Large population of defunct artificial objects in low Earth orbit, known as space debris, poses a serious threat to operational satellites (Murtaza et al., 2020). Removal of existing objects from orbit is needed to prevent the growth of the space debris population (Liou et al., 2010). Thus, the active debris removal (ADR) missions are proposed to capture and remove large space debris from orbit (Bonnal et al., 2013). The European Space Agency (ESA) was working on the e.Deorbit mission to demonstrate the ADR capabilities (Biesbroek et al., 2017). The Clearspace-1 mission is currently under development to capture and remove a selected object from orbit (Biesbroek et al., 2021). Grasping of a target object in-orbit will be the most challenging part of the proposed missions. Various techniques can be used to perform the capture operation (Shan et al., 2016). The most mature approach is based on the use of a gripper attached to a manipulator mounted on a chaser satellite. Such an approach was proposed for the e.Deorbit mission (Estable et al., 2020). Advanced trajectory planning and control algorithms are required to perform the capture operation (Flores-Abad et al., 2014). The control system must take into account the influence of the manipulator motion on the state of the chaser satellite (Dubowsky and Papadopoulos, 1993). Moreover, the manipulator has to avoid collisions with the elements of the target object. Satellites are equipped with appendages (e.g. solar panels), which make the trajectory planning task difficult.
In this paper, we focus on the problem of collision-free trajectory planning of a manipulator mounted on a free-floating chaser satellite. A review of trajectory planning methods applicable for the considered in-orbit capture operation was presented by Rybus (2018). The planning is performed in a high-dimensional configuration space, as the configuration of the satellite-manipulator system is described not only by the angular positions of manipulator joints, but also by the position and orientation of the chaser. The proposed methods include variants of the Rapidly exploring Random Tree (RRT) algorithm (Benevides and Grassi, 2015; Rybus, 2020), the A* algorithm (Gao et al. 2011), and various optimisation techniques (Misra and Bai, 2017; Wang et al., 2018; Rybus et al., 2022). However, these approaches have a high computational cost, which can be a problem due to limited resources of the chaser satellite on-board computer and the short time available for the capture operation. Moreover, the RRT algorithm is based on the random search of the configuration space and it is not deterministic, that is, a different solution is generated in each run. Due to these facts, the possibility of practical application of the aforementioned methods in the ADR missions is limited. Thus, other methods for trajectory planning are being sought.
One of the considered approaches is based on the artificial potential field (APF) method. In the APF method, the manipulator is considered as a particle that is moving in an artificial force field (Khatib, 1986). This force field is composed of the attracting potential of the desired final position of the gripper and the repulsive potential of the obstacles. The APF method is widely used for fixed-base manipulators working on Earth (Lin and Chuang, 2003) and for mobile robots (Kowalczyk et al., 2012). Application of the APF method for a manipulator mounted on a chaser satellite was proposed by Mukherjee and Nakamura (1991). The existence of local minima of the potential field is the main disadvantage of the APF method (Park and Lee, 2003). The trajectory planning fails when the manipulator encounters a local minimum of the field. Yanoshita and Tsuda (2009) considered the trajectory planning problem for a satellite-manipulator system and proposed the potential field based on the Laplace function that allows to partially overcome the problem of local minima. Rybus and Seweryn (2018) modified the classical APF method by introducing a new approach for selection of the direction in which the repulsive force is acting. The modified APF method is more efficient than the classical APF method, that is, it allows to solve the trajectory planning problem in some of the cases, in which the classical APF method is unsuccessful.
Recently, Rybus (2022) proposed the obstacle vector field (OVF) method. In the classical APF method, a scalar potential field is used, while in the OVF method, a vector field is constructed. This vector field surrounds the obstacles and determines the direction in which the link of the manipulator should move to avoid collision with obstacles. The OVF method takes into account the fact that the motion of the manipulator influences the state of the chaser satellite. The OVF method is less efficient than the RRT algorithm, but it is more efficient than the modified APF method. Moreover, it is fully deterministic and it requires significantly less computational time than the RRT algorithm. The OVF method is, to some extent, similar to several concepts known from the literature. Masoud and Al-Shaikhi (2015) proposed an approach in which a vector field guides a mobile robot to the desired position. This field is produced from the harmonic potential and is perturbed by obstacles. The desired trajectory is obtained by solving a gradient dynamical system. Such an approach is more complicated than the OVF method because the algorithm is based on partial differential equations. A trajectory planning algorithm based on parametrised vector potential functions was proposed by Pamosoaji and Hong (2013) for a unicycle vehicle. In this approach, the workspace is split into predefined triangular regions and the vector potential functions are calculated using information on regions’ vertices, the obstacles, and the desired position of the vehicle. Masoud and Bayoumi (1993) used vector potential field for planning a trajectory of a fixed-base manipulator. The guiding vector field is generated by solving a scalar boundary value problem and it has two components: one responsible for driving the gripper toward the desired position and the other one responsible for guiding the gripper around obstacles. A detailed discussion of similarities and differences between these methods and the OVF method was presented by Rybus (2022).
The OVF method was validated in numerical simulations performed for a planar 2 Degrees of Freedom (DoF) manipulator mounted on a free-floating chaser satellite (Rybus, 2022). This method allows to obtain the desired position of the gripper, but it does not allow to obtain the desired orientation of the gripper. This is a major disadvantage as it is necessary to rotate the gripper to a specific orientation to perform the grasping manoeuvre (the gripper has to be fully aligned with a grasping interface on the non-cooperative target satellite).
In this paper, we introduce a modification to the OVF method that allows to obtain not only the desired position of the gripper, but also its desired orientation. We also investigate the possibility of application of the OVF method for a real system operating in microgravity conditions. As a case study, we consider a mock-up of a satellite-manipulator system operated on a planar air-bearing microgravity simulator. Such simulators are widely used for validation of trajectory planning and control algorithms in the field of space robotics (Rybus and Seweryn, 2016). Demonstration of the practical application of the OVF method for a system operating in simulated microgravity is the main contribution of this paper.
The paper is organised as follows. The dynamic equations of a satellite-manipulator system are presented in Section 2. The modified OVF method is described in Section 3. The obtained results are presented in Section 4 and discussed in Section 5. The conclusions are given in Section 6.
To describe the dynamics of a chaser satellite equipped with a manipulator, we follow the approach based on the Generalised Jacobian Matrix (Umetani and Yoshida, 1989). General equations that describe the dynamics of a satellite-manipulator system were presented by Seweryn and Banaszkiewicz (2008) and by Rybus et al. (2022). In this paper, we present application of these equations to a specific case: mock-up of a satellite-manipulator system operated on the planar air-bearing microgravity simulator at the Space Research Centre of the Polish Academy of Sciences (Basmadji et al., 2019). The mock-up of the chaser satellite is equipped with a 3 DoF manipulator. This system is shown in Figure 1. All joints of the manipulator are rotational.
We use the following vector of generalised coordinates (Junkins and Schaub, 1997):
The state of the chaser satellite equipped with a manipulator is described by the following vector:
The dynamic equations of motion have the following form:
The planar air-bearing microgravity simulator is used to emulate microgravity conditions. It provides frictionless motion of the system, hence the influence of external forces can be neglected (Rybus et al., 2019). No gravity acceleration is acting on the system in its plane of motion. The cold gas thrusters are mounted on the chaser satellite (Kindracki et al., 2017), but in the considered scenario, they are not used during the motion of the manipulator. Thus, we assume that
The gripper is located at the end of the last link of the manipulator and its position in Π
By differentiating (7) and (8) with respect to time, we obtain the expressions for the gripper linear and angular velocity,
The general idea of the OVF method is presented in Figure 1. To perform grasping of a non-cooperative target satellite, the gripper mounted on the manipulator has to be moved to a specific position and orientation with respect to the target object. The goal of the trajectory planning is to find a trajectory of the manipulator that allows to obtain the desired position and orientation of the gripper (
In the approach proposed by Rybus (2022), the virtual attractive force is responsible for driving the gripper towards the desired position. Here, we propose a simple extension of the OVF method that allows to obtain both the desired gripper position and the desired gripper orientation. We follow the approach described by Elahres et al. (2021), and we introduce an attractive torque that acts on the last link of the manipulator and allows to obtain the desired orientation of the gripper. The virtual force and torque that act on the gripper are given by
In the OVF method, each obstacle generates a vector field. The repulsive force that acts on the manipulator results from the vector fields generated by multiple obstacles. Only the collisions between the manipulator links and external obstacles located in the manipulator workspace are taken into account. Self-collisions between the manipulator links and collisions between the manipulator links and the elements of the chaser satellite are not considered. The approach presented in this section is based on the original formulation of the OVF method described by Rybus (2022).
The vector field of the obstacle Γ
The following expression is used to calculate the direction perpendicular to the gradient of the potential field:
If the
The
The motion of the manipulator is directed by the virtual attractive and repulsive forces. The following equation is used to calculate the desired motion of manipulator joints under the influence of these forces:
The control torques that result in the desired motion of the manipulator joints are calculated from the following equation:
To apply the OVF method on a real system, it is necessary to implement a closed-loop controller responsible for trajectory tracking. To control the gripper in the Cartesian space, we use the following control law:
The gripper position and orientation error are multiplied by a constant gain
The practical application of the OVF method is demonstrated in experiments performed on the planar air-bearing microgravity simulator operated at the Space Research Centre of the Polish Academy of Sciences. This test-bed consists of a granite plate (2 m × 3 m). The plate is flat and precisely levelled. The mock-up of the chaser satellite is equipped with the 3 DoF planar manipulator. The mock-up and the manipulator are mounted on planar air-bearings that use a thin film of pressurised air to provide frictionless motion on the surface of the granite plate. The microgravity conditions are simulated in the horizontal plane. The air-bearings are supplied with air from a gas canister located on the chaser mock-up. The closed-loop control system is implemented on the main computer (ATSAMA5D36), which is also located on the chaser mock-up. Direct Current (DC) motors are used to drive the joints of the manipulator. Each joint is equipped with a harmonic drive and an absolute optical encoder. All actuators have separate drivers connected to a CAN-bus. External vision system is used to provide measurements of the gripper position and orientation in Π
Mass and geometrical properties of the manipulator
Length | m | 0.45 | 0.45 | 0.31 |
Mass | kg | 2.81 | 2.82 | 4.64 |
CoM position in Π |
m |
|
|
|
Mass moment of inertia | kg · m2 | 0.0637 | 0.0635 | 0.0515 |
The following scenario was selected for demonstration of the OVF method. The origin of Π
Values of the parameters and the constant coefficients in the OVF method were selected by a trial and error method for the given parameters of the system. The following values were used to solve the presented trajectory planning problem:
The test campaign on the planar air-bearing microgravity simulator consisted of 10 repetitions of the experiment. No real obstacle was used in experiments to avoid the risk of a collision in case of problems with the proper realisation of the planned trajectory. The results of six experiments were selected for further analysis and are presented in this section, while the results of four experiments were rejected due to the occurrence of large external disturbances or problems with data recording. The reference gripper trajectory planned with the OVF method and the trajectories obtained in the experiments are shown in Figure 2 on the XY plane. The X and Y components of the gripper position are shown in Figure 3, where the reference position and the desired position (that corresponds to the position of the selected grasping point) are also presented. The orientation of the gripper is shown on the left panel of Figure 4, while the orientation of the chaser satellite mock-up is shown on the right panel of Figure 4. The angular positions of the manipulator joints measured with encoders during the experiments are presented in Figure 5. These positions are compared with the positions obtained in simulations performed for the reference gripper trajectory. Frames captured by the camera of the external vision system are shown in Figure 6. This camera is mounted above the surface of the granite table. Gripper final positioning errors obtained in numerical simulations for the reference trajectory and in the experiments performed on the air-bearing microgravity simulator are presented in Table 2.
Gripper final positioning errors (absolute values)
Position X | m | 0.0008 | 0.0005 | 0.0078 | 0.0043 |
Position Y | m | 0.0018 | 0.0064 | 0.0171 | 0.0106 |
Orientation | deg | 0.1319 | 0.0999 | 1.2999 | 0.5666 |
The OVF method was able to find the collision-free trajectory in the given scenario. As demonstrated by Rybus (2022), the proposed method is able to solve difficult trajectory planning problems. The selected set of parameters and constant coefficients should allow successful planning in a wide range of conditions. However, there is no simple method and no automatic algorithm for selecting the values of parameters and coefficients in the OVF method. The process of selecting these values using the trial and error method is tedious and lengthy. Finding a collision-free trajectory in a different scenario may require a different set of values. This is a disadvantage of the proposed approach. Moreover, if the solution exists for a given problem, it is not guaranteed that the OVF method will be able to find this solution. In very difficult scenarios, the OVF method and other methods based on potential fields may fail to find a solution due to the fact that the considered system is nonholonomic. In such scenarios, other methods should be used, for example, the RRT algorithm (Rybus, 2020). However, the time required by the RRT method is much longer than the time required by the OVF method (Rybus, 2022).
The reference gripper trajectory obtained with the OVF method in the considered scenario is smooth. The results of the test campaign confirm that the OVF method can be successfully applied for a real system operating in simulated microgravity conditions. As it is evident from Figure 3 and from the left panel of Figure 4, the gripper closely followed the reference trajectory. The closed-loop controller implemented on the chaser satellite on-board computer was able to ensure satisfactory trajectory tracking despite disturbances acting on the system (e.g. force resulting from non-perfect levelling of the granite plate). The delay in the trajectory tracking in the Cartesian space results from the application of a simple proportional control law. The maximal position and orientation errors at the end of the trajectory (Table 2) are small and fall within the tolerance of the gripper. Thus, in every experiment, the gripper reached the position and orientation that would allow to perform a successful capture operation. In none of the experiments, the links of the manipulator moved through the area occupied by the obstacle. Thus, no collision would occur in experiments.
The trajectory tracking was performed in the Cartesian space, and closed-loop controller was compensating the gripper position and orientation errors. Thus, due to disturbances acting on the system, there are moderate differences between the manipulator trajectories in the joint space obtained in individual experiments (Figure 5). The positions of the manipulator joints obtained in the experiments and the positions obtained in the numerical simulation performed for the reference gripper trajectory also differ, but these differences are expected (the system was not following the trajectory defined in the joint space).
The influence of the manipulator motion on the position and orientation of the chaser satellite mock-up is clearly visible in Figure 6. Moderate differences between the chaser satellite mock-up orientation obtained in individual experiments result from the disturbances and from the fact that the trajectories in the joint space were different. The changes of the chaser satellite mock-up orientation measured in the experiments are relatively close to the changes of the orientation obtained in numerical simulations performed for the reference gripper trajectory (the right panel of Figure 4). As the satellite-manipulator system was free-floating, these results show the correctness of the system modelling and the identification of its parameters.
The further development of the OVF method will focus on extending the presented approach to a spatial (three-dimensional) case, in which the motion of the system is not limited to one plane. The three-dimensional case is much more complex, as the chaser satellite could rotate around any axis and its orientation should be described by Euler angles or quaternions. In this case, a three-element vector would describe the desired position of the gripper, while another three-element vector would describe its desired orientation. The vector field would have to be constructed in three dimensions and additional parameters would be required to describe this field. It is also planned to take into account the possibility of self-collisions between the manipulator links. Additional scenarios will be considered to better evaluate the performance of the proposed method.
The OVF method takes into account the fact that in the microgravity conditions, the state of the satellite is influenced by the motion of the manipulator. The dynamic equations of the satellite-manipulator system are used during trajectory planning. The OVF method can be applied to plan a collision-free trajectory of a manipulator in difficult scenarios. The modified version of the OVF method presented in this paper allows to obtain the desired position and orientation of the gripper. To perform the grasping manoeuvre, the gripper has to be positioned in a specific point and aligned with the grasping interface. Thus, the modification introduced in the OVF method allows this method to be applied in the planned ADR missions. The OVF method was used to plan a collision-free trajectory of a 3 DoF manipulator mounted on a chaser satellite mock-up. The practical applicability of the presented method was successfully demonstrated in the experiments performed in simulated microgravity conditions. The gripper followed the reference trajectory despite disturbances acting on the system. The final position and orientation of the gripper obtained in experiments were close to the desired position and orientation required for the capture operation. The OVF method may not be able to find the solution of the trajectory planning problem in some very difficult scenarios. However, this method requires less computational time than the RRT algorithm. Moreover, unlike the RRT algorithm, the OVF method is fully deterministic and it will always produce the same result. This is an important issue when assessing the possibility of the practical application of the OVF method in the future orbital missions.