高斯回归策略改进RRT*：GMR-RRT*在移动机器人路径规划中的应用

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"GMR-RRT*：基于采样的高斯回归策略改进RRT算法的路径规划研究" 在机器人自主路径规划领域，高效且优化的算法是实现广泛应用的关键。传统的基于采样的快速探索随机树（RRT）算法在路径规划方面取得了显著成就，但它们往往需要较长的时间来找到最优解，这在实时性和效率上存在挑战。为了解决这一问题，本文提出了一种新的路径规划算法——Gaussian Mixture Regression RRT*（GMR-RRT*），该算法结合了高斯混合回归（GMR）与RRT*算法家族的优点，旨在提高路径规划的质量，同时确保较短的计算时间和更优的路径长度。 Gaussian Mixture Regression（GMR）是一种统计建模技术，用于拟合数据集中的复杂分布。在路径规划中，GMR可以用于学习环境中的障碍物分布，并预测机器人在未知环境中移动时可能遇到的障碍。通过这种方式，GMR-RRT*能够更准确地评估样本点的安全性，从而生成更安全、更平滑的路径。 RRT*（Rapidly-exploring Random Tree Star）是RRT的一种优化版本，它引入了重规划机制来逐步改进初始路径，目标是找到全局最优解。在GMR-RRT*中，GMR的引入不仅加速了路径优化过程，还提高了路径的质量。具体来说，GMR-RRT*通过GMR模型预测未来路径上的潜在碰撞风险，从而在路径搜索过程中更加智能地选择和调整样本点，减少了不必要的探索，提高了路径规划的效率。文章详细阐述了GMR-RRT*算法的设计原理、实现步骤以及性能分析。首先，算法描述了如何构建和更新GMR模型，以适应不断变化的环境。接着，它解释了如何在RRT*框架下集成GMR，包括如何利用GMR预测样本点的安全性，以及如何根据预测结果调整路径。此外，论文还通过仿真和实际环境下的实验，对比了GMR-RRT*与其他传统路径规划算法（如原始的RRT和RRT*）的表现，证明了GMR-RRT*在路径质量、计算时间和鲁棒性方面的优势。 GMR-RRT*算法是机器人路径规划领域的创新性工作，它将机器学习中的GMR方法与经典的RRT*算法相结合，为实时和高效的路径规划提供了一种新途径。这一研究对于提升自动驾驶车辆、无人机和其他移动机器人在复杂环境中的自主导航能力具有重要意义。

2379-8858 (c) 2021 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIV.2022.3150748, IEEE

Transactions on Intelligent Vehicles

IEEE TRANSACTIONS ON INTELLIGENT VEHICLES 3

Chi et al. propose the Comfort and Collision Risk (CCR)

map to model human behavior. Then a sampling-based

algorithm is applied on the CCR map to achieve socially-

complaint path planning. In [15], the Social Force Model

(SFM) is introduced to achieve human-aware navigation in

crowded environments. Later in [16], the Extended SFM

is taken in proactive kinodynamic planning. Recently, in

[17], Wang et al. utilize the potential function to model

the relationship between robots and humans to achieve

socially-compliant path planning. By combing membrane

computing rules and potential led methods, MemEAPF

[18] and MemPBPF [19] are proposed to handle mo-

bile robot path planning. An electrostatic potential eld

method [20] is proposed for mobile robot path planning

in scattered environments. These model-based methods

approximately model human behavior, but the robot still

considers the human as a dierent obstacle. The robot

cannot directly imitate human navigation behavior.

The learning-based method aims at generating trajecto-

ries that account for human behaviors by learning collected

data from real demonstrations. The Inverse Reinforcement

Learning (IRL) technique is often used to nd a novel

cost function to guide the robot path planning. In [21],

Noe et al. use the IRL to learn RRT*’s cost function

from demonstrations. In [22], Henrik et al. consider mod-

eling cooperative navigation behavior of humans using

the IRL technique to achieve socially-compliant mobile

robot navigation. Deep Learning (DL) technique is also

widely applied in this area. Mavrogiannis et al. [23] apply

the Long Short-Term Memory (LSTM) architecture to

learn multi-agent path topologies for socially competent

planning. Convolutional neural network (CNN) is used in

[24] to train a drone to y in the civilian environment

autonomously. Generative Adversarial Network (GAN)

[25] can be also used to learn heuristics for sampling-based

path planning. Meanwhile, a recurrent generative model

[26] is proposed to achieve ecient heuristic generation

for robot path planning. The learning-based method can

help the robot directly imitate human navigation behavior,

which has gained great popularity. However, the learning-

based method requires complex network models and long

oine training sessions.

Unlike the methods mentioned above, the GMR-RRT*

employs the GMR as a learning tool to learn from human

behavior with the Expectation Maximization (EM) algo-

rithm. Then the learned distribution acts as a nonuniform

sampling function to guide the sampling-based path plan-

ner to generate a feasible solution quickly. The simplicity

of the GMR-RRT* allows it to be more ecient and

aordable in practical applications.

III. Algorithm

In this section, we rst introduce how to use the GMR

to learn navigation behavior from human demonstrations.

The learned behavior is then applied in the GMR-RRT*

to achieve fast and high-quality path planning. At last,

we prove that the GMR-RRT* guarantees probabilistic

completeness and asymptotic optimality.

A. Gaussian Mixture Regression

In this subsection, we follow the previous work to give

a preliminary introduction of the GMR [9] and discuss

how to adapt it to the sampling-based path planning.

The GMR utilizes the Gaussian conditioning theorem to

compute the distribution of output given input. Firstly,

the GMM encodes the joint distribution of input and

output using the EM algorithm to calculate the model

parameters. Secondly, the output given the input (spatial

data given temporal data in this paper) is computed

through a linear combination of conditional expectations.

Therefore, the GMR relies on the learned joint distribution

instead of deriving the regression function directly. An

illustration of the GMR calculation process is provided in

Fig. 4.

For each human demonstrated trajectory, the length is

rescaled to a xed value N. The demonstrated trajectory

ξ = {ξ

}

i=1

is dened by N observations ξ

∈ R

. Each

datapoint ξ

= {ξ

, ξ

} encodes time ξ

∈ R and spatial

position ξ

∈ R

. All trajectories are fed into the GMM

with K components, and the probability density function

is dened as

p(ξ

) =

k=1

p(ξ

|k) (1)

k=1

N (ξ

; µ

, Σ

)

k=1

(2π)

−

((ξ

−µ

)

∑

−1

(ξ

−µ

))

where {p

, µ

, Σ

} are the prior, mean, and covariance

matrix of the Gaussian component k. When applying the

GMR to learn the navigation behavior, the temporal value

is used as the query point, and the corresponding spatial

value ξ

is estimated through regression. Therefore, µ

and

can be represented separately as

= {µ

t,k

, µ

s,k

}, Σ



tt,k

ts,k

st,k

ss,k



. (2)

Given ξ

, the conditional distribution of ξ

in each Gaus-

sian model k is

p(ξ

|ξ

, k) = N (ξ

;

s,k

ss,k

s,k

= µ

s,k

+ Σ

st,k

(Σ

tt,k

)

−1

(ξ

− µ

t,k

ss,k

= Σ

ss,k

− Σ

st,k

(Σ

tt,k

)

−1

ts,k

, (3)

where

s,k

is the conditional expectation and

ss,k

the estimated conditional covariance. The complete con-

ditional distribution of ξ

given ξ

p(ξ

|ξ

) =

k=1

N (ξ

;

s,k

ss,k

), (4)

where β

= p(k|ξ

) is the so-called responsibility of the

Gaussian component k

p(ξ

|k)

i=1

p(ξ

|i)

N (ξ

; µ

t,k

, Σ

tt,k

)

i=1

N (ξ

; µ

t,i

, Σ

tt,i

)

. (5)

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