Multi-Objective Trajectory Planning of FFSM Carrying a Heavy Payload

Abstract

Aiming at carrying a heavy payload to a desired pose (including position and orientation), a multi-objective optimization-based approach for maximum-payload trajectory planning of free-floating space manipulators (FFSM) is proposed in this paper. The presented approach corresponds to two typical applications: (i) the manipulator joints attain the desired states; (ii) the inertial pose of the end-effector (pose with respect to the inertial frame) attains the desired values, for which a novel two-stage method is presented. Firstly, for the purpose of reducing computational complexity, dynamics equations are derived using a spatial operator algebra (SOA) method. Secondly, objective functions are defined according to the improvement of load-carrying capacity and pose requirements of the end-effector. Then, the joint trajectories are specified using sinusoidal polynomial functions. Finally, a multi-objective particle optimization (MOPSO) algorithm is employed to obtain a non-dominated solution set, during which process particles that do not satisfy the constraints are eliminated. Simulations are performed for a 7-DOF FFSM, considering three and five objectives for optimization in the two applications, respectively. The results demonstrate that the proposed approach can provide satisfactory joint trajectories and improve load-carrying capacity effectively.

Keywords

free-floating space manipulator multi-objective trajectory planning maximum payload MOPSO algorithm

1. Introduction

In recent years, space manipulators have been playing increasingly important roles in space exploration, especially in on-orbit payload-carrying operations for large structures in motion. Space manipulators are indispensable during construction and maintenance on the space station [1], capturing of objective satellites [2] and servicing of non-cooperative spacecraft [3], for they can assist or even replace astronauts. Considering the limitation of the load-carrying capacity of space manipulators, excessively heavy payload may not only challenge the capability of joints but also result in instability of the base in the free-floating mode. Therefore, in order to avoid the mentioned problems as far as possible, it is necessary to determine the optimal trajectory which can improve the load-carrying capacity of the FFSM.

Load-carrying capacity is usually defined as the maximum payload that the manipulator can repeatedly lift, and it depends on the dynamic motion or trajectory of the end-effector [4]. The previous studies in this field can be categorized into two groups [5]: the first group aims to determine the DLCC (dynamic load-carrying capacity) and corresponding joint trajectories of redundant manipulators along the given path of the end-effector, calculated considering the limitations of joint torques and tracking error [6,7]; the second group aims to solve the optimal-trajectory-planning problem for manipulators in point-to-point tasks, in which joint trajectories that allow the maximum payload to be carried are determined [8–15]. In order to carry a heavy payload to a desired pose, the second type of problem is considered in this paper.

Most of the previous works on trajectory planning during the load-carrying process have been carried out for ground manipulators. Wang and Ravani (1988) adopted an ILP (iterative linear programming) method to solve the constrained nonlinear optimization problem, synthesizing point-to-point dynamic motions with optimum load-carrying capacities for manipulators [8]. A similar computational technique was developed for obtaining optimal trajectory of wheeled mobile manipulators for a given two-end-point task, in which an extended Jacobian matrix concept was used to solve the redundancy and non-holonomic constraints [9,10]. Korayem and Nikoobin (2008) formulated maximum-payload trajectory planning as an optimal control problem, and the application of Pontryagin's minimum principle resulted in a two-point boundary value problem (TPBVP), which was solved numerically by indirect solution of the optimal control problem [11]. Considering payload constraints, Saravanan et al. (2008) presented a new general methodology, in which optimal trajectory planning of a PUMA560 robot was obtained by evolutionary computation of an elitist non-dominated sorting genetic algorithm (NSGA-II) and differential evolution (DE), respectively [12]. Saramago and Ceccarelli (2002) formulated optimal trajectory planning as subject to physical constraints, input torque/force constraints and payload constraints, while the energy of actuators was considered an objective function [13]. Wang et al. (2001) used the dynamics to produce motions that improve the payload capability, and a point-to-point weightlifting motion planner for a PUMA762 robot was developed by converting the governing optimal-control problem into a direct SQP parameter-optimization problem, in which torque limits were formulated as soft constraints that were added into the objective function [14]. In order to determine optimal trajectory of FFSM in a point-to-point task, Jia et al. (2013) proposed a path-planning method to achieve the goal of payload maximization, in which payload-carrying capacity was improved by minimizing the joint torques [15].

However, essentially the studies mentioned above are single-objective optimization problems. The common ground of these studies is merely that they aim to improve load-carrying capacity through optimizing either mechanical energy of the actuators [11–13] or maximum/minimum value of joint torques [8–10,14,15], which are both crucial objects. In addition, when carrying a heavy payload, free-floating mode is necessary to save the fuel of the spacecraft [16]. The resulting non-holonomic constraint (due to the non-integrability of angular momentum) can introduce the disturbance of base attitude during on-orbit operation process [17], and the influence on load-carrying capacity of FFSM cannot be ignored [18]. According to the analysis above, it is indicated that the multi-objective optimization problem (MOP) needs to be solved during trajectory planning.

It is worth noting that multi-objective trajectory-planning problems of manipulators are usually converted into single-objective problems by using a weighting method [12,19–21], which sometimes tends to be trapped in local optimal solution. Besides this, constantly adjusting the weighting coefficients can consume a large amount of computation time. To solve this problem, multi-objective evolutionary algorithms have been adopted in previous studies for mobile and industrial manipulators [22–24], which are also applicable to trajectory optimization of FFSM. To improve load-carrying capacity of free-floating space manipulators through simultaneous minimization of joint torques, base disturbance and total energy, Liu et al. (2014) proposed a MOPSO-algorithm-based load-maximization trajectory-optimization method [25], which has limitations in the following aspects: (i) only the load-carrying process of a 3-DOF planar manipulator is considered, and trajectory planning of FFSM with higher degrees of freedom in the three-dimensional space is more complicated; (ii) the simulations focus on explaining the optimization of joint torques and base disturbance, without adequately illustrating the results of MOP for all three objectives; (iii) computational efficiency of the proposed algorithm is significantly restricted owing to computation complexity of Newton-Euler dynamics equations and redundancy of some constraint functions. Therefore, further research is necessary.

This paper is organized as follows. Section 2 derives the kinematics and dynamics equations of FFSM with a payload; Section 3 formulates the trajectory-planning problem as a constrained MOP, in which constraints and objective functions are proposed; Section 4, explains the procedure of trajectory planning algorithm using MOPSO. Section 5 shows the simulation results. Section 6 presents the conclusions of the work.

2. Mathematical model of FFSM with a payload

As shown in Figure 1, the system considered in this paper consists of abase (link n + 1), a revolute-jointed manipulator (from link n to link 1) which has n degrees of freedom, and a payload which is attached to the end-effector. It is assumed that the components of the system are all rigid bodies. Link 1 and payload can be treated as a single composite rigid body, defined as the coordinate system at its mass centre. Mass and inertia tensor of the composite body can be easily obtained as:

\begin{array}{l} m_{c} = m_{1} + m_{l o a d}, m_{c} p_{c} = m_{1} p_{1} + m_{l o a d} p_{l o a d}, {}^{1}I_{c} = I_{1} + {}^{1}I_{l o a d} \\ {}^{1}I_{l o a d} = {}^{1}R_{L} [I_{l o a d} + m_{l o a d} (p_{l o a d}^{T} p_{l o a d} E_{3} - p_{l o a d} p_{l o a d}^{T})] {}^{1}R_{L}^{T} \end{array}

(1)

where m_load and m_c denote the mass of payload and the composite body, respectively; p_load and p_c denote the vector from Σ₁ to Σ_L and Σ_C, respectively; I_load and ¹I_load denote the inertia tensor of payload with respect to Σ_L and Σ₁ respectively; ¹R_L denotes the rotation matrix of Σ_L with respect to Σ₁; E₃ denotes the third-order identity matrix.

Figure 1.

Simplified model of free-floating space manipulators

Define the symbols as follows:

Σ_I: inertial coordinate system, which is the reference coordinate system of all recursive calculations;

Σ_b: coordinate system of the base;

Σ_k: coordinate system of link k(k=n, n−1,…, 1);

Σ_G: coordinate system at the mass centre of link 1;

Σ_L: coordinate system at the mass centre of payload;

J_k. joint k of the manipulator, which is used to connect link k+1 with link k;

C_k. mass centre of link k;

p _k: vector from J_k to C_k;

l _k,k−1 vector from Σ_k to Σ_k−1;

m_k: mass of link k;

I _k: inertia tensor of link k with respect to Σ_k;

c _bn: vector from C_b,(mass centre of the base) to J_n.

2.1 Kinematics Equations of FFSM

According to [26], the general kinematics equation of space manipulators can be written as:

{\dot{x}}_{e} = J_{b} {\dot{x}}_{b} + J_{m} \dot{q}

(2)

where ${\dot{x}}_{e} = {[v_{e}^{T}, ω_{e}^{T}]}^{T} \in R^{6}$ is the velocity vector of the end-effector with respect to $Σ_{I}$ , ${\dot{x}}_{b} = {[v_{b}^{T}, ω_{b}^{T}]}^{T} \in R^{6}$ is the velocity vector of the base with respect to Σ_I, $Σ_{I}$ , $\dot{q} = {[{\dot{q}}_{n}, {\dot{q}}_{n - 1}, \dots, {\dot{q}}_{1}]}^{T} \in R^{n}$ is the joint angular velocity vector, J _b ∊ R^6×6 is the Jacobian matrix denoting the relationship between the velocity of the base and the velocity of the end-effector, and J_m ∊ R^6×6 is the Jacobian matrix denoting the relationship between the joint angular velocity and the velocity of the end-effector.

It is assumed that the initial linear and angular momentum are equal to zero, and no external forces or torques act on the whole system in the free-floating mode. According to conservation of momentum and angular momentum, we can easily obtain that:

H_{b} {\dot{x}}_{b} + H_{b m} \dot{q} = 0

(3)

where H_b and H_bm denote inertia matrix of base and coupled inertia matrix, respectively [26]. Substituting Eq. (3) into Eq. (2) leads to the relationship as follows:

{\dot{x}}_{e} = J_{f l o a t} \dot{q} = {[\begin{matrix} J_{v e}^{T} & J_{ω e}^{T} \end{matrix}]}^{T} \dot{q}, {\dot{x}}_{b} = J_{b m} \dot{q} = {[\begin{matrix} J_{v b}^{T} & J_{ω b}^{T} \end{matrix}]}^{T} \dot{q}

(4)

where $J_{f l o a t} = J_{m} - J_{b} H_{b}^{- 1} H_{b m}$ denotes the generalized Jacobian matrix of space manipulators; $J_{b m} = - H_{b}^{- 1} H_{b m}$ denotes the relationship between joint angular velocity and velocity of the base; J_ve and J_ωe are the first and the last three rows of J_float, respectively; J_ve denotes the relationship between linear velocity of the end-effector and joint angular velocity, while J_ωe denotes the relationship between angular velocity of the end-effector and joint angular velocity; J_vb and J_ωb are defined as above.

According to Eq. (4), we have:

v_{e} = J_{v e} \dot{q}, ω_{e} = J_{ω e} \dot{q}, v_{b} = J_{v b} \dot{q}, ω_{b} = J_{ω b} \dot{q}

(5)

2.2 Dynamics Equations of FFSM with a Payload

According to the SOA method [27], recursive computation of forces and torques from link k + 1 to link k can be achieved using the spatial transferred operator as follows:

ϕ_{k + 1, k} = (\begin{matrix} E_{3} & {\tilde{l}}_{k + 1, k} \\ 0 & E_{3} \end{matrix}), (k = n, n - 1, \dots, 1)

(6)

where ∼ denotes that $^{\sim}$ denotes that ${\tilde{x}}_{1} x_{2} \equiv x_{1} \times x_{2}$ . The recursive computation of velocity and acceleration from link k + 1 to link k can be achieved using $ϕ_{k + 1, k}^{*}$ , in which * denotes the adjoint matrix.

We define the state-transition matrix of joint k as H_k = [h_k^* O_1×3], in which h_k denotes the unit vector of the axis direction at joint k. $V_{k} = c o l [ω_{k}, v_{k}] = {[ω_{k}^{T}, v_{k}^{T}]}^{T}$ denotes the spatial velocity of link k, in which ω_k and v_k denote angular velocity and linear velocity, respectively; V_n+1 = col[ω_b, v_b]. According to the Newton-Euler Principle, for k = n, n−1, … 1, the velocity of each link can be computed recursively from the base to the end-effector as:

V_{k} = ϕ_{k + 1, k}^{*} V_{k + 1} + H_{k}^{*} {\dot{q}}_{k}

(7)

The acceleration of link k can be obtained by the differential of Eq. (7):

α_{k} = ϕ_{k + 1, k}^{*} α_{k + 1} + H_{k}^{*} {\ddot{q}}_{k} + a_{k}

(8)

where $a_{k} = c o l [ω_{k + 1} \times h_{k} \dot{Θ}, ω_{k + 1} \times (ω_{k + 1} \times l_{k + 1, k})]$ denotes the velocity-dependent Coriolis and centrifugal acceleration.

Then, the inertial forces and torques of each link can be computed recursively from link 1 to link n. The spatial inertial M_k of link k is defined as:

M_{k} = {\begin{matrix} [\begin{matrix} {}^{1}I_{c} & m_{c} {\tilde{p}}_{c} \\ - m_{c} {\tilde{p}}_{c} & m_{c} E_{3} \end{matrix}], (k = 1) \\ [\begin{matrix} I_{k} & m_{k} {\tilde{p}}_{k} \\ - m_{k} {\tilde{p}}_{k} & m_{k} E_{3} \end{matrix}], (k = 2, 3, \dots, n) \end{matrix}

(9)

For k = 1, 2, Ȇ n, the spatial force of each link can be computed as follows:

f_{k} = {\begin{matrix} f_{0} + M_{k} α_{k} + b_{k} (k = 1) \\ ϕ_{k, k - 1} f_{k - 1} + M_{k} α_{k} + b_{k} (k = 2, 3, \dots, n) \end{matrix}

(10)

where $b_{k} = {\begin{matrix} c o l [ω_{1} \times ({}^{1}I_{c} ω_{1}), m_{c} ω_{1} \times (ω_{1} \times p_{c})], (k = 1) \\ c o l [ω_{k} \times (I_{k} ω_{k}), m_{k} ω_{k} \times (ω_{k} \times p_{k})], (k = 2, \dots, n) \end{matrix}$ denotes the velocity-dependent gyroscopic force and $f_{0} = 0$ denotes the velocity-dependent gyroscopic force and f₀=0 denotes external forces or torques acting on the composite body. Then, the driving torque of joint k(k = n, n−1, …, 1) can be obtained as:

T_{k} = H_{k} f_{k}

(11)

The disturbing forces and torques of the base caused by manipulator movement are computed as:

F_{b} = [\begin{matrix} O_{3} & E_{3} \end{matrix}] f_{n}, N_{b} = c_{b n} \times F_{b} + T_{n}

(12)

3. Problem Statement of Multi-objective Trajectory Planning

As formulated in Eq. (1), mass and inertia tensor of the composite body can be very large when carrying a heavy payload. According to the dynamics of FFSM, three aspects should be given great importance when executing given joint trajectories (in contrast with identical operation under light payload condition): first, larger driving torques need to be provided, which challenges the capability of the joints; second, larger disturbance force and torques of the base enlarge the range of movement, thus leading to more fuel consumption to maintain position and orientation; and, third, the total kinetic energy of the whole system increases, which will consume large amounts of energy provided by active forces/torques (only joint torques in this study) according to the principle of the conservation of energy. Therefore, limitations of joint driving torques, disturbance of the base, and total energy of the whole system will restrict the load-carrying capacity of FFSM.

According to the analysis above, payload maximization of FFSM can be solved as an MOP, which in this paper emphatically considers minimization of joint torques, base disturbance and total energy. Meanwhile, for a given trajectory, load-carrying capacity can also be affected by configurations of the manipulator, mechanical limit of joints, etc., which are usually regarded as constraints in previous studies. Considering constraints in the load-carrying process, a mathematical model of the constrained MOP can be established.

3.1 MOP Statement of Trajectory Planning

In this paper, two different types of boundary value problems are considered: the first challenge is to attain the final desired states of the manipulator joints in joint space (which is equivalent to reaching the desired pose of the end-effector with respect to the base) within the specified time and simultaneously maximize the load-carrying capacity of the FFSM; the second challenge is to reach the final desired pose of the end-effector in the Cartesian space (in the inertial frame) within the specified time and simultaneously maximize load-carrying capacity. In practical engineering, when assembling capsules or extravehicular large-scale equipment on the space station, replacing battery packs and experimental modules, or carrying heavy cargo to specified freight compartments, operations in joint space are needed. When putting a payload into a desired orbit, separating the orbiter, or maintaining spacecraft hover, operations in the inertial frame are needed.

Without loss of generality, we define Δt as the step size and t₀ and t_f as the initial and final time, respectively; then, the planning steps can be computed as s = (t_f−t₀)/Δt. Using MOP theory, the FFSM trajectory-optimization problem for the mentioned two types of operations is presented as:

\begin{array}{l} \min y = f (Θ) = {f_{1} (Θ), f_{2} (Θ), \dots, f_{M} (Θ)} \\ s.t. \begin{matrix}  \end{matrix} g_{i, j} (Θ) \leq 0, (i = 1, 2, \dots, k_{1}; j = u, u - 1, \dots, 1) \\ \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix}  \end{matrix} h_{i, j} (Θ) = 0, (i = 1, 2, \dots, k_{2}; j = v, v - 1, \dots, 1) \end{array}

(13)

where $Θ = [q^{T} (t_{0}), q^{T} (t_{0} + Δ t), \dots, q^{T} (t_{f})] \in D^{N}$ denotes the vector of decision variables, D denotes decision space with N = s^*n dimensions; $y = [f_{1}, f_{2}, \dots, f_{M}] \in Y^{M}$ denotes the objective vector, which is the mapping of Θ on M -dimensional objective space Y; f₁(Θ), …, f_M(Θ) are objective functions; g_i,j (Θ) and h_i,j(Θ) denote the constraints in the payload-carrying process, where $g_{i} (Θ) = {[g_{i, u}, g_{i, u - 1}, \dots, g_{i, 1}]}^{T}$ and $h_{i} (Θ) = {[h_{i, v}, h_{i, v - 1}, \dots, h_{i, 1}]}^{T}$

3.2 Constraints in the Payload-carrying Process

The initial and desired joint angle/velocity/acceleration constraints (v=n) are:

\begin{array}{l} h_{1} (Θ) = q (t_{0}) - q_{i n i}, h_{2} (Θ) = q (t_{f}) - q_{d e s} \\ h_{3} (Θ) = \dot{q} (t_{0}) - {\dot{q}}_{i n i}, h_{4} (Θ) = \dot{q} (t_{f}) - {\dot{q}}_{d e s} \\ h_{5} (Θ) = \ddot{q} (t_{0}) - {\ddot{q}}_{i n i}, h_{6} (Θ) = \ddot{q} (t_{f}) - {\ddot{q}}_{d e s} \end{array}

(14)

Joint angle constraints for t_r ∊ [t₀, t_f](u=n) are:

g_{1} (Θ) = q_{\min} - q (t_{r}), g_{2} (Θ) = q (t_{r}) - q_{\max}

(15)

Velocity constraints of the end-effector (u=3) are:

g_{3} (Θ) = v_{e} (t_{r}) - v_{\max}, g_{4} (Θ) = ω_{e} (t_{r}) - ω_{\max}

(16)

For the second type of operation, position- and orientation-deviation constraints of the end-effector (u=6) are:

g_{5} (Θ) = {[| Δ x_{e} |, | Δ y_{e} |, | Δ z_{e} |, | Δ α_{e} |, | Δ β_{e} |, | Δ γ_{e} |]}^{T} - Δ x_{e_\max}

(17)

where Δx_{e_max} ∊ R⁶ denotes the maximum permissible deviation of the end-effector at t_f.

3.3 Objective Functions

In order to improve the FFSM load-carrying capacity, joint driving torques, disturbance of the base and total energy involved in the motion need to be minimized simultaneously for the first type of challenge. In addition, for the second type of challenge, desired inertia pose of the end-effector should be satisfied when the task is completed. Therefore, five objective functions are established as follows.

1. Joint driving torques

According to [4], the limitation of joint driving torques would directly restrict the load-carrying capacity of manipulators. For a given payload-carrying task, space manipulators can be enabled to carry a heavier payload by reducing the peak value of joint torques. Therefore, the objective function minimizing joint driving torques is defined as:

f_{1} = \max {| τ_{1}^{\max} (t) |, | τ_{2}^{\max} (t) |, \dots, | τ_{n}^{\max} (t) |}, t \in [t_{0}, t_{f}]

(18)

2. Disturbance of the base

When the boundary value of planning is determined, space manipulators can carry a heavier payload with the same fuel consumption for attitude maintenance if an appropriate trajectory is chosen minimizing the disturbance of the base [18]. Defining $f_{b} = c o l [N_{b}, F_{b}]$ , $f_{b}^{T} W f_{b}$ can be used to measure the disturbance of the base, where $W = [\begin{matrix} w_{1} & 0 \\ 0 & w_{2} \end{matrix}]$ is the weight matrix. We can then define:

f_{2} = \int_{t_{0}}^{t_{f}} (w_{1} {‖ N_{b} ‖}_{2} / N_{1} + w_{2} {‖ F_{b} ‖}_{2} / N_{2}) d t

(19)

where ${‖ N_{b} ‖}_{2} = {(N_{b}^{T} N_{b})}^{1 / 2}$ , ${‖ F_{b} ‖}_{2} = {(F_{b}^{T} F_{b})}^{1 / 2}$ ; w₁ and w₂ denote weight coefficients, and w₁ + w₂ = 1; N₁ and N₂ denote normalizing parameters, which are used to adjust the magnitude of ||N_b||₂ and ||F_b||₂ to be consistent.

3. Total energy consumption of the whole system

Total energy involved in the motion may significantly increase when f₁ or f₂ is optimized [25], which would restrict load-carrying capacity of FFSM. Considering the extremely strict limitation of energy consumption in the space environment, the objective function minimizing total energy is defined as:

f_{3} = \sum_{i = 1}^{n} \int_{t_{0}}^{t_{f}} {[{\dot{q}}_{i} (t) τ_{i} (t)]}^{2} d t

(20)

4. Position deviation of the end-effector when t = t_f

When the task is completed, the actual position of the end-effector can be obtained by integrating linear velocity as follows:

P_{e f} = P_{e} (t_{0}) + \int_{t_{0}}^{t_{f}} J_{v e} \dot{q} dt

(21)

Defining P_ed as the desired position of the end-effector, the objective function is established as:

f_{4} = {‖ P_{e d} - P_{e f} ‖}_{2}

(22)

5. Orientation deviation of the end-effector when t = t_f

Orientation of the end-effector is represented by a quaternions method. The basic form is:

Q_{e} = {η_{e f}, ξ_{e f}} = η + ξ_{1} \vec{i} + ξ_{2} \vec{j} + ξ_{3} \vec{k}

(23)

where $η^{2} + ξ_{1}^{2} + ξ_{2}^{2} + ξ_{3}^{2} = 1$ According to [28], we have:

[\begin{matrix} {\dot{η}}_{e} \\ {\dot{ξ}}_{e} \end{matrix}] = \frac{1}{2} [\begin{matrix} - ξ_{e}^{T} \\ η_{e} E - {\tilde{ξ}}_{e} \end{matrix}] ω_{e}

(24)

Combined with Eq. (5), the actual orientation of the end-effector when t = t_f can be expressed as:

{η_{e f}, ξ_{e f}} = Q_{e} (t_{0}) + \frac{1}{2} \int_{t_{0}}^{t_{f}} ([\begin{matrix} - ξ_{e}^{T} \\ η_{e} E - {\tilde{ξ}}_{e} \end{matrix}] J_{ω e} \dot{q}) dt

(25)

Defining {η_ed, ξ_ed} as the desired orientation of the end-effector, the orientation deviation of the end-effector when t = t_f can be expressed as follows:

δ Q_{e} = η_{e f} ξ_{e d} - η_{e d} ξ_{e f} - {\tilde{ξ}}_{e f} ξ_{e d}

(26)

The objective function is established as:

f_{5} = {‖ δ Q_{e} ‖}_{2}

(27)

4. Solving the trajectory-planning problem using MOPSO

The trajectory-planning problem proposed in Section 3 is a constrained multi-objective optimization problem, which usually involves simultaneous optimization of several competing objectives. There is no single optimal solution, but rather a set of Pareto optimal solutions. According to the Pareto dominance principle of MOP, the solutions of the optimal-trajectory-planning problem in this paper can be obtained as follows: For any two sets of joint trajectories in D, when the following relationship is satisfied, Θ⁰ dominates Θ¹(Θ⁰ ≻ Θ¹):

\begin{array}{l} f_{m} (Θ^{0}) \leq f_{m} (Θ^{1}), m = 1, 2, \dots, M \\ f_{m} (Θ^{0}) < f_{m} (Θ^{1}), \exists m \in {1, 2, \dots, M} \end{array}

(28)

The solution satisfying Eq. (28) is defined as a non-dominated solution of Eq. (13), which represents a set of trajectory planning that improves payload-carrying capacity as much as possible. Defining {S^*} as the set of all the non-dominated solutions, the mapping of {S^*} 1 on Y is an M -dimensional surface called the Pareto Front. Hence, the key to solving the multi-objective trajectory-planning problem is to achieve as many non-dominated solutions covering all features of MOP as possible. Considering the advantages of fast convergence, lower occupancy of computing resources and excellent diversity (verified by Coello [29]), the MOPSO algorithm is applied for the solution in this section.

4.1 Specifying decision vectors

The joint trajectories need to be parameterized before optimization. However, directly assigning the position of particles using Θ would observably increase the data size of D, especially when Δt is short, leading to substantial computation [22].

In order to obtain smooth and continuous trajectories of $q, \dot{q}, \ddot{q}$ which simultaneously satisfy the constraints in Eqs. (14) and (15), the joint angle is specified as the sinusoidal functions, whose arguments are the polynomial functions:

q_{k} (t) = Δ_{k 1} \sin (\sum_{j = 0}^{7} λ_{k j} t^{j}) + Δ_{k 2} (k = n, \dots, 1; j = 0, \dots, 7)

(29)

where λ_kj. denotes the coefficients of the polynomial, and $Δ_{k 1} = (q_{k}^{\max} - q_{k}^{\min}) / 2$ , $Δ_{k 2} = (q_{k}^{\max} + q_{k}^{\min}) / 2$ , $q_{k}^{\max}$ and $q_{k}^{\min}$ denote the upper and lower limits of joint k, respectively.

Then, angular velocity and angular acceleration are computed as:

{\dot{q}}_{k} (t) = Δ_{k 1} c o s (\sum_{j = 0}^{7} λ_{k j} t^{j}) (\sum_{j = 1}^{7} j λ_{k j} t^{j - 1})

(30)

\begin{array}{l} {\ddot{q}}_{k} (t) = - Δ_{k 1} \sin (\sum_{j = 0}^{7} λ_{k j} t^{j}) \cdot {(\sum_{j = 1}^{7} j λ_{k j} t^{j - 1})}^{2} + \\ \begin{matrix}  \end{matrix} Δ_{k 1} \cos (\sum_{j = 0}^{7} λ_{k j} t^{j}) \cdot (\sum_{j = 2}^{7} j (j - 1) λ_{k j} t^{j - 2}) \end{array}

(31)

For the first type of challenge, it is assumed that q_ini and q_des are already known, and that ${\dot{q}}_{i n i} = {\dot{q}}_{d e s} = {\ddot{q}}_{i n i} = {\ddot{q}}_{d e s} = 0$ . Six equations can be established according to Eq. (14) prior to substituting Eqs. (29–31) into them to obtain solutions as follows:

\begin{array}{l} λ_{k 0} = \arcsin [(q_{k}^{i n i} - Δ_{k 2}) / Δ_{k 1}], λ_{k 1} = λ_{k 2} = 0 \\ λ_{k 3} = - [3 λ_{k 7} t_{f}^{7} + λ_{k 6} t_{f}^{6} - 10 (Δ_{k 3} - Δ_{k 4})] / t_{f}^{3} \\ λ_{k 4} = [8 λ_{k 7} t_{f}^{7} + 3 λ_{k 6} t_{f}^{6} - 15 (Δ_{k 3} - Δ_{k 4})] / t_{f}^{4} \\ λ_{k 5} = - [6 λ_{k 7} t_{f}^{7} + 3 λ_{k 6} t_{f}^{6} - 6 (Δ_{k 3} - Δ_{k 4})] / t_{f}^{5} \end{array}

(32)

where q_kⁱⁿⁱ and q_k^des denote the initial and desired angle of joint k, respectively. $Δ_{k 3} = \arcsin [(q_{k}^{d e s} - Δ_{k 2}) / Δ_{k 1}]$ , $Δ_{k 4} = \arcsin [(q_{k}^{i n i} - Δ_{k 2}) / Δ_{k 1}]$ . $λ_{k 3}, λ_{k 4}, λ_{k 5}$ can be described using λ_k6 and λ_k7. Given $λ = [λ_{n 6}, λ_{n 7}, \dots, λ_{16}, λ_{17}] \in R^{2 n \times 1}$ joint angle, angular velocity and angular acceleration are determined, and h_i(Θ) = 0 is satisfied. Therefore, λ is specified as a decision variable in the first type.

Unlike in the first type, q_des is unknown in the second type. Therefore, only five equations can be established and the solutions are obtained as:

\begin{array}{l} λ_{k 0} = \arcsin ((q_{k}^{i n i} - Δ_{k 2}) / Δ_{k 1}), λ_{k 1} = λ_{k 2} = 0 \\ λ_{k 3} = (21 λ_{k 7} t_{f}^{4} + 12 λ_{k 6} t_{f}^{3} + 5 λ_{k 5} t_{f}^{2}) / 3 \\ λ_{k 4} = - (14 λ_{k 7} t_{f}^{3} + 9 λ_{k 6} t_{f}^{2} + 5 λ_{k 5} t_{f}) / 2 \end{array}

(33)

Similarly, $λ^{'} = [λ_{n 5}, λ_{n 6}, λ_{n 7}, \dots λ_{15}, λ_{16}, λ_{17}] \in R^{3 n \times 1}$ is specified as the decision vector in the second type of operation.

4.2 Obtaining Pareto optimal solution set using MOPSO

In the MOPSO algorithm, g_best (the best position that all the particles have had) is used as guidance to lead the evolution toward the Pareto Front. It cannot be determined immediately, but is replaced by R[h] (a value that is taken from the repository). In this paper, according to the particles' velocity formula which contains the constriction factor, the particle flight in D can be computed as follows:

\begin{array}{l} V_{j} [i] = χ \cdot {V_{j - 1} [i] + c_{1} r_{1} (p_{b e s t}^{j - 1} [i] - P_{j - 1} [i]) \\ \begin{matrix}  \end{matrix} + c_{2} r_{2} (R_{j - 1} [h] - P_{j - 1} [i])} \end{array}

(34)

where P and R denote the population and the repository, respectively; P[i] and V[i] denote the position and velocity of particle i, respectively; j = 1, 2, …, I_max is the iterations of evolutionary computations, initialization when j=0; $χ = 2 / (| 2 - ϕ - \sqrt{ϕ (ϕ - 4)} |)$ is the constriction factor; ϕ = c₁ + c₂, c₁ regulates the step size of the particle flight to p_best[i] (the best position that particle i has had), and c₂ regulates the step size of the particle flight to R[h]; r₁ and r₂ are independent random numbers in the range [0, 1].

For the first type of challenge, the MOPSO algorithm is as follows:

Step 1 Initialize the population P₀:

Define n_P as the number of particles and restrict the search range to [λ_min, λ_max] in each dimension of D.

Generate a random decision vector within the restrictive decision space.

Substitute the vector into Eqs. (29–31) to determine q, $\dot{q}$ and $\ddot{q}$ , then compute v_e and ω_e using kinematics equations.

If Eq. (16) is satisfied, store the decision vector and execute (e). Otherwise, loop to (b).

If the number of stored decision vectors equals n_P, execute (f). Otherwise, loop to (b).

Locate the particles using these decision vectors as coordinates. Each particle's position in D is defined according to the value of its corresponding decision vector.

Initialize the speed of each particle using V₀[i]=0.

Step 2 Initialize the external repository R₀:

Define n_R^I as the maximum allowable number of R₀.

Evaluate each of the particles in P₀: compute $q, \dot{q}, \ddot{q}$ and N_b, F_b, T_k (using the SOA method), which are substituted into Eqs. (18–20) to determine the objective vector of the particle $i (y_{i}^{0} = [f_{1}^{0} (i), f_{2}^{0} (i), f_{3}^{0} (i)])$

Store the positions of particles that represent non-dominated decision vectors (according to Eq. (28)) in R₀.

Generate hypercubes of Y explored so far, and locate the particles using these hypercubes as a coordinate system where the coordinate of particle i is defined according to y_i⁰.

Initialize the best positions of the particles have had using P_best⁰[i]=P₀[i]H stored in R₀.

Step 3 While maximum number of cycles I_max has not been reached, execute evolutionary computations as follows:

In R_j−1 those hypercubes containing more than one particle are assigned a fitness value equal to the result of x / N_P (x > 1 and N_P is the number of particles that these hypercubes contain).

Using these fitness values, roulette-wheel selection is applied to select a hypercube. Then, a random particle within the selected hypercube can be obtained. The position of this particle in decision space is taken as R_j−1[h].

Compute the speed of each particle using Eq. (34), and update the positions of the particles using $P_{j} [i] = P_{j - 1} [i] + V_{j} [i]$

Maintain the particles' flight within the restrictive decision space: when a particle flies beyond the boundaries. Let it take the value of its corresponding boundary, and multiply its velocity by −1.

Evaluate each of the particles in P_j by computing the objective vectors $y_{i}^{j} = [f_{1}^{j} (i), f_{2}^{j} (i), f_{3}^{j} (i)]$ for the particle i), and pick out the particles that represent non-dominated decision vectors (according to Eq. (28)).

Each selected particle which satisfies constraint conditions is defined as a new non-dominated solution. If this new solution is dominated by an individual within the external repository, then such a solution is discarded. If none of the individuals in the external repository dominates the solution, then such a solution is stored in the repository. If there are solutions in the external repository that are dominated by the new element, then such solutions are removed from the repository.

If the number of solutions in the repository has reached n_R, then the adaptive grid method [29] is invoked to produce a well-distributed Pareto Front.

Define the current external repository as R_j.

If P_j[i] is dominated by p_best^j−1[i], then p_best^j[i] = p_best^j−1[i]; otherwise, p_best^j[i] = P_j[i]; if neither is dominated by the other, then one is selected randomly.

Increment the loop counter until j = I_max.

Step 4 Obtain the set of non-dominated solutions {S^*}, which consists of decision vectors of the particles in R_{I
_max}.

However, for the second type of challenge, the proposed scheme cannot be used directly to solve the MOP of f₁ ∼ f₅. If g_i(Θ) ≤ 0 is considered during initialization of the population, it is very difficult to generate a desired decision vector (which satisfies Eq. (17)) because of the indeterminate q(t_f). Consequently, this will cost a large amount of computation time. If g_i(Θ) ≤ 0 is not considered, an invalid repository is obtained when no particle in R₀ (representing a non-dominated individual in P₀) satisfies Eq. (17). In order to avoid these problems, a two-stage method is applied in this paper as follows.

Stage I Solve the MOP of f₄ and f₅ using MOPSO:

Initialize the population P₀^I. Define n_P^I as the number of particles and restrict the search range to [λ_min,λ_max]. Generate decision vectors and store those which satisfy Eq. (16). When the number of stored decision vectors is equal to n_P^I, initialize the position and velocity of each particle.

Initialize the external repository R₀^I. Define n_R^I as the maximum allowable number and store the positions of particles that represent non-dominated decision vectors in R₀^I (according to objective vector [f₄⁰(i), f₅⁰(i)] of the particle i). Then, initialize the best positions that the particles have had.

Adopt a similar procedure to that shown in Step 3 for the first type of challenge to execute evolutionary computations. During the upgrading process of the repository, eliminate the solutions which do not satisfy Eq. (17). When the maximum number of cycles is reached, the set of remaining non-dominated solutions is defined as {S_I^*}.

Stage II Solve the MOP of f₁ ∼ f₅ using MOPSO:

Initialize the population P₀^II. n_P^II equals the number of the solutions in [S_I^* and the search range is restricted to [λ_min/ λ_max]. Each particle is assigned a one-to-one position of the solution in {S_I^*}. Then, initialize the speed of each particle using V₀^II[i]=0.

Initialize the external repository R₀^II. Define n_R^II as the maximum allowable number and store the positions of particles that represent non-dominated decision vectors in R₀^II (according to the objective vector of the particle i:[f₁⁰(i), f₂⁰(i), f₃⁰(i), f₄⁰(i), f₅⁰(i)].) Then, initialize the best positions that the particles have had.

Adopt a similar procedure to that shown in Step 3 for the first type of challenge to execute evolutionary computations. During the upgrading process of the repository, eliminate the solutions which do not satisfy Eq. (16) or Eq. (17). When the maximum number of cycles is reached, the set of remaining non-dominated solutions is defined as {S_II^*}, which is the desired solution set.

5. Simulation results

5.1 Simulation model

In this study, a seven-link space manipulator mounted on a base is considered. A 3D geometrical model of the manipulator is shown in Figure 2(a), and joint frames according to the DH method are shown in Figure 2(b). Relative parameters are listed in Table 1 and Table 2. d₄ = 0.52m, d₇ = d₁ = 1.2m, d₆ = d₅ = d₃ = d₂ = 0.53m, a₅ = a₄ =5.8m. Pose of Σ₈ is [−0.2m, 0m, 2m, 0°, 0°, 0°] with respect to Σ_b, I_zx = −4kg.m².

Figure 2.

7-DOF free-floating space manipulator

Table 1.

Dynamics parameters of 7-DOF space manipulator

Parameters		Base	Link 7	Link 6	Link 5	Link 4	Link 3	Link 2	Link 1	Payload
Mass/(kg)		7.5e +04	30	30	70	75	30	30	40	2.5e +04
ⁱp_i./(m)		0	0	−0.265	2.9	2.7	0	0	0	0
		0	−0.265	0	0	0	0	0	0	0
		0	0	0	0	0.5	0.265	0.265	0.6	3.45
I_k/(kg.m²)	I _xx	7.5e +04	0.98	0.57	1.32	1.91	0.98	0.98	5.18	9.3e +05
	I _yy	7.5e +04	0.57	0.98	197.2	243.4	0.98	0.98	5.18	3.6e +05
	I _zz	7.5e +04	0.98	0.98	197.2	242.9	0.57	0.57	0.75	9.3e +05

Table 1.

D-H parameters of 7-DOF space manipulator

Link i	θ_i (°)	d_i (m)	a_i−1 (m)	α_i−1 (°)
7	θ₇(0)	d ₇	0	90
6	θ₆(90)	d ₆	0	−90
5	θ₅(0)	0	a ₅	0
4	θ₄(0)	d₃ + d₄ + d₅	a ₄	0
3	θ₃(0)	0	0	90
2	θ₂(−90)	d ₂	0	−90
1	θ₁(0)	d ₁	0	0

5.2 Results and discussion

Simulations are performed to find the optimal trajectory of the 7-DOF FFSM in the two cases mentioned in Section 3.1. Fn both cases, the initial pose of the base is zero (t₀=0); the total time of planning is t_f = 100s; the upper and lower limits of joint it k(k = 7, 6, …, 1) are q_k^min = −270° and q_k^max 270°, respectively; maximum absolute linear and angular velocity of the end-effector are limited as v_max = [0.05, 0.05, 0.05]^T(m/s) and ω_max = [0.6, 0.6, 0.6]^T(/s); and the parameters in Eq. (19) are set as w₁ = 0.1, w₂ = 0.9, N₁ = 1 and N₂=10.

Case I: attaining the desired joint angles when carrying a payload.

For this case, the initial and desired angles of the joints are set as q_ini = [−20°, 0, −10°, −140°, 110°, 155°, 70°] and q_des = [−10°, 10°, −30°, −120°, 125°, 135°, 80°], respectively; the range of values for the 14 elements of λ are set as [λ_j^min], λ_j^max] = [−10e-15, + 10e-15]; the parameters of MOPSO are set as n_P = 500, n_R = 2000 and c₁ = c₂ = 2.05. After 800 iterations we obtain {S^*}, whose distribution on objective space is shown in Figure 3. The fitting surface is obtained using the Matlab ‘mesh function’ using the solutions within it. From this figure, it can be seen that all the non-dominated solutions relatively evenly distribute on the surface, which demonstrates that {S^*} can well approach the real Pareto Front. The convergence process of each objective function is shown in Figure 4, which demonstrates that satisfactory convergence effects of the three objective functions are obtained. The number of particles in the repository changes as shown in Figure 5.

Table 1.
Relevant kinematics and dynamics results using extreme solutions and trapezoidal-velocity planning

Results Solution A Solution B Solution C Solution T

Non-dominant Solution/(e-15) [−1.996, −2.529, −4.490, −7.286, −2.472, −9.176, −2.367, 3.214, −3.371, 1.454, −2.852, −6.630, 4.441, −4.933] [−3.322, −2.800, −8.618, −7.555, −3.162, −10, −4.548, 6.895, −7.209, −10, −0.830, −5.930, 2112, −0.663] [−2.609, 0.074, −1.333, −4.127, −0.204, −4.058, 0.703, 0.375, −1.007, 0.447, 2.348, −3.718, 1.729, −0.700] /

Objective Vector [275.0573, 268.5604, 394.8005] [332.4447,204.9874, 594.3254] [335.3355,251.7596, 173.2466] [448.0397,273.005 195.2476]

End-effector Pose, t=t_f Position(m) [−3.4736, −5.0541, 1.6794] [−3.5497, −5.0854, 1.4694] [−3.3003, −5.0906, 1.8358] [−3.0092, −5.0890, 2.2090]

Orientation(rad) [1.6923, −0.3619, −2.8356] [1.666, −0.4259, −2.8191] [1.6878, −0.3585, −2.7989] [1.7018, −0.3130, −2.7498]

Base Pose, t=t_f Position(m) [0.6507,0.4979, −0.1456] [0.6435,0.4927, −0.1033] [0.7118,0.4822, −0.1669] [0.7442,0.4881, −0.2335]

Orientation(rad) [−0.0289, −0.2076, −0.1374] [−0.0606, −0.1828, −0.0753] [−0.0184, −0.1740, −0.1457] [−0.0200, −0.1334 −0.1971]

Maximum Absolute End-effector Velocity Linear Velocity (m/s) [0.0221,0.0366, 0.0163] [0.0295,0.0332, 0.0243] [0.0095,0.0382, 0.0153] [0.0124,0.0304, 0.0097]

Angular Velocity (rad/s) [0.0046,0.0076, 0.0037] [0.0035,0.0079, 0.0038] [0.0025,0.0082, 0.0040] [0.0007,0.0083, 0.0053]

Results	Solution A	Solution B	Solution C	Solution T
Non-dominant Solution/(e-15)	[−1.996, −2.529, −4.490, −7.286, −2.472, −9.176, −2.367, 3.214, −3.371, 1.454, −2.852, −6.630, 4.441, −4.933]	[−3.322, −2.800, −8.618, −7.555, −3.162, −10, −4.548, 6.895, −7.209, −10, −0.830, −5.930, 2112, −0.663]	[−2.609, 0.074, −1.333, −4.127, −0.204, −4.058, 0.703, 0.375, −1.007, 0.447, 2.348, −3.718, 1.729, −0.700]	/
Objective Vector	[275.0573, 268.5604, 394.8005]	[332.4447,204.9874, 594.3254]	[335.3355,251.7596, 173.2466]	[448.0397,273.005 195.2476]
End-effector Pose, t=t_f	Position(m)	[−3.4736, −5.0541, 1.6794]	[−3.5497, −5.0854, 1.4694]	[−3.3003, −5.0906, 1.8358]	[−3.0092, −5.0890, 2.2090]
Orientation(rad)	[1.6923, −0.3619, −2.8356]	[1.666, −0.4259, −2.8191]	[1.6878, −0.3585, −2.7989]	[1.7018, −0.3130, −2.7498]
Base Pose, t=t_f	Position(m)	[0.6507,0.4979, −0.1456]	[0.6435,0.4927, −0.1033]	[0.7118,0.4822, −0.1669]	[0.7442,0.4881, −0.2335]
Orientation(rad)	[−0.0289, −0.2076, −0.1374]	[−0.0606, −0.1828, −0.0753]	[−0.0184, −0.1740, −0.1457]	[−0.0200, −0.1334 −0.1971]
Maximum Absolute End-effector Velocity	Linear Velocity (m/s)	[0.0221,0.0366, 0.0163]	[0.0295,0.0332, 0.0243]	[0.0095,0.0382, 0.0153]	[0.0124,0.0304, 0.0097]
Angular Velocity (rad/s)	[0.0046,0.0076, 0.0037]	[0.0035,0.0079, 0.0038]	[0.0025,0.0082, 0.0040]	[0.0007,0.0083, 0.0053]

Figure 3.

Non-dominated solution set of Case I in objective space

Figure 4.

Optimal values of objective functions during evolution

Figure 5.

Number of particles in the repository during evolution

Solutions A, B and C marked on Figure 3 are the extreme non-dominated solutions, which correspond to the minimum value of f₁, f₂ and f₃, respectively. Given the same parameter as Case I, contrastive simulations are performed where joint velocities are determined by conventional trapezoidal-velocity profile (mark as solution T) [30] (acceleration and deceleration time are both set as 30 seconds). The results listed in Table 3 are obtained by computing kinematics according to the four sets of joint trajectories. Although the motions of the joints, the base and the end-effector are different in these four cases, the manipulator can eventually reach the same state (configuration), while linear and angular velocities of the end-effector do not exceed the limits.

As shown in Table 3, λ_A, λ_B and λ_T do not dominate each other, while λ_C ≻ λ_T. Solution A works best minimizing the driving torque of joints (f₁^A reduced to 56.82% of f₁^A) and solution B works best minimizing the disturbance of the base (f₂^B reduced to 75.65% of f₂^T, but we see a significant increase of f₃ for both solutions (f₃^A = 188.53% f₃^T, f₃^B = 282.86% f₃^T). The decline of the other two objective functions occurs when minimizing the total energy of the whole system, in accordance with the typical characteristics of conflicts in achieving MOP goals [25].

In order to illustrate the effect of the proposed algorithm in improving load-carrying capacity, a non-dominated solution (marked as D) which dominates solution T is selected. f_D = [328.3052,246.5933,175.5356], λ_D = 1.0e-15^*[−1.673, −0.482, −1.344, −4.400, −0.072, −4.943, −0.278,0.7 84, −0.251, 0.886,1.075, −4.171,0.747, −1.253]. As shown in Figure 6 and Figure 7, with respect to trapezoidal-velocity planning, corresponding maximum absolute values of joint torques decreased by about 30%, that of disturbance torques by about 10%, and f₃^D = 89.9% f₃^T. It is assumed that the mass and inertia of the payload are multiplied by a coefficient μ(μ > 1), when μ increases to 1.0723 (m_load increases by 1.81e+03kg although the payload has been great, f_D = [347.5056,264.0548,195.2398]), f₃^D ≍ f₃^T while values of the other two objective functions are superior to that of f_T. Except for the mentioned constraints, we assume that maximum driving torque provided by each joint is 500N.m, and the disturbance forces/torques of the base are limited to 500N/500N.m. Using the same algorithm proposed in [18], the upper boundary of load-carrying capacity can be obtained as: μ_D^max = 1.703, μ_T^max = 1.036. From the above results, the conclusion can be drawn that the trajectory-planning scheme proposed in this paper can obviously improve the load-carrying capacity of FFSM for Case I.

Case II: carrying a payload to the required pose.

For this case, q_ini = [−20°, 0, −10°, −100°, 120°, 180°, 70°]; Δx_{e_max}=[0.01m, 0.01m, 0.01m, 1°, 1°, 1°]; the desired pose of the end-effector is set as x_e(t_f) = [−2.5m, −7.5m, 3m, 1.5rad, Orad, −3rad]; the range of values for the 21 elements of λ is set as $[{λ^{'}}_{j}^{\min}, {λ^{'}}_{j}^{\max}] = [- 10e-16, + 10e-16]$ . Considering that the search space is much smaller than in Case I, the parameters of MOPSO are set as n_p = 300, n_R = 500 and c₁ = c₂ = 2.05 in both stages. After 300 iterations of Stage I and 500 iterations of Stage II, we can obtain {S_II^*}; the relevant results are listed in Table 4. Rectilinear trajectory planning is performed as contrastive simulations, in which linear and angular velocities of the end-effector are determined using trapezoidal-velocity planning (acceleration/deceleration time are both set as 30 seconds).

Figure 6.

Joint trajectories using the mentioned three extreme solutions and trapezoidal-velocity planning

Figure 7.

Joint torques and disturbance forces/torques of the base using solution D and trapezoidal-velocity planning

Table 1.

ETS-MARSE Denavit-Hartenberg modified parameters

Objective function	Best value	Worst value	Rectilinear planning
f ₁	333.6477	337.5971	429.6080
f ₂	373.2347	385.4648	536.9357
f ₃	99.7827	110.3244	154.4616
f ₄	5.3156e04	0.0100	8.6624e07
f ₅	0.0054	0.0144	3.5209e08

The best and worst values of f₁ ∼ f₅ are listed in Table 4. Compared with the corresponding values of rectilinear planning, f₁ ∼ f₃of all the solutions in {S_II^*} are dominated on the premise of satisfying Eqs. (15–17), which demonstrates that maximum payload trajectory can be obtained by selecting an appropriate solution from the obtained non-dominated solution set. We define solution E as the extreme solution which minimizes f₄ best. f_E=[336.2371, 378.2704, 103.2874, 5.3156e-04, 0.0086], λ_E=1.0e-16^* [−2.732, 2.753, 5.428, −2.641, −1.093,4.124, −0.2068, −1.167, −4.362,1.379,2.434, −4.588, 5.147, −0.579, 7.227, 0.383, 0.303, −2.945, − 4.197, −0.081, 5.698]. Computing kinematics using solution E, we can obtain that x_e^E (t_f) = [−2.5019m, −7.4997m, 2.9989m, 1.4935rad, −0.0055rad, −3.0149rad], the deviations are acceptable. f₁^E = 78.27% f₁^R, f₂^E = 70.45% f₂^R, and f₃^E = 66.87% f₃^R, demonstrating that the proposed scheme can obviously improve the load-carrying capacity of FFSM for Case II.

6. Conclusions

In this paper, a MOPSO-algorithm-based scheme is developed to find the optimal joint trajectories of FFSM carrying a heavy payload. In the proposed scheme, maximization of load-carrying capacity and motion requirements of the manipulator can be achieved at the same time. Especially for the point-to-point motion in Cartesian space, a novel two-stage method is proposed to solve the optimization problem considering five objectives. The constraints are met with two measures: (i) decision vectors are specified using sinusoidal polynomial functions; (ii) solutions which do not satisfy the constraints are eliminated. Meanwhile, for the purpose of reducing computational complexity, the SOA method is adopted to compute inverse dynamics. The effectiveness of this scheme in improving load-carrying capacity is verified by simulations. It is worth noting that it is convenient for decision-makers to choose the appropriate solutions (each non-dominated solution in objective space has definite physical meanings). This kind of convenience provides the feasibility of the proposed algorithm in practical engineering. Additionally, the trajectory-planning scheme proposed in this study applies to only point-to-point motion. As another kind of typical on-orbit tasks for FFSM, trajectory tracking of the end-effector in task space exhibits significantly different characteristics and requirements. Corresponding motion-planning strategies need to be considered in future work.

Footnotes

Acknowledgements

This project is supported by the National Natural Science Foundation of China (61175080), the National Key Basic Research Programme of China (2013CB733005) and the Specialized Research Fund for the Doctoral Programme of Higher Education (20130005110009).

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