The energy of a space station is a precious resource, and the minimization of energy consumption of a space manipulator is crucial to maintain its normal functionalities. This paper first presents novel gaits for space manipulators by equipping a new gripping mechanism. With the use of wheels locomotion, lower energy demand gaits can be achieved. With the use of the proposed gaits, we further develop a global path planning algorithm for space manipulators which can plan a moving path on a space station with a minimum total energy demand. Different from existing approaches, we emphasize both the use of the proposed low energy demand gaits and the gaits composition during the path planning process. To evaluate the performance of the proposed gaits and path planning algorithm, numerous simulations are performed. Results show that the energy demand of both the proposed gaits and the resultant moving path is also minimum.
The application of robots in space exploration becomes more advanced. For extravehicular activities (EVA) in low Earth orbit, most of them are designed as a chainlike manipulator which is suitable in performing tasks such as mechanical assistance, capturing operations, monitoring, and satellite maintenance [
For the conventional long limbs design of space manipulators, a larger joint torque is required to drive the arm’s swing motion in conventional gaits. As a result, these conventional gaits are regarded as a high power demand motion. Since the energy source of a manipulator is from the attached space station, the minimization of energy consumption of a space manipulator is crucial to maintain normal functionalities of a space station. Without modifying the joint configuration of existing manipulators, we propose novel gaits for space manipulators by equipping a new gripping mechanism which combines the wheels locomotion in conventional grippers. For linear locomotion, it can eliminate the swing motions in conventional gaits. For turning motion, the range of swing motions is reduced by the wheels locomotion. Different from the Mobile Transporter of Mobile Servicing System (MSS), the proposed concept allows a manipulator to transit between trusses. In fact, several benefits can also be achieved. For example, the complexity of docking process during the exchange of gripping base can be simplified. As a result, the navigation efficiency can be enhanced. Also, the additional redundancy of the manipulator’s mobility allows a fine positioning capability when it moves on a truss.
Based on our previous research, we developed a gripping mechanism, Movable Gripper (MovGrip), which combines the concepts of active wheels in traditional parallel grippers [
With the use of the proposed gaits, we further develop a global path planning algorithm for space manipulators which can plan a moving path on a space station with a minimum total energy demand. For conventional path planning algorithms, they are highlighted by many researchers in the category of planar mobile robots. Fernandez et al. proposed a path planning algorithm for mobile robot navigation with the use of a multicriteria path planar in 1999 [
For the above algorithms, they mainly focus on the path planning problems whose workspace is limited on a 2D plane. Although the properties of a mobile robot and its working environment are studied, the robot’s orientation after each move is not considered. For instance, space manipulators possess a 3D workspace. For a given current and target points pair, different manipulator’s orientations on a truss can result in a different energy demand to reach the target point. Also, most of their works focused on the minimization of the path length as the energy consumption optimization. However, two paths with the same path length may have different energy demands. In fact, it depends on the motions composition in reaching a target position. Since the energy demand of each individual locomotion of a mobile robot can be different, the search of a solution path should also consider the composition of motion primitives.
Besides the planar mobile robots, the optimization problem is also addressed by researchers in climbing robots. For example, Balaguer proposed a path planning algorithm for ROMA which is a frame climbing robot developed to travel into complex metallicbased environments for inspection, assembly, and maintenance [
In this paper, the proposed path planning algorithm considers the composition of the proposed gaits in planning a moving path and the optimization is achieved by GA. To enhance the performance of the conventional GA, a novel genetic operator, PlanarNN is proposed and the design of PlanarNN, is based on the feature of the energy demands of the proposed gaits. The working principle of PlanarNN is to search a solution path with more motion primitives which contribute a lower energy demand. With this, the total energy demand of a solution path not only can be reduced by the use of motion primitives with a minimum individual energy demand, but also can further be minimized by focusing on the composition of motion primitives in the path.
The remaining of this paper is organized as follows. Section
In this section, the design of a novel frame climbing robot (Frambot) is presented which aims to give a brief introduction on the design and use of a movable gripper in truss climbing. The working principle of the movable gripper on Frambot is first described. The major specifications of Frambot are also summarized. After that, the climbing motions of Frambot are illustrated. Details of the design and performance of Frambot are presented in [
Figure
The specifications of frambot.
Weight  1.71 kg 
Dimension 
(72.5, 21, 14.5) cm 
Gripping width  (3.0–4.5) cm 
Maximum gripping force 

Steering angle of wheels  5 degrees 
Diameter of wheels  2 cm 
Vertical climbing speed  2.31 cm/s 
Horizontal climbing speed  4.05 cm/s 
Upsidedown climbing speed  6.2 cm/s 
The prototype of Frambot.
The structure of MovGrip is composed of two major parts, (a) gripper jaw and (b) parallel gripping mechanism (Figure
The design of MovGrip.
For the design of conventional space manipulators, their end effector is mainly designed for grasping and different locomotion can only be performed by the swing motion of their body linkage. With the use of MovGrip, wheels locomotion can be utilized to minimize the arm’s swing motion. For linear locomotion, it is performed by the rotation of wheels in MovGrip and the arm’s swing motion in conventional turnaround and inchworm gaits are therefore eliminated. Figures
The turning motion of Frambot.
The exterior transition of Frambot.
One of the major contributions in this paper is to propose new gaits for space manipulators with an open chain configuration. For its gripping mechanism, it is assumed to be the concept presented in previous section. By this, the gripper not only provides a grasp on a truss, but also allows a linear locomotion along the truss by the embedded active wheels. Under this configuration, both conventional and proposed gaits can be realized and compared unbiasedly. For simplicity, we focus on the motion of a cubic truss structure which is symmetric about its principal axes. With this, the moment of inertia about the three axes is identical which simplifies the energy demand analysis.
Aforementioned, a space station is first regarded as a cubic truss. For the locomotion on a truss, not all the joints of a sevenDOF space manipulator are involved. Therefore, the body of a space manipulator is defined as a fiveDOF chainlike manipulator with its two ends equipped with MovGrip (Figure
The joint configuration of the manipulator.
For linear locomotion, the most commonly used gaits are turnaround and inchworm gaits [
The gait for turn around gait.
For inchworm gait, it is realized by using the three pitch joints (Figure
The gait for inchworm gait.
For the proposed locomotion, we propose the use of wheels locomotion to reduce the swing motions of a space manipulator. Since the manipulator does not move with respect to the gripper, the arm’s swing motion is eliminated. Hence, the energy demand is converted to the rotation of wheels in the grippers.
Figure
The conventional gait for turning.
Figure
The proposed turning motion.
As an alternative of the turning gait, space manipulators can perform inchworm gait to replace the wheels locomotion in the proposed turning gait. Since the notion of inchworm gait is illustrated before, the use of inchworm gait in truss turning is not repeated here. When compared with the conventional turning gaits, there are two common motion steps which are the target posture of the motion steps (c) and (f). For the proposed turning gait, it uses wheels locomotion to reduce the rotation of roll joints in conventional gaits.
Figure
The conventional gait for exterior transition.
Figure
The proposed gait for exterior transition.
Similar to the turning gait, space manipulators can also perform inchworm gait to replace the wheels locomotion in the proposed transition gait and it is not illustrated here. Also, there are two common motion steps for the conventional and proposed transition gaits which are the target posture of the motion steps (Figures
For the energy demand of a gait, it is defined as the energy required to perform that gait. For the joint’s motion, the energy demand, “
Similarly, the energy demand, “
The total energy demand of a gait can be computed as the sum of all individual energy demands. To compare the energy demand of different gaits of a space manipulator, the kinematics, momentums, and dynamics of a space robotic system provide the fundamental approach. Because of this, this section first presents the modeling of a space robotic system. After that, the energy demand of different gaits is analyzed.
Since the formulation of the dynamic coupling effect of a space manipulator with one end fixed on a space station is commonly established [
Figure
A general model of a space robotic system.
For the system modeling, three coordinate frames are first introduced which include (a) inertial frame, (b) manipulator frame, and (c) base frame. For the inertial frame, its origin is fixed in the space. For the manipulator frame and base frame, their origins are located at the mass center of the manipulator and base, respectively. During the modeling, a number of notations are defined as follows:
: joint variable vector (
All vectors are described with respect to the inertial coordinate system unless another coordinate system is mentioned and
In addition, several assumptions are made. First of all, all links are rigid bodies and there are no external forces and torques acting on the system. As a result, the total potential energy of the system is equal to zero. During wheels locomotion, it is assumed that the configuration of the manipulator is maintained (i.e.,
To model the kinematics of the system, the manipulator is first visualized as a prismatic joint and the closed MovGrip is the gripping point on the base. During wheels locomotion, the resultant wheels traction becomes the pushing force of the prismatic joint acting on the base and the gripping point is moving together with wheels locomotion. Since the manipulator does not move with respect to the gripper, the position vector of the manipulator’s mass center can be formulated as follows:
By taking time derivative on both sides, we can derive the system’s kinematics relationship as follows:
where
Aforementioned, it is assumed that there are no external forces and torques acting on the system, and the linear and angular momentums of the system are conserved as
From the kinematics relationship, we express the variables
For the system’s dynamic modeling, the conventional Lagrangian approach is utilized. Since the total potential energy of the system is zero, the total energy of the system during wheels locomotion is the sum of translational and rotational kinetic energy (
Similar to the momentum equations, the energy equations can be expressed in terms of
The matrix
where
For the analysis of energy demand of the proposed gaits, it is assumed that there are no external forces and torques acting on the system. For the space station, it is assumed to be a cubic truss structure which gives a symmetric property for simulations. Length of a side of the truss is 10 meters and the presented system dynamic model is utilized to compute the energy demand.
Both the proposed and conventional gaits are simulated and three different aspects are studied including (a) linear locomotion, (b) turning, and (c) exterior transition. For each of the three aspects, three gaits are considered including (a) turnaround gait, (b) inchworm gait, and (c) wheels locomotion. For fair comparison, the moving distance of different gaits in each of the three aspects is equal. Also, the time to perform different gaits in each of the three cases is fixed. Table
The system parameters for simulations.

0.5 m 



0.5 m 



2 m 



2 m 



0.5 m 



0.5 m 



483.7 kg 

15 kg 

10 kg 

10 kg 

10 kg 

15 kg 
Figure
The space station, base frame, and robot initial and final configuration for wheels locomotion.
For the turning gaits at a truss corner, only the turn left gait is discussed and it is performed from edge A to edge B of the space station in Figure
Figures
The joints’ displacement profiles for the inchworm gait.
The joints’ torque profiles for the inchworm gait.
By using (
The total energy demand of different gaits.
Inchworm  Turn around  Wheels locomotion  

Linear  2.0458 J/m  1.0097 J/m  0.0424 J/m 
Turning  41.2473 J  4.6744 J  3.1689 J 
Exterior transition  17.3267 J  9.3039 J  3.7817 J 
From the nature of the gaits, the arm folding and unfolding motions of the inchworm gait on the one hand require the largest joints movement when compared with the others. On the other hand, the distance travel in one motion step is the smallest. Given a certain operation time to complete the gait, the joints’ torque and angular velocity are much larger than the others. Under a constraint on the operation time, the inchworm gait is regarded as a high power demand motion.
For the wheels locomotion, the wheels’ traction force is in the same direction as the movement. Also, all the mass is moving along a straight line during wheels locomotion. With a long limb design of conventional space manipulator, the use of wheels locomotion can avoid the arm’s swing motion whose joints torque is high. Therefore, the use of wheels locomotion to reduce the joints movement allows an energy efficiency locomotion on a space station.
For the proposed minimum energy demandbased path planning algorithm, the optimization goal is first defined. Given the robot’s initial position and orientation, a list of check points (including the target point) and their position with respect to an inertial frame, the objective is to search a path which passes all the check points once and reaches the target point with a minimum energy demand. This can be visualized as a space manipulator is commanded to follow the solution path and visit a list of points for monitoring or maintenance tasks on a space station. Finally, it is commanded to reach a target point for charging or standby.
Figure
The geometry of a space station.
In conventional GA, a chromosome is the basic element for genetic evolution. For the proposed algorithm, a chromosome represents the path for navigation. It is composed of
In GA, chromosomes with a higher fitness (lower energy demand) with respect to the optimization goal are selected for genetic evolution. To measure the fitness of a chromosome, an evaluation function should be defined. In this algorithm, the optimization goal is to search an optimal path with a minimum energy demand. The fitness function is therefore defined as the total energy demand of the path. Since a chromosome is composed of genes representing the order of check points being visited, the total energy demand of a path is the summation of the individual energy demand of all consecutive gene pairs along the chromosome. Figure
An example illustrating the computation of individual demand.
For the energy demand of a pair of check points, there are three major possible cases. The first one is the case when the line connecting them is parallel to one of the inertial frame axes. The points pair
Assumed that the manipulator is currently at the check point
From the energy demands of the motion primitives in Table
Figure
The genes pair forms a line which is parallel to one of the inertial frame axes.
The genes pair forms a plane which is parallel to the plane formed by any two of the inertial frame axes.
The idea of the proposed operator, PlanarNN.
From Figure
The design of PlanarNN is a greedy search for an optimal solution. To avoid a local optimal solution, three conventional genetic operators are involved in the evolution process. They are (a) genes flip, (b) genes swap, and (c) genes slide. For the working principle of these conventional operators, it is illustrated in [
In this section, the performance of the proposed algorithm is evaluated by simulating a path planning problem on a structural frame. To evaluate the algorithm under different aspects, different simulations are performed. Forty points are first randomly generated on a cubic frame structure and the edge length is 40 meters. It is divided into many smaller cells each with 1 meter long (
The key parameters for the simulations.
Robot initial position  (4, 29, 4) 
Robot initial head vector  (1, 0, 0) 
Robot initial normal vector  (0, 0, 1) 
Robot final position  (31, 26, 25) 
Maximum number of generation  1000 
Number of experiments  5 
The first simulation aims to show the effect of different population sizes on the final solution path. The conventional genetic operators are utilized during the genetic evolution. At the same time, the maximum population size is varied. The simulation is repeated five times and the mean total energy demand of the best solution path is shown in Figure
The total energy demand of the final solution with different population sizes.
In general, the total energy demand of the best solution path searched by the algorithm decreases when the population size increases. It is reasonable because more chromosomes are involved in searching the global optimal path. From the simulation results, the best population size is selected as 80.
In the second simulation, the proposed genetic operator, PlanarNN, is utilized to search a solution path and the result is compared with the results of using conventional genetic operators. The simulation is repeated 10 times and the mean of the ten simulation results is shown in Figure
The effect of different genetic operators on the best path given by the proposed algorithm.
From Figure
A comparison of convergent rate in the first 400 generations.
Fitness at 


0 th  100 th  200 th  300 th  400 th  
Conventional  314.78  254.57  241.46  237.54  234.42 
Proposed  314.78  237.49  235.23  232.55  229.47 
When PlannarNN is involved for genetic evolution, the total energy demand of a path represented by the current best chromosome is decreased below 255 joule in the first 100 generation. Also, the evolution process keeps improving the fitness of the current fittest chromosome and the result is closed to the optimal solution after 400 generations with a variation within 11 joule. To save the computation time, the evolution process can be terminated if the total energy demand of the current best path is satisfied.
To explain these, the nature of the PlanarNN and the hybrid use of genetic operators are considered. The design of PlanarNN is a greedy local search. It groups the genes with a planar or linear relationships. Therefore, the use of PlanarNN can result in a solution path with segments of planar or linear check points. Also, the hybrid use of conventional genetic operators can maintain the concepts of natural evolution in GA and so local optimums can be avoided. Hence, the proposed algorithm can facilitate the search of a minimum energy demand path. Also, it verifies that the composition of motion primitives in a motion path is also important in energy demand minimization. Based on the simulation results, the total energy demand of the best solution path given by the proposed algorithm is 218.79 joule and the corresponding solution path is displayed in Figure
The best path given by the proposed algorithm.
In this paper, we propose the design of new gaits for space manipulators with a lower energy demand which is the major contribution of this research. Based on the system dynamic model, different gaits are simulated and the corresponding energy demand is analyzed. Results show that the proposed gaits require a lower energy demand when compared with conventional gaits. With the use of the proposed gaits, a minimum energy demandbased path planning algorithm is also developed. Different from traditional approaches, the proposed algorithm considers the composition of motion primitives in a path for the energy demand minimization. Based on the framework of conventional GA, we adopted the concept of genetic modification to design a new genetic operator, PlanarNN. Based on the simulation results, it benefits the final solution in terms of fast convergent rate and lower energy demand. Also, it verifies that the composition of motion primitives in a path is also important in energy demand minimization. Although this paper assumes a space station as a cubic truss structure which is different from the existing space station such as International Space Station (ISS), the geometry of the cubic truss can be modified and the concept of nonfeasible edges can be introduced by highly increasing the energy demand of those edges. For the future development, the proposed algorithm will be able to handle uncertainties such as infeasible path and obstacle avoidance.