the Jacobian transfer, which can effectively eliminate the uncertainty of dynamics. Su et al. [25] define an approximately fixed-time convergence terminal sliding model and design a nonsingular sliding model tracking control strategy, but it is carried out in the joint space. In addition, most existing trajectory tracking control laws use position or linear velocity as feedback variables. There are also attempts that use quaternion [26] or dual quaternion [27] as feedback variables, but they are all difficult to apply to the tracking of the continuous time-varying manifold on SE(3).
Above all, this paper will focus on the dynamics modelling under the framework on Lie group SE(3) and the control problem of tracking the continuous time-varying trajectory of the end-effector on SE(3). Highlights of this paper are addressed below. (1)The dynamic model of a multiarm space robot is established by combining the Lie group SE(3) with the Lagrange method. The model has a concise and unified form, which can decrease the difficulty of theoretical derivation and calculation(2)A novel form is proposed to describe the feedback error in the tracking control problem of the space robot. Lie group SE(3) and screw theory are used to build the form, which can assemble the position and attitude, with their derivative, of the end-effector in a unified variable. It can significantly reduce the calculation in the process of closed-loop control(3)A robust fuzzy sliding mode tracking control strategy is proposed to track the continuous and time-varying trajectory, including position and attitude together, of the end-effector. The tracking control strategy has good control accuracy, good robustness, and no chatter
2. Dynamic Modelling
2.1. Dynamic Equations of the Space Robot System
For a dual-arm space robot, as provided in Figure 1, the kinetic energy of the link of the arm in the barycentre-fixed inertial coordinate system is where , , , and are the body angular velocity, the inertia, the mass, and the velocity, respectively. refers to the base.
According to the definition of body velocity in screw theory, formula (1) can be transformed into the following form: where is the body velocity, and represents the attitude rotation matrix in the inertial coordinate system.
According to the relationship between body velocity and spatial velocity, formula (2) is transformed into where is the spatial velocity, is the adjoint transformation, represents the antisymmetric matrix of the corresponding vector, and means generalized inertia matrix.
Therefore, the total kinetic energy of the dual-arm space robot is, where represents the total momentum of the dual-arm space robot.
As is known, is workable in the case of a free-floating base. Substituting GJM into formula (4) yields where are the GJM of the link and the base, respectively. is the angular velocity vector of all joints. represents the generalized inertia tensor of the dual-arm space robot, the specific expression is as follows: where
If the gravity is ignored and the joint elasticity is not considered, this system satisfies the Lagrange equation in the following form:
Substituting formula (5) into the above formula, then
It can be rewritten as where which represents the centrifugal force and Coriolis force.
For a single-arm space robot, the above derivation process is also applicable.
2.2. Static Force Jacobian Matrix
Static force Jacobian matrix is also a fundamental concept in the dynamics of space robots. Similar to the speed Jacobian matrix, it describes the mapping relationship between the output force on the end effector and the joint torque. This derivation is based on the equivalence principle of work and energy. Taking a single-arm space robot as an example, and assuming that the output force on the end effector is , then the work is where is the spatial velocity of the end effector. Simultaneously, the work done by the joint torque is
If joint friction is not considered, then and should be equal and time-independent. Thus,
As is known, , where is the speed GJM of the end-effector for the joints. So where is the static force Jacobian matrix.
2.3. Dynamic Equations in the Workspace
When performing trajectory tracking control of the end effector, it is also necessary to establish dynamic equations in the workspace. Firstly, according to the GJM of the end-effector, there are
Accordingly,
Substituting formula (17) into formula (10),
It can be rewritten as
Then, substituting formula (15) into the above formula,
where and . It can be proved that the above dynamic equation satisfies, (1) is a symmetric positive definite matrix(2) is an antisymmetric matrix
2.4. Feedback Calculation Model Based on SE(3)
As the desired trajectory is expressed on SE(3), compatible feedback should be designed first. Supposing that and are the actual and desired pose of the end effector, respectively, then the pose error can be written as where .
The tracking error in exponential coordinate is presented as where represents the logarithmic mapping from Lie group SE(3) to its Lie algebra se(3).
The above equation can be written in the form of screw coordinates as
It is used as feedback at the position level. It should be noted that is with respect to the body-fixed coordinate system of the end effector.
The feedback at the velocity level can be expressed as where is the actual body velocity, and is the desired body velocity.
In fact, the following relationship exists:
The expression of in the above formula is presented in [28] and omitted here.
Accordingly, the feedback of pose expressed on SE(3) can be expressed as
Transforming it into inertial coordinate system yields
3. Continuous Trajectory Tracking Control Strategy
3.1. PD Controller Based on Jacobian Matrix
According to the feedback obtained in the previous section, the following PD control law is designed: where is the output force on the end effector with respect to the inertial coordinate system, and and are the position and velocity gains, respectively.
Then, the joint control torque can be computed as
Accordingly, detailed steps of a closed-loop tracking control strategy are given as follows (as shown in Figure 2):
Step 1. Set the initial joint angle of the arm and the attitude of the base. Set the desired pose
Step 2. Calculate the current pose of the end effector according to the kinematics equation, and then obtain the force Jacobian matrix
Step 3. Calculate the current state feedback according to formula (27)
Step 4. Calculate the joint control torque at the current moment according to formula (29)
Step 5. Calculate the joint angle and joint angle rate at the next moment according to the dynamic equation, and then calculate the base attitude angle at the next moment
Step 6. Repeat steps 2-5 till to the end time.
The controllers proposed in this research include the PD controller, the robust sliding mode controller (RSMC), and the robust fuzzy sliding mode controller (RFSMC). RSMC or RFSMC can replace the PD controller module in Figure 2, and then the corresponding control block diagram is generated.
3.2. Robust Sliding Mode Tracking Controller
The PD control has a simple form and can meet general needs. However, the control accuracy will be significantly reduced when the state changes fast. Considering the advantages of sliding mode control, such as the high control accuracy and strong robustness against external disturbances, a controller based on robust sliding mode control theory with the feedback calculation model above is proposed in this section.
3.2.1. Linear Reaching Law-Based Sliding Mode Control Law
According to the tracking feedback at the velocity level, the sliding mode surface is designed as follows: where is a positive definite diagonal matrix. A control law can be given where is a positive definite diagonal matrix.
Theorem 1. For the dynamics of a space robot system, applying sliding mode surface (30) and control law (31), the system will gradually converge to the equilibrium point.
Proof. Define the Lyapunov function as follows: According to the sliding mode surface (30), Calculating the derivation of and substituting (28) into it yields where . is the smallest eigenvalue of , and is the largest eigenvalue of . It can be seen that the sliding mode surface converges to zero driven by the control law (31). If , there is , then progressively converge to the equilibrium point. Proof completed.
3.2.2. Robust Sliding Mode Control Law
When the dynamic model can be accurately established, the sliding mode control law (31) can be substituted into the PD control strategy to replace the PD control law. However, as the dynamic model of the space robot is very complicated, if control law (31) is directly adopted not only will the modelling be complex but also the amount of calculation will be huge. Let
Regard this term as a disturbance and assume that the term is bounded and satisfies the following inequality:
In order to overcome the disturbance term, the following sliding mode control law is designed: where is the robust coefficient and and is the robust term to overcome the disturbance.
Theorem 2. For the dynamics of a space robot system, applying sliding mode surface (30) and control law (37), the system will gradually converge to the equilibrium point.
Proof. Define the Lyapunov function as follows:
Similar to Theorem (1),
It can be proven like Theorem (1).
Proof completed.
3.3. Robust Fuzzy Sliding Mode Tracking Controller considering Uncertainty
The robust sliding mode tracking controller has better control accuracy; however, when there are uncertainties with the parameters of the system model, the controller must choose a more significant robust coefficient or amplify the gain to ensure robust performance and control accuracy which may lead to the obvious chattering of the control torque. In order to solve this problem, fuzzy ideas are introduced into to sliding mode control law to approximate the dynamic model with uncertain parameters, which can realize robust and stable control of the space robot.
3.3.1. Fuzzy Approximation Method
Taking into account the unavoidable external interference, introduce the interference force to the dynamic model as follow. The following fuzzy sliding mode control law is designed where is the estimated value of the fuzzy system. Substituting the control law into equation (40) to obtain where and .
The fuzzy approximation can be used to estimate the function in equation (35), where is the optimal weight matrix of the fuzzy approximation system, is the approximation error of the system, and is the basis function.
Let be the estimation of and let the sliding mode surface as the input variable of the basis function, and then the dynamic compensation part of the space robot control law is estimated to be
3.3.2. Robust Fuzzy Sliding Mode Tracking Control Law
Firstly, assuming that (1)The output of the fuzzy approximation system is bounded, and the total external disturbance of the estimated system is bounded, which is expressed as(2)The estimation error of the fuzzy approximation system is bounded, which is expressed aswhere is a positive constant. (3)The estimated weight of the fuzzy approximation system is bounded, which is expressed aswhere is a positive constant.
The input variable of the fuzzy system is designed as
Then, the control law can be reshaped as where is a robust term used to overcome approximation errors and interference. Let be the estimation error of the optimal weight matrix.
According to the above control law, there is where , and .
The robust term is designed as
The adaptive updating law of the optimal weight matrix is taken as where is a positive constant.
Theorem 3. For the dynamics of a space robot system considering interference, the fuzzy approximation method is used to estimate the uncertainty of the system, and the weight matrix updating law is set as formula (52), then the system under the action of the control law (49) will gradually converge to the equilibrium point.
Proof. Define the Lyapunov function as follows:
Derivation of the above formula is
Substituting equation (50) into the above equation,
Substituting the updating law (52) into the above equation,
due to
Then, there is
If , then . According to the LaSalle invariance principle, the closed-loop system has the property of gradual stability as , . Thus, the error of the pose in exponential form and the velocity will gradually converge to zero as ; so the pose and velocity of the end-effector will gradually converge to desired values as .
Proof completed.
4. Simulation Example
Considering a planar, three-DOF space robot as a simulation case and verifying the effectiveness and control performance of the proposed control strategies. The initial pose and the final pose are given as follows:
The desired trajectory is planned based on the drive transformation method and screw theory.
4.1. Simulation Results of PD Controller
Simulations of the PD controller are performed in MATLAB/Simulink, and the results are obtained as follows. The state error varying with time is shown in Figure 3(a), as is seen in which the maximum error is about , and the steady state error is less than . Figures 3(b)–3(d) show the comparison between the actual and desired pose components. It can be seen that the tracking process is ideal.
(a)
(b)
(c)
(d)
The output force on the end effector and the joint control torque obtained by the PD control law is shown in Figures 4 and 5 separately. The joint control torque is relatively small, and no saturation occurs. The space robot’s joint motions are shown in Figures 6 and 7. It can be seen that the joint motions are continuous and smooth without sudden change. To sum up, the designed PD controller performs well under the gentle motion state.
4.2. Simulation Results of the Robust Sliding Mode Controller
The simulation object remains the same as in the previous section, and the initial and final poses remain unchanged. The desired trajectory is replanned to move faster. The parameters of the sliding mode controller are set as , , and .
The PD control strategy and the robust sliding mode tracking control strategy are simulated, respectively, and simulation results are obtained as follows. From comparisons of Figures 8(a) and 3(a), it can be seen that accuracy of PD control has been significantly reduced, the maximum error has reached 0.013, and the steady-state error also increases. In the same situation, the accuracy of the robust sliding mode control is higher, the maximum error is about , and the steady-state error remains within as in Figure 8(b). Figures 9(a) and 9(b) show the joint control torque of the two control algorithms. In contrast, the control torque of sliding mode control has a specific chattering phenomenon.
(a) PD controller
(b) Robust sliding mode controller
(a) PD controller
(b) Robust sliding mode controller
4.3. Simulation Results of the Robust Fuzzy Sliding Mode Controller
The simulation object remains the same as in the previous section, and the initial and final poses remain unchanged. Set the parameters of the robust sliding mode controller as ,,and . Seven fuzzy subjection functions of the fuzzy sliding mode controller are as follows:
The robust sliding mode tracking control strategy and the robust fuzzy sliding mode tracking control strategy are simulated, respectively, and simulation results are obtained as follows. It can be seen in Figures 10(a) and 10(b) and Figures 11(a) and 11(b) that, after increasing the gain and robust coefficient of the robust sliding mode control, the control accuracy is improved. However, the joint control torque has serious chattering. In contrast, fuzzy sliding mode control can effectively eliminate chattering while ensuring similar accuracy, and the peak value of joint control torque is also significantly better. Figure 12 shows the output force of the end effector and the corresponding estimated value obtained during the fuzzy sliding-mode control algorithm process.
(a) Robust sliding mode controller
(b) Robust fuzzy sliding mode controller
(a) Robust sliding mode controller
(b) Robust fuzzy sliding mode controller
(a) Output force of end-effector
(b) Corresponding estimated value
4.4. Simulation Results of Robust Fuzzy Sliding Mode Controller with Interference Force
In order to verify the robust performance of the fuzzy sliding mode controller, the interference force is set as follows:
The simulation is performed under the same conditions as Section 4.3, and the following results are obtained. After the interference force is added, the state feedback error and joint control torque are shown in Figures 13(a) and 13(b). Comparing with Figures 10(b) and 11(b), respectively, it can be seen that the interference force only has an impact on the system within the first 1 s, which makes the error and joint torque amplify compared with when there is no disturbance, but keep it unanimous afterward. It demonstrates that the fuzzy sliding mode tracking controller has good robustness.
(a) Tracking error
(b) Joint control torque
Furthermore, another simulation is performed with a pulse interference force set as follows:
Following results are obtained. It can be seen in Figure 14(a) that the error keeps very similar, while the joint control torque fluctuates slightly in the third and fifth seconds in Figure 14(b); however, the system can still keep stable after the pulse, which demonstrates the robustness of the fuzzy sliding mode tracking controller again.
(a) Tracking error
(b) Joint control torque
5. Conclusion
Firstly, based on the Lagrange principle and the generalized Jacobian matrix, the dynamic equations in the joint space and workspace are established separately. Then, a feedback calculation model is proposed based on the screw theory. Furthermore, the PD and robust fuzzy sliding mode control laws are designed separately. Finally, the closed-loop control strategies are constructed by integrating control laws, the GJM, a dynamic model, and a kinematic model. The effectiveness of each control law is verified by simulations, and the control accuracy and performance are compared and analyzed. It can be summarized that the proposed dynamic modelling method is feasible and efficient. The PD controller is proven to have relatively high control accuracy. Nevertheless, the accuracy reduces when the system state changes more dramatically. The robust sliding mode controller is able to overcome such disadvantages, and the fuzzy sliding mode controller could solve the chattering problem.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This work is funded by the National Science Foundation of China. Grant No.11902362.