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

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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

4 0 determines the deceleration magnitude when

the robot joints approach the joint-velocity limits

. Coefﬁcients

and

should be chosen such that the feasible region of

€

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

(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

. Thus, (25) and (26) can be further improved into the

following expression:



¼ maxf

ðð



Þ

Þ,



Þ,

€



¼ minf

ðð



Þ

Þ,



Þ,

€

i.e., the ones in Theorem 2, where

4 0 is a small constant vector

used to scale the safety region, with each element

(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

:¼fu9u



r ur u

gR

n þ m

such that

ðuu

ðMu

þqÞZ 0 8uA

, ð27Þ

where the primal-dual decision variable vector uA R

n þ m

and its

upper and lower bounds u

are deﬁned, respectively, as

u ¼

, u



: ð28Þ

Appearing in the above equation, y A R

is a dual decision variable

vector deﬁned corresponding to equality constraint (22),

¼½1, ..., 1

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 mÞðn þ mÞ

and vector qA R

n þ m

are deﬁned as

M ¼

W J

J 0

, q ¼

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

, 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]:

ðuðMuþqÞÞu ¼ 0, ð29Þ

where P

ðÞ is a simple projection operator from R

n þ m

onto set

and the ith output of P

ðuÞ is deﬁned as



if u

o u



if u



r u

if u

4 u

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

ðIþ M

ÞðP

ðuðMuþqÞÞuÞ: ð30Þ

In this paper, by canceling the scaling term ðIþM

Þ of LVI-PDNN

(30), we obtain the following simpliﬁed LVI-based primal-dual

neural network:

u ¼

ðP

ðuðMuþqÞÞuÞ: ð31Þ

In (30) and (31),

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

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

. In addition, the exponential convergence rate is

proportional to

Proof. Can be generalized from [23,37]. &

Corollary 3.1. The existence of equilibrium point u

of the S-LVI-

PDNN model (31) is equivalent to the nonempty of the feasible region

made by constraints (22) and (23), denoted by

ðxÞ. In the robotic

QP formulation (21)–(23),

ðxÞa | if and only if the end-effector

path-tracking task is feasible.

Contrary to Corollary 3.1, the nonexistence of u

(or x

) 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

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