Repetitive Motion Planning of Free-Floating Space Manipulators

Abstract

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 free-floating 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:

The ability of tracking closed trajectory is required.

Each joint angle should reset as far as possible when the task ends.

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 non-redundant 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.

Figure 1.

The general model of a space manipulator system

Σ_I: the inertia frame, the subsequent vectors are described in the frame, if no special instruction

Σ_E: the end-effector frame of the manipulator

Σ_k (k = 0, …, n): the k^th link frame of the space manipulator system

C_k (k = 0, …, n): the centroid point of k^th link

L_k (k = 0,1, …, n): the k^th link of the space manipulator system, L₀ denotes the spacecraft body

J_k(k=1, …, n): the joint that connects L_k−1 and L_k

a_k ∈ R³(k=1, …, n): the position vector from J_k to C_k

b_k ∈ R³(k=1, …, n): the position vector from C_k to J_k+1,

b_n denotes the position vector from C to the end-effector

r_k ∈ R³(k=0, …, n): the position vector of the k^th link centroid point C_k

p_k ∈ R³(k=1, …, n): the position vector of the k^th joint J_k; ^mR_k ∈ R^3×3: the rotation matrix of the frame Σ_k with respect to the reference frame Σ_m (when frame Σ_m is the inertia frame Σ_I, the superscript m can be missed)

r_g ∈ R³: the position vector of the space manipulator system centroid point

p_e ∈ R³: the position vector of the end-effector

R_e ∈ R^3×3: the rotation matrix of the end-effector frame Σ_E with respect to the inertia frame Σ_I

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 p_e and the rotation matrix R_e of the end-effector can be expressed respectively as:

p_{e} = r_{0} + b_{0} + \sum_{k = 1}^{n} (a_{k} + b_{k})

(1)

R_{e} = R_{0} \cdot^{0} R_{1} \cdot \dots \cdot^{n - 1} R_{n} \cdot^{n} R_{e}

(2)

Differentiating Eq. (1) and Eq. (2) with respect to time, the linear velocity and angular velocity of the end-effector can be obtained:

v_{e} = {\dot{p}}_{e} = v_{0} + ω_{0} \times (p_{e} - r_{0}) + \sum_{k = 1}^{n} [k_{k} \times (p_{e} - p_{k})]

(3)

ω_{e} = ω_{0} + \sum_{k = 1}^{n} k_{k} {\dot{θ}}_{k}

(4)

where v₀ and ω₀ are the linear velocity and angular velocity of base and k_k is the unit vector describing the rotation direction of J_k.

Assume the base velocity ${\dot{x}}_{0} = {[v_{0}^{T}, ω_{0}^{T}]}^{T} \in R^{6}$ and the end-effector velocity ${\dot{x}}_{e} = {[v_{e}^{T}, ω_{e}^{T}]}^{T} \in R^{6}$ , the relationship between the end-effector velocity ẋ_e and joint velocity $\dot{Θ} = {{\dot{θ}}_{1}, \dots, {\dot{θ}}_{n}}^{T}$ as well as base velocity ẋ₀ can be determined:

{\dot{x}}_{e} = J_{0} {\dot{x}}_{0} + J_{m} \dot{Θ}

(5)

According to the Eq. (3) and (4), the Jacobian matrices J₀ and J_m dependent on the base and manipulator are:

J_{0} = (\begin{matrix} E_{3} & - {\hat{r}}_{0 e} \\ 0_{3} & E_{3} \end{matrix}) \in R^{6 \times 6}, r_{0 e} = p_{e} - r_{0}

(6)

J_{m} = (\begin{matrix} k_{1} \times (p_{e} - p_{1}) & \dots & k_{1} \times (p_{e} - p_{1}) \\ k_{1} & \dots & k_{1} \end{matrix}) \in R^{6 \times n}

(7)

where E_n and 0_n respectively represent the n dimension identity matrix and zero matrix, and r̂ 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_{1} {\dot{x}}_{0} + H_{2} \dot{Θ} = 0

(8)

where, the matrices H₁ and H₂ are the inertia matrix of the base and the coupling inertia matrix, respectively.

H_{1} = [\begin{matrix} M E_{3} & M {\tilde{r}}_{0 g}^{T} \\ M {\tilde{r}}_{0 g}^{​} & H_{w} \end{matrix}] \in R^{6 \times 6}, r_{0 g} = r_{g} - r_{0}

(9)

H_{2} = [\begin{matrix} J_{T w} \\ H_{w ϕ} \end{matrix}] \in R^{6 \times n}

(10)

where, M represents the mass of the whole system and the specific expressions of matrices H_w, J_Tw and H_wϕ 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:

v_{0} = - {\tilde{r}}_{0 g}^{T} ω_{0} - \frac{J_{T w}}{M} \dot{Θ} = {\tilde{r}}_{0 g}^{​} ω_{0} - \frac{J_{T w}}{M} \dot{Θ}

(11)

In a similar way, according the last three rows of Eq. (8), the relationship between the angular velocity ω ₀ of the base and joint velocity is given:

H_{s} ω_{0} + H_{θ} \dot{Θ} = 0

(12)

where the matrices H_s and H_θ can be solved as:

{\begin{matrix} H_{s} = M {\tilde{r}}_{0 g} {\tilde{r}}_{0 g} + H_{w} \\ H_{θ} = H_{w ϕ} - {\tilde{r}}_{0 g} J_{T w} \end{matrix}

(13)

It can be proved that the matrix H_s is nonsingular. According to Eq. (12), the following relationship can be obtained:

ω_{0} = - H_{s}^{- 1} H_{θ} \dot{Θ} = J_{0 m_ω} \dot{Θ}

(14)

Substitute Eq. (14) into Eq. (11) and the linear velocity v₀ of the base is updated as follows:

v_{0} = - (\frac{J_{T w}}{M} + {\tilde{r}}_{0 g} H_{s}^{- 1} H_{θ}) \dot{Θ} = J_{0 m_v} \dot{Θ}

(15)

By combining Eq. (14) with Eq. (15) the relationship between the velocity ẋ₀ of base and the joint velocity $\dot{Θ}$ is given:

{\dot{x}}_{0} = [\begin{matrix} v_{0} \\ ω_{0} \end{matrix}] = [\begin{matrix} J_{0 m_v} \\ J_{0 m_ω} \end{matrix}] \dot{Θ} = J_{0 m} \dot{Θ}

(16)

Substitute Eq. (16) into Eq. (5) and the general kinematic equation can be obtained:

{\dot{x}}_{e} = J_{g} (x_{0}, Θ) \cdot \dot{Θ}

(17)

where, J_g (x₀, Θ) is the Generalized Jacobian Matrix (GJM) [20] and the specific expression is J_g = J_m + J₀ · J_0m

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 Σ_I and the base frame Σ₀ are denoted as x_e = {P_e, ϕ_e}^T and ⁰x_e = {⁰P_e, ⁰ϕ_e}^T, respectively. The base pose coordinate, with respect to the inertia frame Σ_I, is denoted as x₀ = {P₀, ϕ₀}^T. The homogeneous transformation matrixes corresponding to the pose coordinates are:

T_{e} = [\begin{array}{l} R_{e} & P_{e} \\ 0_{3 \times 1}^{T} & 1 \end{array}],^{0} T_{e} [\begin{array}{l} ^{0} R_{e} & ^{0} P_{e} \\ 0_{3 \times 1}^{T} & 1 \end{array}], T_{0} [\begin{array}{l} R_{0} & P_{0} \\ 0_{3 \times 1}^{T} & 1 \end{array}]

Then, the relationship described in (18) can be obtained:

T_{e} = T_{0} \cdot^{0} T_{e}

(18)

When the space manipulator is in some configuration, the homogeneous transformation matrix ⁰T_e from the end-effector frame to the base frame can be represented as:

^{0} T_{e} (Θ) = [\begin{matrix} ^{0} R_{e} (Θ) & ^{0} P_{e} (Θ) \\ 0_{1 \times 3} & 1 \end{matrix}] = f (Θ)

(19)

where, Θ = {θ₁, ⃛, θ_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 Σ₀.

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.

{\begin{array}{l} P_{e} = P_{0} + R_{0} \cdot^{0} P_{e} (Θ) \begin{matrix} ​ \end{matrix} \\ R_{e} = R_{0} \cdot^{0} R_{e} (Θ) \begin{matrix} \begin{matrix} ​ & ​ \end{matrix} \end{matrix} \end{array}

(20)

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 P_e 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 Θ_end → Θ_int (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 ⁰P_e(Θ) and ⁰R_e(Θ) 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 R_e 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.

{\begin{matrix} Condition1 : P_{e}, Θ \begin{matrix} ​ \end{matrix} and \begin{matrix} ​ \end{matrix} R_{0} \begin{matrix} ​ \end{matrix} are \begin{matrix} ​ \end{matrix} optimized \\ Condition2 : P_{e}, Θ \begin{matrix} ​ \end{matrix} and \begin{matrix} ​ \end{matrix} R_{e} \begin{matrix} ​ \end{matrix} are \begin{matrix} ​ \end{matrix} optimized \end{matrix}

In 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:

\dot{Θ} = J_{g}^{†} {\dot{x}}_{e} + (E_{n} - J_{g}^{†} J_{g}) \cdot ϕ

(21)

where, ϕ 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 ϕ through the projection operator (E_n−J_g⁺J_g)·φ, is added to its minimum-norm solution $J_{g}^{†} {\dot{x}}_{e}$ . It is worth noting that during the task performing period, dynamic singularity may arise, which will make the determinant value of matrix J_gJ_g^T approach zero. With regard to this issue, the Damped Least-Squares Scheme [22] is employed to avoid dynamic singularity by replacing J_g with J_g^T(J_gJ_g^T + λ²E_n)⁻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. ϕ 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:

H_{R} (θ) = ρ_{R} \sum_{i = 1}^{n} \frac{{(θ_{i_c} - θ_{i_f})}^{2}}{(θ_{i_c} - θ_{i_n}) (θ_{i_n} - θ_{i_f})}

(22)

where, θ_{i_c} represents the maximum value of the i^th joint virtual range, θ_{i_f} represents the minimum value of i^th joint virtual range, θ_{i_n} is the current i^th joint angle and ρ_R is the weight coefficient.

Meanwhile, the relationship below exists according to the scheme mentioned above:

θ_{i_int} = (θ_{i_c} + θ_{i_f}) / 2

(23)

The virtual range β_{i_r} can be determined by Eq. (24):

θ_{i_r} = {\begin{matrix} 2 [k_{i_θ} (θ_{i_n} - θ_{i_int}) + θ_{c o n s t}] \begin{matrix} ​ & θ_{i_n} > θ_{i_int} \end{matrix} \\ 2 [k_{i_θ} (θ_{i_int} - θ_{i_n}) + θ_{c o n s t}] \begin{matrix} ​ & θ_{i_n} \leq θ_{i_int} \end{matrix} \end{matrix}

(24)

where, k_{i_θ} is the width coefficient of the joint virtual operation range and θ_const is a constant representing joint offset.

According to Eq. (23) and (24), the expressions of θ_{i_c} and θ_{i_f} can be obtained:

{\begin{matrix} θ_{i_c} = θ_{i_int} + \frac{θ_{i_r}}{2} \\ θ_{i_f} = θ_{i_int} - \frac{θ_{i_r}}{2} \end{matrix}

(25)

It is worth noting that the value of k_{i_θ} in Eq. (24) is subject to the following two constraints.

In order to make the current joint angle θ_{i_n} in the virtual range, k_{i_θ} > 1.

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 ≤ k_{i_θ} ≤ 1.5.

The joint offset constant θ_const is designed to prevent the denominator in Eq. (22) approaching zero when θ_{i_n} – θ_{i_int} → 0, which will avoid algorithm singularity. However, the value of θ_const 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 θ_const 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 H_R(θ) 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 H_R(θ) 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 H_R(θ).

α_{J_R} (θ) = {\frac{\partial H_{R} (θ)}{\partial θ_{1}}, \frac{\partial H_{R} (θ)}{\partial θ_{2}}, \dots, \frac{\partial H_{R} (θ)}{\partial θ_{n}}}

(26)

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 δϕ₀. between the current base attitude and its initial attitude needs to be optimized to make it as small as possible, namely δϕ₀ → 0. We use Z – Y – X Euler angles to express the attitude; so δϕ₀ can be calculated as follows:

δ ϕ_{0} = E u l (R_{0_n}^{- 1} \cdot R_{0_int}) \to 0

(27)

where, R_{0_int} is the initial rotate matrix of the base attitude and R_{0_n} represents the current rotate matrix of the base attitude. On the basis of Eq. (27), we introduce the additional absolute angular velocity ω_{0_f} of the base as follows:

ω_{0_f} = T_{ω_E} (ϕ_{0}) \cdot (δ ϕ_{0} / δ T)

(28)

where, δT is the control cycle and T_{ω_E}(ϕ₀) is the transformation matrix from Euler angular velocity to absolute angular velocity. If ϕ = {α,β,γ}, the matrix T_{ω_E}(ϕ) can be expressed as:

T_{ω_E} (ϕ) = [\begin{matrix} 0 & - \sin α & \cos α \cos β \\ 0 & \cos α & \sin α \cos β \\ 1 & 0 & - \sin β \end{matrix}]

(29)

When the deviation δϕ₀ is produced by a space manipulator's motion, an optimization operator, which can make the base have an opposite velocity to ω_{0_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:

α_{0 ϕ_R} (θ) = J_{0 m_ω}^{†} \cdot ω_{0_f}

(30)

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 ω_{e_f} of the end-effector can be calculated by the deviation δϕ_e between the current attitude and the initial attitude of the end-effector as follows:

ω_{e_f} = T_{ω_E} (ϕ_{e}) \cdot (δ ϕ_{e} / δ T)

(31)

On this basis, take out the last three rows of J_g in Eq. (17) to construct a matrix J_{g_ω}, 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:

α_{e ϕ_R} (Θ) = J_{g_ω}^{†} \cdot ω_{e_f}

(32)

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:

α_{R 1} (Θ) = ρ_{J_R} * α_{J_R} (Θ) + ρ_{0 ϕ_R} * α_{0 f_R} (Θ)

(33)

α_{R 2} (Θ) = ρ_{J_R} * α_{J_R} (Θ) + ρ_{e ϕ_R} * α_{e ϕ_R} (Θ)

(34)

where, ϱ_{J_R}, ϱ_{0ϕ_R} and ϱ_{eϕ_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:

H_{s} ω_{0} + H_{θ} (J_{g}^{†} {\dot{x}}_{e} + (E_{n} - J_{g}^{†} J_{g}) ϕ) = 0

(35)

The solution is:

ϕ = {(H_{θ} (E_{n} - J_{g}^{†} J_{g}))}^{†} \cdot (- H_{s} ω_{0} - H_{θ} J_{g}^{†} {\dot{x}}_{e})

(36)

Considering the requirement of base attitude maintenance, assume ω ₀ = {0,0,0}^T and substitute this condition into Eq. (36), then the base attitude maintenance optimization operator is as follows:

α_{ϕ 0_m} = - {(H_{θ} (E_{n} - J_{g}^{†} J_{g}))}^{†} H_{θ} J_{g}^{†} {\dot{x}}_{e}

(37)

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:

\dot{Θ} = J_{g}^{†} {\dot{x}}_{e} - (E_{n} - J_{g}^{†} J_{g}) \cdot (ρ_{R 1} α_{R 1} (Θ) + ρ_{ϕ 0_m} α_{ϕ 0_m})

(38)

Or:

\dot{Θ} = J_{g}^{†} {\dot{x}}_{e} - (E_{n} - J_{g}^{†} J_{g}) \cdot (ρ_{R 2} α_{R 2} (Θ) + ρ_{ϕ 0_m} α_{ϕ 0_m})

(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 ρ_R1 (ρ_R2) and ρ_ϕ0–m as follows to ensure the continuity and smoothness of the joint angles:

ρ_{R 1} = {\begin{matrix} 1 \begin{matrix} ​ & ​ & ​ & ​ & ​ & ​ & 0 \leq t \leq T_{R} \end{matrix} \\ 0 .5*(Sin (τ_{1} + \frac{π}{2}) + 1) \begin{matrix} ​ & T_{R} < t \leq T_{m} \end{matrix} \\ 0 \begin{matrix} ​ & ​ & ​ & ​ & ​ & ​ & T_{m} < t \leq T_{A} \end{matrix} \end{matrix}

(40)

ρ_{ϕ 0_m} = {\begin{array}{l} 0 \begin{matrix} ​ & ​ & ​ & ​ & ​ & ​ & 0 \leq t \leq T_{R} \end{matrix} \\ 0 .5*(Sin (τ_{1} - \frac{π}{2}) + 1) \begin{matrix} ​ & T_{R} < t \leq T_{m} \end{matrix} \\ 1 \begin{matrix} ​ & ​ & ​ & ​ & ​ & ​ & T_{m} < t \leq T_{Z} \end{matrix} \\ 0 .5*(Sin (τ_{2} + \frac{π}{2}) + 1) \begin{matrix} ​ & T_{Z} < t \leq T_{A} \end{matrix} \end{array}

(41)

T_A 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 ρ_R1 (ρ_R2) and ρ_{ϕB_m}.

Figure 2.

The curves of coefficient functions ρ_R1 (ρ_R2) and P_{ϕ0_m}

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.

Table 1.

The values of DH parameters

Link i	θ _i	d_i(m)	a_i–1(m)	α _i–1
1	θ₁(0)	0	0	90°
2	θ₂(0)	0.52	0	−90°
3	θ₃(0)	0	5.5	0
4	θ₄(0)	1.04	5.5	0
5	θ₅(0)	0	0	90°
6	θ₆(0)	0.52	0	−90°
7	θ₇(0)	0.52	0	0

Table 2.

The values of dynamical parameters

		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
ⁱP_i (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
ⁱI_i (kg·m²)	I_xx	800	0.98	0.98	20.95	66.93	0.98	0.98	0.98
	I_yy	800	0.95	0.95	160.86	206.84	0.95	0.95	0.95
	I_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
	I_xz	0	−0.01	−0.01	−11.40	−14.71	−0.01	−0.01	−0.01
	I_yz	0	−0.03	−0.03	0.03	0.03	−0.03	−0.03	−0.03

Figure 3.

DH coordinate systems of the studied manipulator

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.

Θ = {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 T_A = 60s, the acceleration time T_s=10s and the control cycle δT=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: T_R = 40s, T_m = 45s, T_Z = 55s, ρ_{J_R} = 3 and ϱ_{eϕ_R} = 0.01.

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.

Figure 4.

The velocities curves obtained by the method proposed in this paper

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 (δ^* denotes the simulation results using the method in this paper).

When the task ends, the reset deviation of the end-effector pose is:

{\begin{array}{l} δ r_{e} = {0.96 mm, - 8.39 mm, 0.64 mm} \\ δ r_{e}^{*} = {4.58 m m, 0.31 m m, 0.93 m m} \\ δ ϕ_{e} = {- {0.01}^{o}, {0.04}^{o}, {0.03}^{o}} \\ δ ϕ_{e}^{*} = {{0.16}^{o}, - {0.23}^{o}, {0.38}^{o}} \end{array}

the reset deviations of the joint angles are:

{\begin{array}{l} δ θ = {{5.73}^{o}, - {9.00}^{o}, - {18.95}^{o}, {5.97}^{o}, - {3.31}^{o}, {10.74}^{o}, {5.86}^{o}} \\ δ θ^{*} = {{0.62}^{o}, {0.06}^{o}, - {0.82}^{o}, {0.32}^{o}, - {0.61}^{o}, - {0.10}^{o}, {0.00}^{o}} \end{array}

the reset deviation of the base pose is:

{\begin{array}{l} δ r_{0} = {- 50.22 mm, 57.06 mm, 42 .97mm} \\ δ r_{0}^{*} = {- 4.65 mm, 1.79 mm, 1.48 mm} \\ δ ϕ_{0} = {- {16.76}^{o}, {6.64}^{o}, {7.82}^{o}} \\ δ ϕ_{0}^{*} = {- {1.16}^{o}, {0.44}^{o}, {0.38}^{o}} \end{array}

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).

Figure 5.

The disturbance of the base attitude during the task using two methods

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.

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:

The disturbance of the base attitude during the task is low, which will ensure the energy harvesting and communications.

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).

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