基于简化LVI的神经网络解决冗余平面机器人加速度级重复运动规划
110 浏览量
更新于2024-07-15
收藏 1015KB PDF 举报
本文主要探讨了冗余平面机器人在关节加速度级的重复运动规划问题，提出了一种基于简化LVI（Linear Velocity Inequality，线性速度不等式）的混合神经网络求解方法。这项创新的研究由杨云龙等人进行，他们隶属于中山大学信息科学技术学院和深圳中山大学研究院，地址分别位于广州和深圳。
传统的重复运动规划（Repetitive Motion Planning, RMP）通常关注关节角度和速度控制，然而，这篇论文着重于加速级别，即关节加速度的约束，这在实际工业应用中具有重要意义，例如精密装配、精细操作或高精度重复任务中，对机器人的动态性能有着更高的要求。为了实现这一目标，作者借鉴了张等人提出的神经动力学方法，该方法能够有效地处理多链接平面机器人系统中的复杂动力学特性。
张等人通过简化LVI基础的原-对偶神经网络结构，设计了一套适用于冗余平面机器人系统的加速级RMP（Accelerated RMP, ARMP）方案。这个方案的关键在于将关节角度限制、关节速度限制和关节加速度限制整合到一个统一的数学框架中，通过二次优化的形式表达，使得运动规划更加精确且鲁棒。这种方法的优势在于它能够在保证运动质量的同时，考虑到冗余度带来的灵活性，提高了机器人执行复杂轨迹的能力。
ARMP方案的实施步骤可能包括以下几个阶段：
1. **模型建立**：首先，构建冗余平面机器人动力学模型，考虑关节加速度作为决策变量。
2. **神经网络设计**：利用简化LVI，设计神经网络架构，以便在网络内部处理速度不等式约束，确保运动的可行性。
3. **优化算法**：将关节加速度级的RMP转化为一个二次优化问题，利用神经网络的非线性映射能力求解最优加速度路径。
4. **收敛性与稳定性分析**：证明简化LVI的神经网络方法在求解过程中保持收敛性和稳定性，确保规划出的轨迹是可执行的。
5. **实验验证**：通过数值仿真和实际机器人实验验证ARMP方案的有效性和优越性，展示其在冗余平面机器人上的实际应用效果。
总结来说，这篇研究论文提出了一个新颖的、实用的重复运动规划策略，不仅提升了机器人在复杂运动任务中的性能，还为理论与实际应用之间的桥梁搭建了坚实的基础。该成果对于推动机器人技术在制造业和其他领域的应用具有重要价值。
joints enter the aforementioned critical area, and the intensity
coefficient
k
v
4 0 determines the deceleration magnitude when
the robot joints approach the joint-velocity limits
_
y
8
. Coefficients
k
p
and
k
v
should be chosen such that the feasible region of
€
y
converted by joint-angle limits (18) and joint-velocity limits (19)
is normally not smaller than the original one made by joint-
acceleration limits (20) .
However, in practical planar robot arms, the safety devices are
always used, in addition to the indefiniteness of
y
8
(i.e., the
values can be positive, zero or negative). If a joint of the robot arm
approaches or reaches its physical limit, the joint would be locked
by the physical safety device throughout the task duration, which
would lead to the failure of the end-effector task execution. It is
thus necessary to leave some ‘‘safety region’’ to the joint physical
limits
y
7
. Thus, (25) and (26) can be further improved into the
following expression:
z
i
¼ maxf
k
p
ðð
y
i
þ
W
i
Þ
y
i
Þ,
k
v
ð
_
y
i
_
y
i
Þ,
€
y
i
g,
z
þ
i
¼ minf
k
p
ðð
y
þ
i
W
i
Þ
y
i
Þ,
k
v
ð
_
y
þ
i
_
y
i
Þ,
€
y
þ
i
g,
i.e., the ones in Theorem 2, where
W
4 0 is a small constant vector
used to scale the safety region, with each element
W
i
(i ¼ 1, ..., nÞ
set to be, for example, 0.1745 rad (i.e., 10 degrees) in the ensuing
simulations of this paper. Thus, we have the QP reformulation
(21)–(23), and the proof is now complete. &
It is worth mentioning that the QP formulation of the ARMP
scheme cannot be derived by using the gradient-based neural
network (GNN) method. For details of the mathematical proof,
refer to Appendix B. In this sense, such novel derivations and
comparison also show the efficacy and superiority of the ZNN
approach on robotic problems solving and schemes design.
3. S-LVI-PDNN solver
As we know, the dual neural network approach requires online
inversion of coefficient matrix W [5]. In this section, to overcome
the weakness and improve the efficacy of the online solution of
QP (21)–(23), we present a simplified LVI-based primal-dual
neural network by using the following design method [23].
First, we convert QP problem (21)–(23) to a linear variational
inequality (LVI) problem. That is, to find a primal-dual equili-
brium vector u
n
A
O
:¼fu9u
r ur u
þ
gR
n þ m
such that
ðuu
n
Þ
T
ðMu
n
þqÞZ 0 8uA
O
, ð27Þ
where the primal-dual decision variable vector uA R
n þ m
and its
upper and lower bounds u
7
are defined, respectively, as
u ¼
x
y
"#
, u
þ
¼
z
þ
$
1
v
"#
, u
¼
z
$
1
v
"#
: ð28Þ
Appearing in the above equation, y A R
m
is a dual decision variable
vector defined corresponding to equality constraint (22),
1
v
¼½1, ..., 1
T
denotes an appropriately dimensioned vector com-
posed of ones, and
$
b 0 is sufficiently large to replace þ1
numerically for simulation and implementation purposes. Coeffi-
cient matrix MA R
ðn þ mÞðn þ mÞ
and vector qA R
n þ m
are defined as
M ¼
W J
T
J 0
"#
, q ¼
~
z
d
:
In summary, we can have the following important result (with
proof omitted due to space limitation and generalizable from
[22,28]).
Lemma 1. With the existence of optimal solution x
n
, the QP problem
(21)–(23) can be converted to the LVI problem (27).
Second, it is known that LVI problem (27) is equivalent to the
following piecewise-linear equation [5,34,36]:
P
O
ðuðMuþqÞÞu ¼ 0, ð29Þ
where P
O
ðÞ is a simple projection operator from R
n þ m
onto set
O
,
and the ith output of P
O
ðuÞ is defined as
u
i
if u
i
o u
i
,
u
i
if u
i
r u
i
r u
þ
i
,
u
,
if u
i
4 u
þ
i
:
8
>
<
>
:
8iA f1, 2, ..., nþmg
Third, to solve (29), guided by design experience [5 ,34], we can
adopt the following dynamics [which is actually the LVI-PDNN
solver for QP (21)–(23)]:
_
u ¼
g
ðIþ M
T
ÞðP
O
ðuðMuþqÞÞuÞ: ð30Þ
In this paper, by canceling the scaling term ðIþM
T
Þ of LVI-PDNN
(30), we obtain the following simplified LVI-based primal-dual
neural network:
_
u ¼
g
ðP
O
ðuðMuþqÞÞuÞ: ð31Þ
In (30) and (31),
g
A R is a positive design parameter used to adjust
the convergence rate of the neural networks. Besides, we can have
the following theoretical results about the S-LVI-PDNN (31)
[23,28].
Theorem 3. Assume the existence of optimal solution x
n
to the
strictly convex QP problem depicted in (21)–(23). Being the first n
element of state u(t), output x(t) of the simplified LVI-based primal-
dual neural network (31) is globally exponentially convergent to
optimal solution x
n
. In addition, the exponential convergence rate is
proportional to
g
.
Proof. Can be generalized from [23,37]. &
Corollary 3.1. The existence of equilibrium point u
n
of the S-LVI-
PDNN model (31) is equivalent to the nonempty of the feasible region
made by constraints (22) and (23), denoted by
U
ðxÞ. In the robotic
QP formulation (21)–(23),
U
ðxÞa | if and only if the end-effector
path-tracking task is feasible.
Contrary to Corollary 3.1, the nonexistence of u
n
(or x
n
) means
the no-solution case. A large number of randomly generated
numerical tests [23] show that, although state y may usually be
divergent, the output x(t) of S-LVI-PDNN (31) is always conver-
gent. This property is very useful in robotic applications. That is,
even if the end-effector path-tracking task is not feasible, the
output x(t) of S-LVI-PDNN (31) is always convergent and provides
an approximate solution to the ARMP scheme.
Remarks. Most controllers are digital in the robotic industry.
To implement the S-LVI-PDNN in practice, in addition to analog
circuit implementations, MATLAB routine ‘‘ODE’’ as an alternative
approach can be employed to generate discrete-time solutions of
such a neural network. In this paper, ODE15SðODEFUN, ½0 : 0:01 :
T, Y0, OPTIONSÞ is adopted. The solutions generated by such an
‘‘ODE15s’’ solver are discrete-time with a fixed time gap being
0.01 s, and can meet the requirement of digital controllers. Thus,
the S-LVI-PDNN is realizable and applicable in the robotic
industry.
4. Application to three-link planar arm
In order to test the proposed ARMP scheme (16)–(20 ),computer
simulations are performed based on a physically constrained three-
link planar robot arm by using the simplified LVI-based primal dual
Y. Zhang et al. / Robotics and Computer-Integrated Manufacturing 29 (2013) 328–343 331
剩余15页未读，继续阅读
2021-05-21 上传
2014-12-24 上传
2021-02-11 上传
2021-05-25 上传
2021-05-03 上传
2021-05-12 上传
2021-05-30 上传
2010-08-05 上传
2021-05-16 上传
weixin_38545517
- 粉丝: 2
- 资源: 957
我的内容管理
展开
最新资源
- MATLAB实现小波阈值去噪：Visushrink硬软算法对比
- 易语言实现画板图像缩放功能教程
- 大模型推荐系统: 优化算法与模型压缩技术
- Stancy: 静态文件驱动的简单RESTful API与前端框架集成
- 掌握Java全文搜索：深入Apache Lucene开源系统
- 19计应19田超的Python7-1试题整理
- 易语言实现多线程网络时间同步源码解析
- 人工智能大模型学习与实践指南
- 掌握Markdown：从基础到高级技巧解析
- JS-PizzaStore: JS应用程序模拟披萨递送服务
- CAMV开源XML编辑器：编辑、验证、设计及架构工具集
- 医学免疫学情景化自动生成考题系统
- 易语言实现多语言界面编程教程
- MATLAB实现16种回归算法在数据挖掘中的应用
- ***内容构建指南：深入HTML与LaTeX
- Python实现维基百科“历史上的今天”数据抓取教程
