﻿ 自由漂浮空间机器人路径优化的Legendre伪谱法
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 力学学报  2016, Vol. 48 Issue (4): 823-831  DOI: 10.6052/0459-1879-16-158 0

### 引用本文 [复制中英文]

[复制中文]
Ge Xinsheng , Chen Kaijie . PATH PLANNING OF FREE FLOATING SPACE ROBOT USING LEGENDRE PSEUDOSPECTRAL METHOD[J]. Chinese Journal of Ship Research, 2016, 48(4): 823-831. DOI: 10.6052/0459-1879-16-158.
[复制英文]

### 文章历史

2016-06-06 收稿
2016-06-17网络版发表

1. 北京信息科技大学理学院, 北京 100192 ;
2. 北京信息科技大学机电工程学院, 北京 100192

1 空间机器人动力学方程

 图 1 空间机械臂示意图 Fig.1 Space manipulator sketch
 ${{r}_{i}}={{r}_{0}}+\sum\limits_{i=1}^{i}{{{\rho }_{i}}},\quad i=1,2,\cdots ,n$ (1)
 ${{\Omega }_{j}}={{\Omega }_{0}}+\sum\limits_{i=1}^{j}{{{\Omega }_{i-1,i}}},\quad j=1,2,\cdots ,n$ (2)

 图 2 邻接刚体示意图 Fig.2 Sketch of contiguous bodies
 ${{\rho }_{i}}={{b}_{i-1}}+{{d}_{i}}$ (3)

 ${{m}_{0}}{{r}_{0}}+\sum\limits_{i=1}^{n}{{{m}_{i}}}{{r}_{i}}=\mathbf{0}$ (4)

 ${{r}_{0}}=-\frac{1}{m}\sum\limits_{i=1}^{n}{{{m}_{i}}}\sum\limits_{j=1}^{i}{{{\rho }_{j}}}=-\frac{1}{m}\sum\limits_{j=1}^{n}{(}\sum\limits_{i=j}^{n}{{{m}_{i}}}){{\rho }_{j}}$ (5)

 ${{H}_{c}}={{I}_{0}}{{\Omega }_{0}}+{{m}_{0}}{{\tilde{r}}_{0}}{{\dot{r}}_{0}}+\sum\limits_{i=1}^{n}{{{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{I}}}_{i}}}{{\Omega }_{\text{c},i}}+\sum\limits_{i=1}^{n}{{{m}_{i}}}{{\tilde{r}}_{i}}{{\dot{r}}_{i}}$ (6)

 ${{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{I}}_{i}}={{R}_{0,i}}{{I}_{i}}R_{0,i}^{\text{T}}$ (7)

 ${{r}_{0}}=-\frac{1}{m}\sum\limits_{j=1}^{n}{\left[ (\sum\limits_{i=j}^{n}{{{m}_{i}}})({{R}_{0,i-1}}{{b}_{i-1}}+{{R}_{0,i}}{{d}_{i}}) \right]}$ (8)

 \begin{align} & {{{\dot{r}}}_{0}}=\frac{1}{m}\sum\limits_{j=1}^{n}{\left[ (\sum\limits_{i=j}^{n}{{{m}_{i}}})({{R}_{0,i-1}}{{{\tilde{b}}}_{i-1}}R_{0,i-1}^{\text{T}}+{{R}_{0,i}}{{{\tilde{d}}}_{i}}R_{0,i}^{\text{T}}) \right]}{{\Omega }_{0}}+ \\ & \frac{1}{m}\sum\limits_{j=1}^{n-1}{\left[ \sum\limits_{j=k}^{n-1}{(}\sum\limits_{i=k+1}^{n}{{{m}_{i}}}){{R}_{0,i}}{{{\tilde{b}}}_{i}}R_{0,i}^{\text{T}} \right.}+ \\ & \left. \sum\limits_{j=k}^{n}{(}\sum\limits_{i=k}^{n}{{{m}_{i}}}){{R}_{0,i}}{{{\tilde{d}}}_{i}}R_{0,i}^{\text{T}} \right]{{\Omega }_{j-1,j}}+ \\ & \frac{1}{m}{{m}_{n}}{{R}_{0,m}}{{{\tilde{d}}}_{m}}R_{0,m}^{\text{T}}{{\Omega }_{n-1,n}} \\ \end{align} (9)

 \begin{align} & {{{\dot{r}}}_{i}}={{{\dot{r}}}_{0}}-\sum\limits_{j=1}^{n}{({{R}_{0,i-1}}{{{\tilde{b}}}_{i-1}}R_{0,i-1}^{\text{T}}+{{R}_{0,i}}{{{\tilde{d}}}_{i}}R_{0,i}^{\text{T}}}){{\Omega }_{0}}- \\ & \sum\limits_{j=1}^{n-1}{\left[ \sum\limits_{i=j}^{n-1}{{{R}_{0,i-1}}{{{\tilde{b}}}_{i-1}}R_{0,i-1}^{\text{T}}+\sum\limits_{i=j}^{n}{{{R}_{0,i}}{{{\tilde{d}}}_{i}}R_{0,i}^{\text{T}}}} \right]}{{\Omega }_{j-1,j}} \\ & -{{R}_{0,i}}{{{\tilde{d}}}_{i}}R_{0,i}^{\text{T}}{{\Omega }_{i-1,i}} \\ \end{align} (10)

 ${{H}_{\text{c}}}={{J}_{0}}{{\Omega }_{0}}+\sum\limits_{i=1}^{n}{{{J}_{i}}}{{\Omega }_{i-1,i}}$ (11)

 \begin{align} & {{J}_{0}}={{I}_{0}}+\sum\limits_{j=1}^{n}{{{R}_{0,j}}{{I}_{j}}R_{0,j}^{\text{T}}+\frac{1}{m}({{m}_{0}}{{{\tilde{r}}}_{0}}+\sum\limits_{i=1}^{n}{{{m}_{i}}}{{{\tilde{r}}}_{i}})}\times \\ & \sum\limits_{k=1}^{n}{\left[ (\sum\limits_{i=k}^{n}{{{m}_{i}}})({{R}_{0,k-1}}{{{\tilde{b}}}_{k-1}}R_{0,k-1}^{\text{T}}+{{R}_{0,k}}{{{\tilde{d}}}_{k}}R_{0,k}^{\text{T}}) \right]}+ \\ & \sum\limits_{k=1}^{n}{\left[ (\sum\limits_{i=k}^{n}{{{m}_{i}}}{{{\tilde{r}}}_{i}})({{R}_{0,k-1}}{{{\tilde{b}}}_{k-1}}R_{0,k-1}^{\text{T}}+{{R}_{0,k}}{{{\tilde{d}}}_{k}}R_{0,k}^{\text{T}}) \right]} \\ \end{align} (12)
 \begin{align} & {{J}_{i}}=\left\{ \sum\limits_{j=1}^{n}{{{R}_{0,j}}}{{I}_{j}}R_{0,j}^{\text{T}}+\frac{1}{m}({{m}_{0}}{{{\tilde{r}}}_{0}}+\sum\limits_{i=1}^{n}{{{m}_{i}}}{{{\tilde{r}}}_{i}}) \right.\times \\ & \left[ \sum\limits_{k=1}^{n-1}{(}\sum\limits_{i=k-1}^{n}{{{m}_{i}}}){{R}_{0,k}}{{{\tilde{b}}}_{k}}R_{0,k}^{\text{T}} \right.+ \\ & \sum\limits_{k=1}^{n}{(}\sum\limits_{i=k}^{n}{{{m}_{i}}}){{R}_{0,k}}{{{\tilde{d}}}_{k}}R_{0,k}^{\text{T}}- \\ & \left. \sum\limits_{k=1}^{n-1}{(}\sum\limits_{i=k-1}^{n}{{{m}_{i}}}{{{\tilde{r}}}_{i}}){{R}_{0,k}}{{{\tilde{b}}}_{k}}R_{0,k}^{\text{T}} \right]- \\ & \left. \sum\limits_{k=1}^{n}{(}\sum\limits_{i=k}^{n}{{{m}_{i}}}{{{\tilde{r}}}_{i}}){{R}_{0,k}}{{{\tilde{d}}}_{k}}R_{0,k}^{\text{T}} \right\}{{R}_{0,k}}, \\ & i=1,2,\cdots ,n-1 \\ \end{align} (13)
 \begin{align} & {{J}_{n}}=[{{R}_{0,n}}{{I}_{n}}R_{0,n}^{\text{T}}+\frac{1}{m}({{m}_{0}}{{{\tilde{r}}}_{0}}+\sum\limits_{i=1}^{n}{{{m}_{i}}}{{{\tilde{r}}}_{i}})\times \\ & {{m}_{n}}{{R}_{0,n}}{{{\tilde{d}}}_{n}}R_{0,n}^{\text{T}}-{{m}_{n}}{{{\tilde{r}}}_{n}}{{R}_{0,n}}{{{\tilde{d}}}_{n}}R_{0,n}^{\text{T}}]{{R}_{0,n}} \\ \end{align} (14)

2 最优控制问题

 $J=\phi (x({{t}_{f}}),\ {{t}_{f}})+\int_{{{t}_{0}}}^{{{t}_{f}}}{G(x(t),\ u(t),\ t)}dt$ (20)

 $\dot{x}(t)=f(x(t),u(t),\ t)$ (21)

 $\varphi (x({{t}_{0}}),\ x({{t}_{f}}),\ {{t}_{0}},\ {{t}_{f}})=0$ (22)

 $C(x(t),\ u(t),\ t)\le \mathbf{0}$ (23)

 $J(u,\ t)=\int_{{{t}_{0}}}^{{{t}_{f}}}{\langle u(}t),\ u(t)\rangle dt$ (24)

 $\left| {{u}_{i}}(t) \right|<{{u}_{m}},i=1,2,\cdots ,n$ (25)

 $x({{t}_{0}})={{x}_{0}},x({{t}_{f}})={{x}_{f}}$ (26)
3 Legendre伪谱法

Legendre伪谱法属于一类典型的直接法，该方法使用正交多项式的根为配点，采用全局插值多项式作为有限基，将状态和控制变量在LGL点离散，通过Lagrange插值多项式逼近状态和控制变量，得到节点上的离散动力学方程.从而将状态方程的微分运算和性能函数中的积分运算转化为代数运算，最终将OCP转化为以节点处的状态、控制变量为待优化参量的NLP.

 $t=\frac{{{t}_{f}}+{{t}_{0}}}{2}-\frac{{{t}_{0}}-{{t}_{f}}}{2}\tau$ (27)

 ${{L}_{N}}(\tau )=\frac{1}{{{2}^{N}}N!}\frac{{{d}^{N}}}{d{{x}^{N}}}{{({{\tau }^{2}}-1)}^{N}}$ (28)

 ${{\Phi }_{i}}(\tau )=\frac{1}{N(N+1){{L}_{N}}({{\tau }_{i}})}\frac{({{\tau }^{2}}-1){{{\dot{L}}}_{N}}({{\tau }_{i}})}{\tau -{{\tau }_{i}}}$ (29)

 $x(\tau )\approx X(\tau )=\sum\limits_{i=0}^{N}{{{\Phi }_{i}}(\tau ){{X}_{i}}}$ (30)
 $u(\tau )\approx U(\tau )=\sum\limits_{i=0}^{N}{{{\Phi }_{i}}(\tau ){{U}_{i}}}$ (31)

 $\dot{x}({{\tau }_{k}})\approx {{\dot{X}}_{k}}=\sum\limits_{i=0}^{N}{{{{\dot{L}}}_{i}}({{\tau }_{k}}){{X}_{i}}}=\sum\limits_{i=0}^{N}{{{D}_{ki}}{{X}_{i}}}$ (32)

 ${{D}_{ki}}=\left\{ \begin{array}{*{35}{l}} \frac{{{L}_{N}}({{\tau }_{k}})}{{{L}_{N}}({{\tau }_{j}})}\cdot \frac{1}{{{\tau }_{k}}-{{\tau }_{j}}}, & k\ne j \\ -\frac{N(N+1)}{4}, & k=j=0 \\ \frac{N(N+1)}{4}, & k=j=N \\ 0, & \text{other} \\ \end{array} \right.$ (33)

 $\sum\limits_{i=0}^{N}{{{D}_{ki}}{{X}_{i}}}-\frac{{{t}_{f}}-{{t}_{0}}}{2}f({{X}_{k}},\ {{U}_{k}},\ {{\tau }_{k}})=\mathbf{0}$ (34)

 $\varphi ({{X}_{0}},\ {{X}_{N}},\ {{\tau }_{0}},\ {{\tau }_{N}})=0$ (35)
 $C({{X}_{k}},\ {{U}_{k}},\ {{\tau }_{k}})\le \mathbf{0}$ (36)

 $J=\phi +\frac{{{t}_{f}}-{{t}_{0}}}{2}\sum\limits_{i=0}^{N}{{{w}_{i}}G({{X}_{i}},\ {{U}_{i}},\ {{\tau }_{i}})}$ (37)

 ${{w}_{i}}=\int_{-1}^{1}{{{\Phi }_{i}}(\tau )d\tau =}\frac{2}{N(N+1)}{{\left[ {{L}_{N}}\left( {{\tau }_{i}} \right) \right]}^{-2}}$ (38)

 $\left| {{U}_{ik}} \right|\le {{U}_{\max }}$ (41)

4 仿真算例

 图 3 带空间机械臂的刚体航天器简图 Fig.3 Rigid spacecraft with space manipulators diagram

 $\dot{x}=\left[ \begin{matrix} {{E}_{3\times 3}} \\ -{{\Gamma }^{-1}}J_{0}^{-1}[{{J}_{1}}{{e}_{1}}{{J}_{2}}{{e}_{2}}{{J}_{3}}{{e}_{3}}] \\ \end{matrix} \right]u$ (42)

 \begin{align} & {{x}_{0}}={{[-\pi /2-\pi /2-0.5-1-11]}^{\text{T}}} \\ & {{x}_{f}}={{[\pi /2-\pi /2-0.5-1-11]}^{\text{T}}} \\ \end{align}

 图 4 空间机械臂关节角运动优化轨迹 Fig.4 Space manipulator joint angle motion optimization trajectory
 图 5 航天器载体姿态运动优化轨迹 Fig.5 Spacecraft attitude motion optimization trajectory
 图 6 空间机械臂最优控制输入规律 Fig.6 Optimal control input of space manipulator

 \begin{align} & {{x}_{0}}={{[-\pi /2-\pi /2-0.51-1-1]}^{\text{T}}} \\ & {{x}_{f}}={{[-\pi /2-\pi /2-0.5111]}^{\text{T}}} \\ \end{align}

 图 7 空间机械臂关节角运动优化轨迹 Fig.7 Space manipulator joint angle motion optimization trajectory
 图 8 航天器载体姿态运动轨迹 Fig.8 Spacecraft attitude motion optimization trajectory
 图 9 空间机械臂最优控制输入规律 Fig.9 Optimal control input of space manipulator

5 结论

(1) 应用Legendre 伪谱法求解了空间机器人的最优路径规划问题.在已知系统初末位形的情况下，文中以机械臂关节最小能量为待优化目标函数，同时考虑实际应用中的控制输入受限，将空间机器人的最优路径规划问题离散化为非线性参数规划问题，从而应用成熟的非线性规划算法进行求解.

(2) 伪谱法通过对系统状态空间和最优空间的全局逼近，将状态变量和控制变量同时离散为一组未知参数，使得非线性规划问题的未知量数目较多.但考虑到伪谱法可以利用少量的节点而获得"谱精度"，因此在获得同等精度条件下，Legendre 伪谱法的未知量的个数要远小于基于有限元配置的直接法，使得Legendre伪谱法的计算速度和计算精度都要优于传统的基于有限元配置的直接法.

(3) 本文的结果主要针对非完整系统模型，应用伪谱法得到的自由漂浮空间机器人最优路径能够较好地满足各种约束条件，并具有精度较高和计算速度快的优点，为解决非完整运动规划问题提供了一条新的途径.

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PATH PLANNING OF FREE FLOATING SPACE ROBOT USING LEGENDRE PSEUDOSPECTRAL METHOD
Ge Xinsheng1, Chen Kaijie2
1. School of Applied Science, Beijing Information Science & Technology University, Beijing 100192, China ;
2. School of Mechanical & Electrical Engineering, Beijing Information Science & Technology University, Beijing 100192, China
Abstract: Based on the Legendre pseudospectral method, the optimal control of free floating space robots path planning problems are studied. Free floating is the working status for the space robots in task and path planning is the foundation for them to fulfil a complex space task. Because the space robots have no fixed pedestal and there are nonholonomic constraints between the manipulator and the carrier, and it makes the path planning for free floating space robots different from those on the ground. In this paper, the Legendre pseudospectral method which can realize the optimal control of free floating space robots path planning problem is presented. Firstly, a dynamic model of free floating space robots is established by using multi-body dynamics theory. The path planning problem of Bolza with certain initial and terminal stance is then obtained. Here, we select the minimum joint dissipation as performance index and consider the actual control input saturation. Then, by applying the Legendre pseudospectral method, the state and control variables are discretized at Legendre-Gauss-Lobatto (LGL) point and Lagrange interpolation polynomials are constructed to approximate the state and control variables. The problem of continuous path optimization is discretized to solve a nonlinear programming problem. Finally, results of the numerical simulation show that using the application of Legendre pseudospectral method to solve the problem of free floating space robot path planning can get the optimal trajectory of manipulator and carrier. It can fit various constraint conditions well. And this method is with fast simulation calculation, high accuracy and good real-time performance.
Key words: space robot    Legendre pseudospectral    nonholonomic constraint    path planning    optimal control