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International Journal of Advanced Robotic Systems
Repetitive Motion Planning of Free-Floating Space Manipulators
Regular Paper
Gang Chen^*, Long Zhang1, Qingxuan Jia1, Ming Chu1 and Hanxu Sun1
1 School of Automation, Beijing University of Posts and Telecommunications, Beijing, China * Corresponding author E-mail: buptcg@gmail.com
Received 17 Dec 2012 ; Accepted 18 Mar 2013 DOI: 10.5772/56402
© 2013 Chen et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract In this paper, a repetitive motion-planning scheme of free-floating space manipulators is presented. Repetitive motion means when one task ends, the end-effector pose, the joint angles and the base pose should reset (return to their initial value), which will facilitate the subsequent tasks. First, due to the lack of DOF , an order of priority to the given tasks is introduced. Second, the joint reset optimization operator, the base attitude reset optimization operator and the end-effector attitude reset optimization operator are designed. Then, the repetitive motion scheme is proposed by combining the three optimization operators above in a creative way. Finally, to make the optimization of repetitive motion obvious, the base attitude maintenance is also considered. Simulation results verify the correctness and the validity of the repetitive motion planning method of space manipulator.
Keywords Free-Floating Manipulator, On-Orbit Servicing, Task Priority, Repetitive Motion Planning
1. Introduction
In continuous space exploration, the on-orbit operations, such as services, construction and maintenance for
satellites and space stations, are becoming increasingly complex. Considering the space environment, with characteristics such as intense radiation, high temperature difference and super vacuums, it's extremely dangerous for astronauts to do extravehicular activities. Therefore, it is advisable to employ space manipulators to assist or even replace astronauts for on-orbit operation tasks, which can also improve efficiency and reduce expenses.
In order to facilitate on-orbit operations and task assignment, a berthing configuration of space manipulators, which is the initial configuration for any task, is given. In the process of on-orbit operation, the manipulator is not only required to start from its berthing configuration, but also to return to the same one when the task ends to facilitate subsequent task performance. In the spacecraft (base), a berthing attitude, which has a large influence on energy harvesting and communications, is also needed. When space manipulators work in freefloating mode (the position and attitude of the spacecraft are not actively controlled), the disturbance of the spacecraft results from the motion of manipulators. Therefore, the attitude of the spacecraft usually doesn't coincide with its berthing attitude when a task is done, which will have an impact on the stability of the space
manipulator system. However, the adjustment of the spacecraft's attitude consumes relatively large amounts of fuel, limiting the useful life of the system. In other words, the planning and control of space manipulators have some additional problems beyond those on earth due to the dynamic coupling between space manipulators and their spacecraft.
In Cartesian space, the end-effector of a manipulator should track a continuous trajectory or complete a point-to-point manœuvre to meet different task requirements. However, when a task ends, the end-effector of the space manipulator needs to return to its initial pose in either of the conditions above. In other words, a closed trajectory of the end-effector is formed when any task is completed. Meanwhile, in order to reduce energy consumption and meet the requirements of space manipulators performing on-orbit tasks, an appropriate planning method, which relies on the motion of the space manipulator itself, should be adopted to meet the following requirements :
a. The ability of tracking closed trajectory is required.
b. Each joint angle should reset as far as possible when the task ends.
c. The pose (position and attitude) of the spacecraft also should reset as far as possible when the task ends.
Much effort in the research community has been spent on dealing with path planning in space manipulators. Considering requirements a) and c) mentioned above simultaneously, Vafa and Dubowsky (1990) proposed a cyclic motion method that uses the cyclic motion of the manipulator joints to adjust the base attitude [1]. However, this method has to assume that the cyclic motion is small enough to neglect the nonlinearity of an order greater than two, leading to many cycles making a small change in base attitude. Meanwhile, it also requires the total change of base attitude to be an integer multiple of the one produced in every cyclic motion. Dubowsky and Torres (1991) planned the path of a space manipulator with Enhanced Disturbance Map (EDM) technology, which achieves low disturbance of the spacecraft [2]. However, the EDM of high DOF (degrees of freedom) manipulators, is obtained difficultly. Yoshida et al. (2001) put forward the Zero Reaction Manœuvre (ZRM) concept [3], in which the attitude of spacecraft will not be disturbed by the motion of the space manipulator. However, the existence of the ZRM is very limited for a non-redundant manipulator. Papadopoulos et al. (2005) parameterized the joint trajectory using polynomial functions and the trajectory can be adjusted by determining the unknown parameters to make the base achieve its desired attitude and the end-effector of the space manipulator reach its desired position simultaneously [4]. However, this method only takes a 2-DOF manipulator as an example, for space manipulator systems with a higher DOF, it needs further research. Xu
et al. (2007) parameterized the joint trajectory using five-order polynomial functions and normalized the parameterized trajectory for a 6-DOF space manipulator. The unknown parameters are solved by the Iterative Newtonian method and the Particle Swarm Optimization algorithm respectively [5-6]. Both the approaches are effective, the desired pose of the end-effector can be achieved and the planned path is very smooth in controlling the space manipulator. However, the base attitude is not considered in either approach and the Iterative Newtonian method is affected by the initial values of the parameters. To overcome the shortcomings above, Xu et al. (2008) parameterized the joint trajectory using seven-order polynomial functions and solved the unknown parameters using a genetic algorithm and the Particle Swarm Optimization algorithm respectively [7-8]. The proposed approaches can achieve the desired pose of the end-effector and the desired base attitude simultaneously, but the computational cost might be large and the convergence time will be long. Yoshida et al. (1991) proposed a method for using the motion of one manipulator to compensate for the base attitude disturbance created by the motion of the other [9]. It has proved useful, but may augment the cost and control algorithm complexity.
With the purpose of meeting requirements b) and c) synchronously, Nakamura and Mukherjee (1991) employed a bidirectional approach based on the Lyapunov function to plan the path for controlling both the manipulator configuration and the spacecraft attitude
[10]. This method does not neglect the nonlinearity of the space manipulator system and has achieved much success, but the control law fails to prove the stability of the system and smooth feedbacks do not exist. To control the spacecraft attitude and manipulator configuration at the same time, Fernandes et al. (1994) presented a near-optimal path planning approach based on internal motion
[11]. Although the approach can be automated computationally, symbolic manipulation software to obtain the Jacobian matrix is essential to the algorithm.
As to the requirements which satisfy a) and b) synchronously, some scholars proposed the repetitive path planning method. Repetitive path planning means when the end-effector traces a closed path in its workspace, the path in the joint space is also closed. In other words, the joint angles should return to their initial ones. For non-redundant ground manipulators, a closed path of the end-effector in Cartesian space yields a closed path in joint space. However, for redundant manipulators, the conventional resolved motion rate control method will produce drift in joint space when a closed trajectory task is performed in Cartesian space. Of course, we can readjust the manipulators' configuration by the self-motion of redundant manipulators, but this would be inefficient and undesirable.
Klein et al. (1983) first studied the problem of redundant robot repeatability. They used pitch and arc length to predict the joint angle drift and the drift limit [12-13]. Shamir (1988) analysed the question of repeatability of redundant manipulators by considering integral surfaces for a distribution in joint space and judged whether the repeatability could be achieved only by pseudo-inverse in terms of Lie brackets [14]. However, the method can only judge whether the repeatability of a redundant manipulators exists and doesn't give a method for repetitive path planning. Chevallereau (1988) introduced the null space solution to extend the pseudo-inverse kinematics solution to take into account other criteria, which provides the theory basis for repetitive path planning [15]. Baillieul (1985) proposed an Extend Space Method, which can transform the non-square Jacobian matrix into a square matrix by extending the dimensions of the task space. Then the rest of the work is to solve the inverse kinematics solution of a nonredundant manipulator, which can ensure the repeatability [16]. Both the methods that Chevallereau and Baillieul proposed are hard to deduce and compute, especially for a high DOF manipulator system. Zhao et al. (1999) presented a pseudo-inverse Reset Initial Value Approach [17]. This approach can make the joint angles of a 7-DOF manipulator system return to their initial angles when the end-effector task ends. Nevertheless, it cannot eliminate the joint angle drift produced by the first cycle motion ; what's more, the initial value and the time to reset the value have a large influence upon the smoothness of the joint trajectory. Zhang et al. (2008) proposed a primal-dual neural network based on the inequalities of linear variation (LVI) for repetitive motion planning in a PUMA 560 manipulator, which resolves the problem of joint drift. As a quadratic programming (QP) solver, the LVI-based primal-dual neural network involves simple piecewise-linear dynamics and global exponential convergence to optimal solutions [18-19]. Although this method can ensure the repetitive motion of a manipulator, it takes a long time and its complexity is high.
However, for on-orbit tasks, space manipulators should meet requirements a), b) and c) simultaneously. For this kind of problem, no method had been seen so far. In order to meet these requirements, a repetitive motion-planning scheme of free-floating space manipulators is presented in this paper. First, due to the lack of DOF , an order of priority to the given tasks is introduced. Second, the joint reset optimization operator, the base attitude reset optimization operator and the end-effector attitude reset optimization operator are designed. Then, the repetitive motion scheme is proposed by combining the three optimization operators above in a creative way. Finally, to make the optimization of repetitive motion obvious, the base attitude maintenance is also considered. With the proposed method, the free-floating space manipulator can achieve its repetitive motion.
This paper is organized as follows. Section 2 derives the kinematic equations of a space manipulator system.
Section 3 discusses the problem of repetitive motion in space manipulators and designs the optimization operators. Section 4 includes the scheme of the repetitive motion of free-floating space manipulators. In Section 5, the computer simulation results are shown. Section 6 presents the conclusions of the work.
2. Modelling of a Space Manipulator System
2.1 Symbols and Variables
A general model of a space manipulator system with a single arm, which is composed of a n - DOF manipulator and a spacecraft, is shown in Fig. 1. The symbols and variables in Fig. 1 are defined as follows.
Xj: the inertia frame, the subsequent vectors are
described in the frame, if no special instruction
XE : the end-effector frame of the manipulator
Xk(k = 0,...,n): the kth link frame of the space
manipulator system
Ck(k = 0,...,n) : the centroid point of kth link Lk(k = 0,1,...,n): the kth link of the space manipulator system, L0 denotes the spacecraft body Jk(k=1,...,n) : the joint that connects Lk-1 and Lk ak e R3(k=1,.. ,,n) : the position vector from Jk to Ck bk e R3(k=1,...,n): the position vector from Ck to Jk+1, bn denotes the position vector from Cn to the end-effector rk e R3(k=0,...,«) :the position vector of the kth link centroid point Ck
pk e R3(k=1,.,«): the position vector of the kth joint Jk; mRk e R3x3 : the rotation matrix of the frame Xk with respect to the reference frame Xm (when frame Xm is the inertia frame Xj, the superscript m can be missed) rg e R3 : the position vector of the space manipulator system centroid point
pe e R3 : the position vector of the end-effector
Re e R3x3 : the rotation matrix of the end-effector frame
XE with respect to the inertia frame Xj
Figure 1. The general model of a space manipulator system
2.2 Kinematic Equations of a Free-Floating Robotic System
According to the general model of a space manipulator shown in Fig. 1, the position vector pe and the rotation matrix Re of the end-effector can be expressed respectively as:
ME3 Mr0Tg
MÏQg Hw
E R-6,r0g = rg - r0
J Tw Hw^
Pe =r0 +b0 + E (ak + bk)
Re = R0 • 0 R1 •
• n-1Rn • " Re
(1) (2)
Differentiating Eq. (1) and Eq. (2) with respect to time, the linear velocity and angular velocity of the end-effector can be obtained :
where, M represents the mass of the whole system and the specific expressions of matrices Hw, JTw and Hw^ can be obtained in reference [10].
Substitute Eq. (9) and Eq. (10) into Eq. (8) and then take the first three rows of Eq. (8), the linear velocity v0 of the base can be expressed as follows:
= Pe = V0 + «0 X (Pe - r0) + E[kk X (Pe - Pk)] (3)
v0 = -r0T «0 - Jtw (9 = r0 «0 - Jtw ©
0 °g 0 M 0g 0 M
:«0+E M k
In a similar way, according the last three rows of Eq. (8), (4) the relationship between the angular velocity m0 of the base and joint velocity is given:
where v0 and w0 are the linear velocity and angular velocity of base and kk is the unit vector describing the rotation direction of Jk .
Hs«0 + H.© = 0
where the matrices Hs and H0 can be solved as:
T T t /r
Assume the base velocity X0 = [v°,«°] eR and the
T* T T
end-effector velocity Xe = [v°, ^ ] e R , the relationship between the end-effector velocity Xe and joint velocity © = {91,^,0n|T as well as base velocity X0 can be determined:
X e = J0X 0 + Jm©
According to the Eq. (3) and (4), the Jacobian matrices J0 and Jm dependent on the base and manipulator are:
T _| E3 r0e |E R6x6 r _ P r
J0 _[03 E3 |E R 'r0e _ Pe - r0
k1 X (Pe - Pl) k.
k1 X (Pe - Pl) 1E R6xn (7)
where En and 0n respectively represent the n dimension identity matrix and zero matrix, and r is the cross-product operator.
Because no external forces and torques act on the space manipulator system, the linear and angular momentum of the whole system are conserved. Assume that the initial momentum of the system is zero, the momentum equation can be obtained as follows :
H,x0 + H20 _ 0
where, the matrices H1 and H2 are the inertia matrix of the base and the coupling inertia matrix, respectively.
Hs _ Mr0gr0g + Hw He _ Hw^ - Ï0gJTw
It can be proved that the matrix Hs is nonsingular. According to Eq. (12), the following relationship can be obtained:
-H-1He(3 _ J0m
Substitute Eq. (14) into Eq. (11) and the linear velocity v0 of the base is updated as follows:
V0 _-( jM^ + Ï0gH-1He)0 _ J0
By combining Eq. (14) with Eq. (15) the relationship between the velocity X0 of base and the joint velocity © is given:
V0 J0m_v
W0 _ J0m_ffl
0 _ J0m0
Substitute Eq. (16) into Eq. (5) and the general kinematic equation can be obtained :
Xe _ Jg(X0,©) • ©
where, Jg(x0,©) is the Generalized Jacobian Matrix (GJM) [20] and the specific expression is Jg = Jm + J0 • J0m
3. The optimization operators of repetitive motion of space manipulators
3.1 Problem description of repetitive motion of space manipulators
When space manipulators work in free-floating mode, the motion of manipulators will disturb the pose of the base. Thereby, differing from ground manipulators, the repetitive motion of space manipulators requires that the end-effector pose, the base pose and the joint angles return to their initial values, while the end-effector accomplishes a closed trajectory tracking task. However, there is some relationship between the end-effector pose and the base pose, as well as the joint configuration and the analysis is in the following.
The pose coordinate of the end-effector with respect to the inertia frame X and the base frame X0 are denoted as xe = |Pe,^ejT and 0xe = {0Pe,0^ejT, respectively. The base pose coordinate, with respect to the inertia frame Xj , is denoted as x0 = {P0,^0}T . The homogeneous transformation matrixes corresponding to the pose coordinates are:
J Te =
0 R 0 P
0T3x1 1
Then, the relationship described in (18) can be obtained:
Te = T0 • U Te
When the space manipulator is in some configuration, the homogeneous transformation matrix 0Te from the end-effector frame to the base frame can be represented as:
0 Te(©) =
0 Re(0) °Pe(0)
01x3 1
= f(®)
where, © = {^1,-,6»n}T and f(-) means the direct kinematic equation. Eq. (19) shows the relationship between the joint angles and the pose of the end-effector with respect to the base frame X0 .
According to Eq. (19), the following relationship between the pose of the end-effector and the joint angles, as well as the base pose, can be obtained.
Pe = Po + Ro • 0 Pe(®) Re = Ro • 0 Re(®)
On this basis, in order to meet the requirements of repetitive motion of space manipulators, we analyse the optimization object from two aspects: position and attitude. In terms of position, the end-effector of the manipulator has to trace a closed path, so the terminate
coordinate Pe of the end-effector can be determined. However, for redundant manipulators, the uniqueness of the inverse kinematic solution means that a closed trajectory of the end-effector doesn't necessarily yield a closed joint trajectory. Therefore, to achieve the repetitive motion, the joint angles reset has to be an optimization object, namely ©end ^ ©^ (the subscript 'end' means the final value of joint angle and the subscript 'int' means the initial value of joint angle). Thereafter, the relative pose 0 Pe(©) and 0 Re(©) in Eq. (20) can be determined.
From the attitude constraint condition in Eq. (20), we can see that the attitude can be ensured if only R0 or Re is an optimization object. On this basis, R0 becomes a known matrix, thus P0 can be determined, which means the position requirement for repetitive motion of space manipulators is also met.
Through the analysis above, if either of the following conditions is satisfied, the requirements of the repetitive motion of space manipulators can be met.
iConditionl: Pe,© and R0 are optimized
1 Condition2 : Pe, © and Re are optimized
jn other words, the repetitive motion problem of space manipulators can be transformed to either of the two problems.
Problem1: The attitude of the end-effector and the joint angles reset simultaneously, while the end-effector accomplishes a closed trajectory tracking task. Problem2: The base attitude and the joint angles reset simultaneously when the end-effector accomplishes a closed trajectory tracking task.
3.2 The scheme of space manipulators' repetitive motion
Each of the equivalent two problems above has three groups of constraints, which are subjected to n + 6 constraint equations (the end-effector needs 3 constraint equations to accomplish a closed trajectory tracking task, the joint angles need n constraint equations to reset and the attitude of end-effector or the base attitude reset needs 3 constraint equations). Unfortunately, free-floating space manipulators only have n control variables (joint angles) and the lack of control variables makes it very hard to achieve its repetitive motion exactly. Meanwhile, additional algorithmic singularities arise when the constraints above conflict with each other. These conflicts can be handled in the framework of the task-priority strategy by suitably assigning an order of priority to the given tasks and then satisfying the lower priority task only in the null space of the higher priority task [21]. In a typical case, the end-effector task is always considered as the primary task. Combined with the kinematics knowledge we can obtain the non-minimum-norm
solutions to Eq. (17), based on the Jacobian pseudoinverse in the general form:
Meanwhile, the relationship below exists according to the scheme mentioned above:
® = Jgxe + (En - JgJg) • *
^i_int = (^i_c +^i_f)/2 (23)
The virtual range 0i r can be determined by Eq. (24):
2IXe(0l n -0i_tat) + econst ] ^n >3 _e
n i _ int
2 |ki _ e^int -^i_n) + econst ] ^n ^ijnt
where, ki e is the width coefficient of the joint virtual operation range and econst is a constant representing joint offset.
and ei f can be obtained:
e c=e int
ei f =e mt-
where, * is an n -dimensional arbitrary joint velocity vector. In Eq. (21), a homogeneous term, which is obtained by filtering the null-space velocity components of * through the projection operator (En - jgJg), is
added to its minimum-norm solution J+ X . It is worth
noting that during the task performing period, dynamic singularity may arise, which will make the determinant value of matrix Jg JgT approach zero. With regard to this issue, the Damped Least-Squares Scheme [22] is
employed to avoid dynamic singularity by replacing Jg According to Eq. (23) and (24), the expressions of e with JgT ( Jg JgT +^2En)-1 in Eq. (21).
Because null-space velocities produce a change in the configuration of the manipulator without affecting its velocity at the end-effector, they can be exploited to achieve additional goals, for example, the joint angles or the base attitude return to their initial values. * can be chosen as the optimization operator to achieve the goals. Hence, the joint reset optimization operator, the base attitude reset optimization operator and the end-effector attitude reset optimization operator are designed as follows.
3.3 Joint Reset Optimization Operator
In order to make the joint angles return to their initial angles when the end effector accomplishes its closed trajectory tracking task, a joint reset optimization operator is designed based on the Zghal criterion, which was proposed to avoid joint limit [23]. The specific scheme is as follows. Set the virtual operation range of each joint (the range is a subset of the limit operation range of the joint) to make sure the current joint is in the virtual range and close to the limit of the virtual range. Then set the initial joint angle as the intermediate value of the virtual range. Thereafter, using the criterion of avoiding the joint limit to optimize the path in the joint space. The joint angles in every control cycle will approach the intermediate value of the virtual range (their initial ones). As the joint angles approach to the initial ones, the virtual range narrows gradually and the end joint angles will reset when the task ends. On the basis of this scheme, the joint reset optimization function can be written as:
HR(e) = Pr L
(ei c -ei f)2
i=1 (ei c - ei n)(ei n -ei f)
where, ei c represents the maximum value of the ith joint virtual range, ei f represents the minimum value of ith joint virtual range, ei n is the current ith joint angle and pR is the weight coefficient.
It is worth noting that the value of ki e in Eq. (24) is subject to the following two constraints.
1. In order to make the current joint angle ei n in the virtual range, ki e > 1.
2. Considering the optimization effect of the joint reset, the current joint angle needs to be closed to the limit of the virtual range. In other words, the deviation between the current angle and the initial angle should be greater than the one between the current angle and the maximum of the virtual range/the minimum of the virtual range. Combining the two aspects, we suggest 1.2 < ki e< 1.5 .
The joint offset constant econst is designed to prevent the denominator in Eq. (22) approaching zero when ei n - ei int ^ 0 , which will avoid algorithm singularity. However, the value of econst should be chosen reasonably. If the value is too large, it will slow down the optimization speed of joint reset. On the contrary, if it is too small, the joint angle velocity might be too large to obtain a smooth path in joint space. For this reason, it is suggested that the value of econst should be chosen according to joint angle velocity limit.
From Eq. (22) to (24), we can see that the current joint angle is close to the limit of the virtual range in every control cycle, so the value of optimization function HR(e) in Eq. (22) is relatively large. In fact, the value will become smaller if the current joint angle is close to its initial angle. That is to say, the joint angles will return to their initial ones if the value of HR(e) is controlled to be as small as possible in every control cycle. The gradient of
the optimization function can be obtained through the partial derivative of HR(6>).
a j R(0) =
8Hr(0) 8Hr(0) 8Hr(0)
where, 8Hr0) _ (0i_c - 0_f)2(20i_n -0_c -0_f)
(0 c -0i n)2(0 n -0i f)2
Eq. (26) is the expression of the joint reset optimization operator. Substitute it into Eq. (21), and then the optimization of joint configuration reset can be realized.
3.4 Base Attitude Reset Optimization Operator
In order to make the base attitude return to its initial one
when a task ends, the deviation
between the current
base attitude and its initial attitude needs to be optimized to make it as small as possible, namely S$0 ^ 0 . We use Z - Y - X Euler angles to express the attitude; so S$0 can be calculated as follows:
r0 ■
_ Eul(R-1n • R0 int) ^ 0
where Ro_int attitude and R
is the initial rotate matrix of the base 0 n represents the current rotate matrix of the base attitude. On the basis of Eq. (27), we introduce the additional absolute angular velocity w0 f of the base as follows:
°0 f _ To
e(^O) • (<%/ &T)
where, 6T is the control cycle and Ta E(0O) is the transformation matrix from Euler angular velocity to absolute angular velocity. If 0 = |a,p,yj, the matrix Ta E(0) can be expressed as:
T e(0_
0 - sin a cos a cosß
0 cos a sin a cosß
1 0 - sinß
When the deviation S0O is produced by a space manipulator's motion, an optimization operator, which can make the base have an opposite velocity to w0 f needs to be introduced to eliminate the base attitude deviation. According to the relationship between the base angular velocity and the joint angles described in Eq. (14), the base attitude reset optimization operator can be obtained:
a0i R(0) _ J.
0m _ m "'0_f
By substituting this into Eq. (21) the base attitude reset can be achieved.
3.5 End-Effector Attitude Reset Optimization Operator
The end-effector attitude reset optimization is similar to the base attitude reset optimization, the difference is that the optimization of the end-effector attitude reset relies on the relationship between the end-effector angular velocity and the joint angles.
The additional angular velocity we f of the end-effector can be calculated by the deviation S0e between the current attitude and the initial attitude of the end-effector as follows:
j _ Ta_e(«U • (&J ÔT)
On this basis, take out the last three rows of Jg in Eq. (17) to construct a matrix Jg m, which represents the relationship between the angular velocity of the end-effector and the joint angular velocity. Then the end-effector attitude reset optimization operator can be obtained:
(®) _ Jg
So far, we have achieved the joint reset optimization operator, the base attitude reset optimization operator and the end-effector attitude reset optimization operator.
3.6 Repetitive Motion Optimization Operator
According to the analysis above, it is known that the repetitive motion of space manipulators needs to solve either of the problems mentioned in Section 3.1. Therefore, two corresponding repetitive motion optimization operators can be described as the following:
i(®) _ Qj_R * aJ_R(©) + Ç
R * a0f R(0)
2(®) =
J * aJ_R(0) + R * R
(©) (34)
where, Qj_r, and pe0_R represent joint reset
optimization coefficient, base attitude reset optimization coefficient and the end-effector attitude reset optimization coefficient respectively.
4. Repetitive Motion Planning of Space Manipulators
The repetitive motion of space manipulators requires the end-effector pose, the base pose and the joint angles to reset simultaneously when the end-effector accomplishes a closed trajectory tracking task. In the task process, the end-effector tracing a closed trajectory is considered as the primary task and the secondary task is to achieve the repetitive motion. In the motion process of a space manipulator, the conflicts between the end-effector task and the optimization object of repetitive motion may exist. However, at the terminal time, the primary task object of the end-effector position reset coincides with the
second task object of repetitive motion. Therefore, it only needs to introduce the optimization operators during the end phase instead of the whole process.
Aimed at n - DOF space manipulators, its primary task will cost 3 DOF. According to the analysis in Section 3.2, the secondary task requires n + 3 DOF, while in fact it has to rely on the left n - 3 DOF , which will limit the ability of repetitive motion. In order to make the effect of repetitive motion optimization obvious, it is necessary to reduce the drift of joint configuration and the base attitude during the task to relieve the burden of repetitive motion optimization. According to Eq. (21), the primary task is solved by the minimum-norm solution, which will reduce the drift of joint configuration to some extent, while the reduction of the base attitude drift needs to be achieved by the introduction of the base attitude maintenance optimization operator.
In order to achieve the optimization of the base attitude maintenance, substitute Eq. (21) into Eq. (12), and then Eq. (35) can be obtained:
0 < t < T
0.5*(Sin(ri + -) +1) Tr < t < Tm (40)
Tm < t < TA
S0_m '
0.5*(Sin(z"i --) +1) Tr < t < Tm
Tm < t < TZ
0.5*(Sin(r2 + -) +1) Tz < t < Ta
where, ^ = -£<L-!k2, T2 = -fÇ-M and Tm , TR, TZ,
TA denote the start time of repetitive motion, the end time of base attitude maintenance optimization, the start time of ending the repetitive motion optimization and the whole planning time, respectively.
Fig.2 shows the curves of coefficient functions pR1 ( pR2 ) and p,B m .
Hs«0 + H,(Jgxe + (En - JgJgW) = 0 (35)
The solution is:
* = (H,(En - JgJg))+ • (-Hs«0 - H.jgxe) (36)
Considering the requirement of base attitude maintenance, assume «0 = {0,0,0}T and substitute this condition into Eq. (36), then the base attitude maintenance optimization operator is as follows:
= -(H,(En - Jg Jg))+H.Jgxe
On this basis, the optimization of the base attitude maintenance can be achieved by substituting Eq. (37) into
Eq. (21).
Through the analysis above, by substituting Eq. (33) or Eq. (34) and Eq. (37) into Eq. (21) simultaneously we can achieve the repetitive motion planning of space manipulator:
Figure 2. The curves of coefficient functions pRi (pR2 ) and
5. Simulation Study 5.1 Study Object
In order to verify the correctness and the validity of the repetitive motion planning method of space manipulators, without losing generality, we took a 7 DOF space manipulator as an example to carry out the validation. The DH coordinate systems of the studied manipulator are shown in Fig. 3 and the values of the DH parameters and dynamical parameters are listed in Table 1 and Table 2, respectively.
® = Jgxe - (En * JgJg) • (PR1«R1(®) + >
® = Jgxe - (En * JgJg) • (PR2«R2(®) + >
J (38)
J (39)
Considering the fact that we introduce the base attitude maintenance optimization operator in the beginning phase and introduce the repetitive motion optimization operator in the end phase, we design the coefficient functions pR1 ( pR2 ) and p^0 m as follows to ensure the continuity and smoothness of the joint angles:
Figure 3. DH coordinate systems of the studied manipulator
Link i 0 di(m) ai_l(m) ai1
1 0(0) 0 0 90°
2 0,(0) 0.52 0 -90°
3 0,(0) 0 5.5 0
4 04(0) 1.04 5.5 0
5 05(0) 0 0 90°
6 06(0) 0.52 0 -90°
7 07(0) 0.52 0 0
Table 1. The values of DH parameters
5.2 Simulation example
The end-effector is planned to trace a closed circular curve, which is determined by the initial point of the end-effector and any other two points in its workspace. Assume that the inertial frame coincides with the initial base frame and the initial values of variables are set as follows.
x0 = {0,0,0,0,0,0}T is the initial pose coordinate of the base.
0 = {15° ,120°,-100° ,80° ,20° ,10° ,30°} are the initial joint angles and the corresponding initial position coordinate of the end-effector is P1 = {-1.5,-9.0,-1.1} . Any other two points are P2 = {-2,-8.2,-0.7} and P3 = {-1.8, -8.5,-1.2} respectively. Thereafter, a circle with 1.16m diameter is determined by the three points.
The end-effector velocity is planned by a symmetric trapezium curve with parabolic transition to make the acceleration change continuously. Considering the limit of joint velocity, we assume that the whole planning time TA = 60s, the acceleration time Ts=10s and the control cycle 5T=50ms. On this basis, the repetitive motion-planning operator of space manipulators expressed as Eq. (38) is employed to carry out the validation. The related parameters are given as: TR=40s, Tm=45s, TZ=55s,
P] R = 3 and = 0.01.
Base Link 1 Link 2 Link 3 Link 4 Link 5 Link 6 Link 7
Mass (kg) 1000 48 48 72 72 48 48 48
i Pi (m) 0 0.003 0.003 0.992 0.992 0.003 0.003 0.003
0 0.018 0.018 0.011 0.011 0.018 0.018 0.018
0 0.638 0.233 0.393 0.838 -0.313 0.202 0.202
i Ii (kg • m2) xx 800 0.98 0.98 20.95 66.93 0.98 0.98 0.98
yy 800 0.95 0.95 160.86 206.84 0.95 0.95 0.95
zz 800 0.94 0.94 141.62 141.62 0.94 0.94 0.94
I xy 0 0.01 0.01 0.70 0.70 0.01 0.01 0.01
xz 0 -0.01 -0.01 -11.40 -14.71 -0.01 -0.01 -0.01
yz 0 -0.03 -0.03 0.03 0.03 -0.03 -0.03 -0.03
Table 2. The values of dynamical parameters
1.0 0.5
0.5 1.0 1.5
(a) The velocities of joint 1~4 Figure 4. The velocities curves obtained by the method proposed in this paper
(b) The velocities of joint 5~7
The curves of joint velocities during the task are shown in Fig. 4. In light of the figures, we can see that the joint velocities curves are smooth and steady, which means this method can be applied to actual operation.
In addition, in order to prove the validity of the repetitive motion planning method of space manipulators, we compare it with the joint angle minimum-norm method in the following aspects:
When the task ends, the reset deviation of the end-effector pose, the reset deviations of joint angles, the reset deviation of the base pose and during the task, the disturbance of the base attitude (S denotes the simulation results using the method in this paper).
When the task ends, the reset deviation of the end-effector pose is:
ôre = {0.96mm, -8.39mm,0.64mm} ôr* = {4.58mm, 0.31mm, 0.93mm}
= {-0.01° ,0.04° ,0.03°} = {0.16°, -0.23° ,0.38°}
the reset deviations of the joint angles are:
\se = {5.73°, -9.00°, -18.95° ,5.97°, -3.31° ,10.74° ,5.86°} [ Se* = {0.62°,0.06°,-0.82°,0.32°,-0.61°,-0.10°,0.00°}
the reset deviation of the base pose is:
Sr0 = {-50.22mm, 57.06mm, 42.97mm} Sr0* = {-4.65mm, 1.79mm, 1.48mm} Sf0 = {-16.76° ,6.64° ,7.82°} Sf* = {-1.16° ,0.44° ,0.38°}
The disturbance of the base attitude during the task is shown in Fig. 5 (In the legend, M-NM represents the Minimum-Norm Method and RMPM represents the Repetitive Motion Planning Method of Space Manipulator, which is proposed in this paper).
Through comparison with the joint angle minimum-norm method, we can see the advantages of the proposed method. The reset deviations of the end-effector attitude, the joint angles and the base pose are very small, although we loss a little accuracy of the end-effector pose. From the curves in Fig. 5, it can be seen that during the task, the disturbance of the base attitude is very low, which will ensure energy harvest and communications.
b) The disturbance of the base attitude in Y direction
a) The disturbance of the base attitude in X direction
c) The disturbance of the base attitude in Z direction Figure 5. The disturbance of the base attitude during the task
using two methods
Therefore, we can draw the conclusion that the optimization operators designed in this paper are correct and the repetitive motion planning method of a space manipulator proposed in this paper is effective.
6. Conclusions
For free-floating redundant manipulators, when the end-effector traces a closed trajectory in Cartesian space, the path in joint space is usually not a closed one and its end-effector attitude and base pose will also produce drift at the end time, which will present an inconvenience for the subsequent tasks. In order to solve these problems, a repetitive motion-planning scheme of free-floating space manipulators is presented in this paper. With this method, the reset deviations of the end-effector attitude, joint angles and the base pose become very small, although we loss a little accuracy of the end-effector pose. Therefore, the free-floating redundant manipulator can achieve its repetitive motion. In addition, this planning method has the following other advantages:
a. The disturbance of the base attitude during the task is low, which will ensure the energy harvesting and communications.
b. The effective optimization operators can be used to accomplish some other sub-missions.
7. Acknowledgements
This project is supported by the National Key Basic Research Program of China (2013CB733005) and the National Natural Science Foundation of China (Grant No. 61175080).
8. References
[1] Vafa Z and Dubowsky S (1990) On the Dynamics of
Space Manipulator Using the Virtual Manipulator with Application to Path Planning. J. Astronaut. Sci. 38: 441-472.
[2] Dubowskys S and Torres M (1991) Path Planning for
Space Manipulators to Minimize Spacecraft Attitude Disturbance. Proc. IEEE Int. Conf. on Robotics and Automation, Piscataway, NJ: 2522-2528.
[3] Yoshida K, Hashizume K and Abiko S (2001) Zero
Reaction Manoeuvre: Flight Validation with Est-Vii Space Robot and Extension to Kinematically Redundant Arm. Proc. IEEE Int. Conf. on Robotics and Automation, Piscataway, NJ: 441-446.
[4] Papadopoulos E, Tortopidis I, Nanos K (2005) Smooth Planning for Free-Floating Space Robots Using Polynomials. Proc. IEEE Int. Conf. on Robotics and Automation, Barcelona, Spain: 4272-4277.
[5] Xu W. F, Li C, Bin L et al. (2007) Path Planning of Free-
Floating Robot in Cartesian Space Using Direct Kinematics. International Journal of Advanced Robotic Systems 4(1): 17-26.
[6] Xu W F, Li C, Bin L et al. (2008) The Cartesian Path
Planning of Free-Floating Space Robot using Particle Optimization. International Journal of Advanced Robotic Systems 5(3): 301-310.
[7] Xu W F, Liu Y, Liang B, et al. (2008) Non-holonomic
Path Planning of a Free-Floating Space Robotic System Using Genetic Algorithm. Advanced Robotics 22: 451-476.
[8] Xu W F, Li C, Wang X Q, et al. (2009) Study on Non-
holonomic Cartesian Path Planning of a Free-Floating Space Robotic System. Advanced Robotics 23: 113143.
[9] Yoshida K, Hashizume R, Umetani Y (1991) Dual Arm
Coordinate in Space Free-Floating Robot. Proc. IEEE Int. Conf. on Robotics and Automation, Piscataway, NJ: 2516-2521.
[10] Nakamura Y, Mukherjee R (1991) Nonholonomic Path Planning of Space Robots via a Bidirectional Approach. IEEE Trans. Robotics Automat 7: 500-514
[11] Fernandes C, Gumits L, Li Z X (1994) Near-optimal Nonholonomic Motion Planning for a System of
Coupled Rigid Bodies. IEEE Trans. On Automatic Control 39(3): 50-63.
[12] Klein C A, Huang C H (1983) Review of Pseudoinverse Control for Use with Kinematically Redundant Manipulators. IEEE Trans SMC 13(3): 245-250.
[13] Klein C A, Kee K B (1989) The Nature of Drift in Pseudoinverse Control of Kinematically Redundant Manipulators. IEEE Trans. On Robotics and Automation 5(2): 231-234.
[14] Shamir Y Y (1988) Repeatability of Redundant Manipulators: Mathematical Solution of the Problem. IEEE Trans Automatic Control, AC-33: 1004-1009.
[15] Chevsllereau W. K (1988) A New Method for the Solution of the Inverse Kinematics of Redundant Robots. IEEE Int Conf Robot Automat: 34-42.
[16] Baillieul J (1985) Kinematic Programming Alternatives for Redundant Manipulators. IEEE Int Conf Robot Automat: 722-728.
[17] Zhao J, Bai S X (1999) The Study on Joint Motion Repeatability of Spatial 7R Redundant Robots. Mechanical Design 2: 25-27.
[18] Zhang Y N, Lv X J, Li Z H (2008) Repetitive Motion Planning of PA10 Robot Arm Subject to Joint Physical Limits and Using LVI-based Primal-Dual Neural Network. Mechatronics 18(9): 475-485.
[19] Zhang Y N, Guo D S, Cai B H et al. (2011) Analysis and Verification on Repetitive Motion Planning Scheme of Redundant Manipulators Using New Performance Index, Journal of Wuhan University of Technology 35(1): 67-71.
[20] Umetani Y, Yoshida K (1989) Resolved Motion Rate Control of Space Manipulators with Generalized Jacobian Matrix, IEEE Trans. On Robotics and Automation 5: 303-314.
[21] Chiaverini S (1997) Singularity-Robust Task-Priority Redundancy Resolution for Real-Time Kinematic Control of Robot Manipulators. IEEE Trans. On Robotics and Automation 13(3): 398-410.
[22] Stefano C, Bruno S, Olav E (1994) Review of the Damped Least-Squares Inverse Kinematics with Experiments on an Industrial Robot Manipulator. IEEE Trans. On Control System Technology 2(2): 123134.
[23] Zghal H, Dubey R V, Euler J A (1990) Efficient Gradient Projection Optimization for Manipulators with Multiple Degrees of Redundancy. IEEE Trans. On Robotics and Automation 2: 1006-1011.
