\section{Relation to concrete mathematics}





	
	\section{Introduction}\label{intro}
	
	Autonomous robot navigation with a complete map has been well studied~\cite{planning} in the last few years. 
	However, maps that have global knowledge are not always available in many situations, such as in search and rescue scenarios or rapidly changing environments. 
	Conventional path planning approaches mostly rely on accurate, static models of the environments, and they often fail to function when such models (i.e., maps) are not known. 
	As a result, robust navigation approaches in unknown environments are needed to help navigate the robot to its target using local perception.
	
	
	
	In recent years, deep reinforcement learning (DRL) algorithms have been successfully applied to navigation in unknown environments~\cite{tai2017virtual,ral2018mapless,icra2020discrete}. These data-driven methods learn their navigation policy by trial-and-error explorations combined with carefully designed reward signals. The policy usually directly maps the perception inputs, like 2D laser data or local costmap to low-level control commands, such as the translational and rotational velocities that the robot aims to achieve. 
	However, existing methods hardly focus on the local minimum problem~\cite{lmp}, which is more common and difficult in unknown environments. As a result, they only handle relatively simple environments without dead ends.
	In this paper, we investigate solving this problem with DRL by formulating the navigation task differently.

	
	

	
	Before illustrating our method, we introduce two important parameters in conventional navigation methods---the forward simulation time (FST) and the \emph{control interval}. FST is the amount of time to forward-simulate trajectories, which can be considered as how far forward the robot plan.
	\emph{control interval} is the reciprocal of control frequency, which means executing each command for how long. 
	Clearly, the smaller the \emph{control interval} is, the more timely the robot can react to its surroundings. Therefore, the \emph{control interval} is usually set to be small, which depends on the features of hardware, like the computing power of the platform. 
	On the contrary, there usually exists a ``sweet value range'' for the choice of FST. If the FST is smaller than this range, the robot will do short-term planning and be more likely to be trapped in local minimums. When the FST is longer, it will be computationally expensive and hard for the robot to find its way in environments filled with obstacles. 
	In this sense, FST is a critical parameter for the performance of most classical navigation methods~\cite{dwa,teb}. 
	
	On the other hand, most existing DRL-based navigation methods identify the two parameters (FST and \emph{control interval}) during their training in simulation environments. 
	In the setting of most DRL methods, the FST is equal to the \emph{control interval}. These methods usually set their \emph{control interval} to a fixed small value and their FST is as small as the fixed \emph{control interval}.
	As mentioned above, there are problems with FST being too large or too small. Thus, we have a natural idea---introducing adaptive FST in DRL navigation.
	
	
	Similar to the idea of adaptive FST, there are existing works in reinforcement learning that adjust the duration of actions\footnote{In reinforcement learning, FST can also be considered as the \emph{control interval}, which are both equal to the notation of action duration}, which is called dynamic action repetition. 
	It allows the agent to calculate an action once and repeat the same action for multi-steps. The number of steps adjusts dynamically w.r.t the surroundings. Therefore, they can accomplish the task with lower planning frequency and achieve better real-time performance~\cite{learning2repeat}. 
	Moreover, action repetition can be considered as a way to introduce temporal abstractions in the state transition, which enables the transitions between temporally distant advantageous states~\cite{durugkar2016deep, lakshminarayanan2017dynamic}.
	These transitions are exactly what we need to lead our robot to escape from the local minimum. 
	
	
	Inspired by dynamic action repetition, we propose Adaptive Forward Simulation Time (AFST), a DRL-based navigation method that can handle complex unknown environments with local minimums by dynamically adjusting the FST.
	To suitably model this planning problem with different duration of the actions, we propose to formulate the navigation task as a Semi-Markov Decision Process (SMDP)~\cite{rl4cmdp}, each of whose actions consists of three components: translational velocity, rotational velocity, and duration. 
	

	
	To efficiently adapt the FST, we face the following two main challenges: the larger volume of the continuous action space that is required to be sampled, as well as the difficulty of obtaining stable improvement despite the diverse action duration. 
	To address the first challenge, we devise a transformation that converts our 3D action space to a 2D trajectory parameter space (TP-space)~\cite{blanco2008extending}.  
	In the paper, we will show that such 3D-to-2D conversion can reduce the action space for the training of navigation policies. In specific, we will group possible actions of the robot into multiple disjoint sets, choose the most ``efficient'' action for each set, and train the navigation policy by only considering these ``efficient'' actions.
	Notice that, this reduction can greatly improve the efficiency of the DRL training.


	We address the second challenge by improving Distributed Proximal Policy Optimization (DPPO) algorithm~\cite{dppo} in the SMDP problem. 
	We obtain stable improvement by a better estimation of the policy gradient for the SMDP.
	The default policy estimator of DPPO is Generalized Advantage Estimation (GAE)~\cite{schulmanetal2016high}, which can substantially reduce the variance of policy gradient estimates at the cost of acceptable bias. However, the original GAE focuses on MDP problems, and we revise the discount rate in the policy gradient estimates to suit our SMDP problem. 
	The extended GAE (EGAE) can also make a trade-off between variance and bias. Besides, we prove that EGAE can theoretically introduce no bias under certain circumstances.
	
	We evaluate our approach under multiple scenarios both in the simulator and the real world. 
	We compare AFST with a classical motion planner and three DRL-based approaches.
	The experimental results show that AFST can efficiently accomplish navigation tasks in unknown environments.
	We also conduct ablation experiments to demonstrate the effectiveness of our components. 
	Our main contributions are summarized as follow:
	\begin{itemize}
		\item We propose the first DRL-based navigation method modeled by a SMDP with continuous action space.
		\item We improve the DPPO algorithm for the specified SMDP problem by modifying its GAE to better estimate the policy gradient in SMDPs.
		\item We provide a transformation that converts the
		3D action space to a 2D TP-space and reduce the training difficulty.
	
	\end{itemize}
	
	
	
	\section{Related Work}\label{related work}
	The navigation problem has been formulated as a SMDP problem before. In previous work such as~\cite{mahadevan1999robust,saha2017real}, SMDP models were used. However, they all use tabular reinforcement learning algorithms, which limits their state space and action space. Hence, their generalization and practicability cannot be compared with DRL-based methods. 
	


	
	
	DRL-based navigation approaches can be roughly
	divided into two categories: methods with agent-level inputs
	and methods with sensor-level inputs. In specific, methods
	with agent-level inputs~\cite{chen2017decentralized, everett2018motion} require expensive motion data of other robots (like velocities) for the states of the DRL model and methods with sensor-level inputs~\cite{tai2017virtual,ral2018mapless,icra2020discrete,chen2020distributed} consider sensor inputs as the states of the DRL model.
	Our method follows sensor-level inputs.
	
	Many existing methods with sensor-level inputs directly map the sensor-level inputs into the robot's low-level control commands.
	Thus, only transitions between temporally adjacent states are allowed in their framework, which makes it difficult to escape the local minimums.
	A few approaches~\cite{brito2021go,chane2021goal} try to solve long horizon tasks by selecting subgoals as actions and reaching subgoals with a fine-tuned prior policy. 
	These methods use a hierarchical architecture which is an efficient way to decompose the difficult tasks.
	However, their performance heavily depends on the fixed prior policy which cannot be improved during training and requires an extensive amount of engineering effort.
	Differently, our policy is entirely learned from interactive experience automatically with little prior knowledge.
	
	The difference between our method and normal DRL-based methods is that we introduce the time dimension to the action space. This duration of executing an action has a significant impact on the exploration efficiency and the variance of value function estimation in DRL, which has been illustrated in~\cite{durugkar2016deep, learning2repeat,lakshminarayanan2017dynamic}. Another novelty of our method is the
	dimensionality reduction of the action space.
	We found no other work that considers how to reduce the
	action space for RL-based robot navigation.
	
	
	
	\section{Preliminaries}\label{prelim}
	
	\subsection{Semi-Markov Decision Process.}
	
	To treat temporal abstraction as a minimal extension of the
	reinforcement learning framework, the theory of semi-Markov Decision Process (SMDP)~\cite{rl4cmdp} is built. A SMDP is a tuple $(S, A, P, R, T, \gamma)$, where $S$ is the state space, $A$ is the action space, $T$ is the transition time space, and $\gamma \in(0,1]$ is the discount factor. 
	We assume the environment has transition dynamics $P\left(s^{\prime}, \tau \mid s, a\right)$ which is unknown to the agent, where $\tau$ represents the time between taking action~$a$ in observed state~$s$ and arriving in the next state~$s^{\prime}$ and can take a new action. 
	In general, we are given the reward function $r$ for the reward after observing~$s^{\prime}$.
	The goal throughout is to learn a policy maximizing long-term expected rewards $\mathbb{E}\left[\sum_{i=0}^{L-1} \gamma^{t_{i}} r_{t_{i}}\right]$
	with a time horizon $T=t_{L}$, where $r_{t_{i}}$ is the reward from action $a_{t_{i}}$, $t_0 = 0$, $t_{i}=\sum_{j=0}^{i-1} \tau_{t_{j}}$, and $\tau_{t_{j}}$ is the duration of action $a_{t_{j}}$ which is executed from $t_{j}$.
	
	
	\subsection{Formulation of Navigation in SMDP.}\label{formulation}
	
	In this paper, we specify the navigation task as a SMDP problem.
	Specifically, an action~$a$ is formulated as a triple $(v, \omega, d)$, where $v$ and $\omega$ denote the required translational and rotational velocities for the robot, and $d$ denotes the execution duration for both velocities (i.e. FST). 
	A state~$s$ consists of the pose of the robot, i.e., $(x, y, \theta)$ for the robot's position $(x,y)$ and the orientation $\theta$, as well as the corresponding observation of the robot at the position.
	If there is no collision during the execution, then the robot would execute the velocities $v$ and $\omega$ for the time interval $d$, which results in the next state $s'$.
	However, the robot may collide and end an action before completing its given execution duration, which results in the transition dynamics $P\left(s^{\prime}, \tau \mid s, a\right)$, with $\tau \in [0, d]$ denoting the actual execution time.
	A policy $\pi(a\mid s)$ specifies the probability of taking action~$a$ at state $s$. The reward function $r$ encourages the navigation policy to avoid collisions and arrive at the target position as soon as possible. 
	
	\subsection{Trajectory Parameter Space with Velocities}\label{TP-space}
	To decrease the number of samples typically required, we convert our 3D action space to a 2D trajectory parameter space (TP-space)~\cite{blanco2008extending} for planning and convert the action back to 3D for actual execution. 
	
	A trajectory parameter space is a 2D space where each point corresponds to a drivable trajectory for a robot.
	The basic idea is that a sequence of robot's pose in 3D space $(x, y, \theta)$ can be charted employing 2D manifolds in the parameter space of a family of trajectories.
	
	In this paper, we consider differential-drive robots, which only move along circular trajectories due to their movement restrictions.
	Then we improve the original circular parameter trajectory generator (C-PTG)\footnote{The original C-PTG assumes that the robot moves along the path with the maximum translational velocity all the time, which ignores the kinematic constraint of rotational velocities. Our revision not only considers this constraint but also removes the assumption of translational velocity.}~\cite{blanco2008extending} and use a 2D virtual action $(v^\textit{TP}, \omega^\textit{TP})$ to parameterize the trajectories with a predefined fixed time scale $\tau^\textit{TP}$. We call it virtual action because $v^\textit{TP}$ and $\omega^\textit{TP}$ are virtual translational and rotational velocities in TP-space, which are not limited by the actual robot's maximum translational velocity $v_m$ and rotational velocity $\omega_{m}$.
	
	Assume the robot starts from the initial point $(0, 0)$ and executes the virtual action $(v^\textit{TP}, \omega^\textit{TP})$ for a time interval $\tau^\textit{TP}$. 
	The path it moves along is the same as the paths of the set $\Phi$ of executable actions, such that
		\[
		\Phi = \left\{(v, \omega, d)\mid v\, d = v^\textit{TP}\, \tau^\textit{TP} \text{ and } \frac{v}{\omega}=\frac{v^\textit{TP}}{\omega^\textit{TP}} \right\}.
		\]

	If we want the robot to move along this path in the shortest possible time, we can convert the virtual action $(v^\textit{TP}, \omega^\textit{TP})$ to an unique executable action $(v, \omega, d)$, i.e., 
	{\small
		\begin{align}\label{2Dto3D}
			v &= \frac{v^{\textit{TP}}}{k}, &
			\omega & = \frac{\omega^{\textit{TP}}}{k}, &
			d &= k \, \tau^\textit{TP}, 
	\end{align}}%
	where $k$ is a real number, with $k=\max(\frac{v^{\textit{TP}}}{v_{m}},\,\frac{\omega^{\textit{TP}}}{\omega_{m}})$. 
	In this way, we only need to consider such most efficient action $(v, \omega, d)$ for the navigation policy instead of all possible actions in $\Phi$. Hence, we reduce the action space.
		
	
	\section{Extending GAE for SMDP}\label{EPG}
	
	In reinforcement learning, events (e.g., a robot collides) in the far future are weighted less than events in the immediate future. In our scheme, an action's execution duration can determine when an event will take place in the future, and thus also determine the event's weight.
	However, original generalized advantage estimation (GAE)~\cite{schulmanetal2016high} focuses on MDPs and does not consider such impact of execution duration.
	To address this issue, we extend the policy gradient (PG) theory for SMDPs\footnote{The detailed process is specified in the appendix at \url{http://book.baige.me/view/YbY}.The appendix material mentioned below can also be found at this link.} and improve GAE to estimate the policy gradient in SMDPs.
	We prove that EGAE can theoretically introduce no bias.
	
	
	
	\begin{comment}
	First, we introduce the definitions of the state value function $V^{\pi_\theta}$, the state-action value function $Q^{\pi_\theta}$, and the advantage function $A^{\pi_\theta}$ in SMDP, i.e.,
	{\small
	\begin{align*}
	V^{\pi_\theta}(s_{t_{i}}) &= \mathbb{E}_{\rho \sim p_{\theta}(\rho_{s_{t_i}})}\Big[\sum_{j=0}^{\infty} \gamma^{z_{i}^{j}} r_{t_{i+j}}\Big],\\
	Q^{\pi_\theta}(s_{t_{i}},a_{t_i}) &= \mathbb{E}_{\rho \sim p_{\theta}(\rho_{{s_{t_i}, a_{t_i}}})} \Big[\sum_{j=0}^{\infty}\gamma^{z_{i}^{j}} r_{t_{i+j}}\Big],\\
	A^{\pi_\theta}(s_{t_{i}},a_{t_i}) &= Q^{\pi_\theta}(s_{t_i}, a_{t_i}) - V^{\pi_\theta}(s_{t_i}),
	\end{align*}}%
	where $\rho_{s_{t_i}}$ (resp. $\rho_{s_{t_i}, a_{t_i}}$) denotes the episode starting from the state $s_{t_{i}}$ (resp. the state $s_{t_{i}}$ and the action $a_{t_i}$), and $z_{i}^{j}=t_{i+j}-t_{i}=\sum_{k=i}^{k=i+j-1} \tau_{t_k}$.
	
	
	An advantage estimator $\hat{A}_{\rho}$ denotes the estimation of the advantage function by considering the episode $\rho$.
	Following the notion $\gamma$-just in~\cite{schulmanetal2016high}, we can define E-just for the advantage estimator $\hat{A}_{\rho}$ in SMDP, so that it is an estimator that does not introduce bias when we use it in place of $A^{\pi_\theta}$.
	\begin{definition}
	The estimator $\hat{A}_{\rho}$ is {\em E-just} if 
	{\small
	\begin{multline*}
	\mathbb{E}_{\rho \sim p_{\theta}(\rho(s_{t_0}, a_{t_0}))}
	\left[\hat{A}_{\rho}(s_{t_i}, a_{t_i}) \nabla \log {\pi}_{\theta}\left(a_{t_i} \mid s_{t_{i}}\right)\right]\\
	= \mathbb{E}_{\rho \sim p_{\theta}(\rho(s_{t_0}, a_{t_0}))} \left[A^{\pi_\theta}(s_{t_{i}},a_{t_i}) \nabla \log {\pi}_{\theta}\left(a_{t_i} \mid s_{t_{i}}\right)\right].
	\end{multline*}}
	\end{definition}
	\end{comment}
	
	Given an approximate state value function $\hat{V}$, for each $k\geq 1$ we define the advantage function of $k$ step
	{\small
		\begin{multline*}
			\hat{A}_\rho^{(k)}(s_{t_i}, a_{t_i}) = \sum_{j=0}^{k-1} \gamma^{z_i^j} \delta_{t_{i+j}}^{\hat{V}} =\\
			-\hat{V}(s_{t_{i}})+r_{t_{i}} + \gamma^{z_i^1} r_{t_{i+1}}+\cdots+ \gamma^{z_{i}^{k-1}} r_{t_{i+k-1}} +\gamma^{z_{i}^{k}} \hat{V}(s_{t_{i+k}}),
	\end{multline*}}%
	where $z^j_i=t_{i+j} - t_i$ and
	$\delta_{t_{i+j}}^{\hat{V}}=  r_{t_{i+j}}+\gamma^{z_{i+j}^1} \hat{V}(s_{t_{i+j+1}}) -\hat{V}(s_{t_{i+j}})$.
	In particular,
	{\small
		\begin{align*}
			\hat{A}_\rho^{(1)}(s_{t_i}, a_{t_i}) &= \delta_{t_i}^{\hat{V}},\\
			\hat{A}_\rho^{(\infty)}(s_{t_i}, a_{t_i}) &= -\hat{V}(s_{t_{i}})+\sum_{j=0}^{\infty} \gamma^{z_{i}^{j}} r_{t_{i+j}}.
	\end{align*}}%
	Then we define EGAE $\hat{A}_\rho^{\text{EGAE}}$
	{\small
		\begin{align*}
			&\hat{A}_\rho^{\text{EGAE}}(s_{t_i}, a_{t_i})\\ 
			&= (1-\lambda) \left(\hat{A}_\rho^{(1)}(s_{t_i}, a_{t_i})+\lambda \hat{A}_\rho^{(2)}(s_{t_i}, a_{t_i})+\cdots\right) \\
			&= (1-\lambda)\left(\delta_{t_i}^{\hat{V}}+\lambda\left(\delta_{t_i}^{\hat{V}}+\gamma^{z_{1}^{i}} \delta_{t_{i+1}}^{\hat{V}}\right)+\cdots\right)\\
			&= \sum_{j=0}^{\infty} \gamma^{z_{i}^{j}} \lambda^{j} \delta_{t_{i+j}}^{\hat{V}},
	\end{align*}}%
	where $\lambda\in [0, 1]$ is a hyperparameter representing the compromise between bias and variance. Similar to the discussion in~\cite{schulmanetal2016high}, the increase of $\lambda$ results in the increase of the variance and the decrease of the bias. In the settings of SMDPs, we prove that EGAE can theoretically be an estimator that introduce no bias when $\lambda=1$ or $\hat{V}$ is accurate. The detailed derivation process and more research on the properties of EGAE are presented in the appendix.
	\begin{comment}
	\begin{theorem}\label{them:1}
	The estimator EGAE $\hat{A}_\rho^{\text{EGAE}}$ is E-just when $\lambda =1$ or $\hat{V}$ is accurate, i.e., $\hat{V} = V^{\pi_\theta}$.
	\end{theorem}
	
	Theorem~\ref{them:1}\footnote{The proof is specified in Appendix Material at \url{http://staff.ustc.edu.cn/~jianmin/supple/appendix.pdf} due to space limitations.} 
	provides the theoretical guarantee for EGAE that it could be unbiased when we use it in place of $A^{\pi_\theta}$.
	\end{comment}
	
	
	\section{Approach}\label{approach}
	
	In this section, we present details on how to apply Distributed Proximal Policy Optimization (DPPO)~\cite{dppo} to the SMDP problem with the adoption of EGAE.
	We consider DPPO in the paper as PPO and its extensions are widely applied in DRL-based robot
	navigation. More importantly, it is a policy-based DRL algorithm, which can learn stochastic policies.
	Notice that, for a navigation policy that drives the robot in unknown environments, it would frequently face aliased states due to its partial observation of the environment, while a stochastic policy can help the robot to address the problem~\cite{silver2015lecture}.


	
	\subsection{Reinforcement Learning Components.}\label{sec:drl}
	
	\begin{figure}[tp]
		\centerline{\includegraphics[width=1\linewidth]{figure/network.png}}
		\caption{We use DPPO to solve the navigation problem modeled in a SMDP. Above is the architecture of our policy network. The input of the network is $s_{t_i}$, which consists of the local grid map $M_{t_i}$ and the relative target pose $g_{t_i}$. The output of the last layer is added by a Gaussian noisy $a_{t_i}^{logstd}$ and get the raw action $a_{t_i}^{raw}$. Then we use a specified activation function with an ELU to process it and get the 2D virual action $a_{t_i}^{TP}$. Finally, we  convert $a_{t_i}^{TP}$ to 3D action $a_{t_i}$ for execution according to Eq.~\eqref{2Dto3D}}
		\label{fig2}
	\end{figure}
	

	\subsubsection{\bf \emph{State}}
	a state $s_{t_i}$ consists of the egocentric local grid map~$M_{t_i}$ and the relative target pose $g_{t_i}$ of the robot as shown in the Fig.~\ref{fig2}. 
	In our experiment, $M_{t_i}$ is generated from the data of a 2D laser scan with a 180-degree horizontal Field of View (FOV), which encodes the robot's shape and observable appearances
	of nearby obstacles. We construct the egocentric local grid map $M_{t_i}$ in the same way as~\cite{chen2020distributed}.
	
	\subsubsection{\bf \emph{Action}}
	an executable action $a_{t_i}$ for the robot is a triple $(v_{t_i}, \omega_{t_i}, d_{t_i})$.
	We adopt the conversion illustrated in Sec.~\ref{TP-space} to reduce the action space.
	Here, we introduce the processing process from 2D raw actions to 3D executable actions as shown in Fig.~\ref{fig2}. Firstly, the output of the last layer is added by a Gaussian noisy $a_{t_i}^{logstd}$ for exploration. Then, we get the raw action $a_{t_i}^{raw}$. 
	Secondly, we specify an activation function with a modified exponential linear unit (ELU)~\cite{ELU} to convert the raw action $a_{t_i}^{raw}$ to a 2D virtual action $a_{t_i}^{TP}$, where $a_{t_i}^{raw}=(v^\textit{raw}_{t_i}, \omega^\textit{raw}_{t_i})$, $a_{t_i}^{TP}=(v^{TP}_{t_i}, \omega^{TP}_{t_i})$,
	{\small
		\begin{equation}\label{activate}
			\begin{aligned} 
				v^{TP}_{t_i}&=\left\{
				\begin{array}{ll}
					0.2 e^{5 v^{\textit{raw}}_{t_i}-1} & \text { if }v^{\textit{raw}}_{t_i}<0.2, \\
					v^{\textit{raw}}_{t_i} & \text { otherwise, }
				\end{array}\right.\\
				\omega^{TP}_{t_i}&=\omega^{\textit{raw}}_{t_i}.
			\end{aligned} 
	\end{equation}}%
	We adopt the ELU as it can improve the robustness to noise when the robot needs subtle movements to pass through dense obstacles.
	As shown in Fig.~\ref{fig7}, our ELU gradually saturate to zero when the argument gets smaller. We can observe that the smaller the argument is, the slower the gradient vanishes, which preserves ELU's property of refining the granularity of small values. This is why the ELU has the characteristic of soft saturation and improves the robustness to noise.
	\begin{wrapfigure}{r}{0.2\textwidth}
		\centering
		\includegraphics[width=0.2\textwidth]{figure/ELU.png}
		\caption{Our modified ELU}
		\label{fig7}
	\end{wrapfigure}
	The parameters of the ELU influence the probability distribution of the different actions sampled by the initial policy. Tested by experiments, we find that the ELU's parameters in Eq.~\eqref{activate} are preferable.
	
	Finally, $a_{t_i}^{TP}$ can be converted to $a_{t_i}$ for execution by Eq.~\eqref{2Dto3D}.
	
	\subsubsection{\bf \emph{Reward}}
	the reward $r_{t_{i}}$ consists of four components, i.e.,
	{\small
		\begin{align*}
			r_{t_{i}} &=r^{app}_{t_{i}}+r^{arr}_{t_{i}}+r^{col}_{t_{i}}+r^{step}_{t_{i}},\\
			r^{app}_{t_{i}} &=\varepsilon_{a}\left(\left\|{l}_{t_{i-1}}-l_{g}\right\|-\left\|{l}_{t_{i}}-l_{g}\right\|\right), \\
			r^{arr}_{t_{i}} &=\left\{\begin{array}{ll}
				r_{arr}  & \text { if }\left\|{l}_{t_{i}}-l_{g}\right\|<0.3,\\
				0 & \text{ otherwise,}
			\end{array}\right.\\
			r^{coll}_{t_{i}} &=\left\{\begin{array}{ll}
				r_{col} & \text { if collision}, \\
				0 & \text { otherwise,}
			\end{array}\right.\\
			r^{step}_{t_{i}} &= -\varepsilon_{t} \tau_{t_{i}} - \varepsilon_{\tau} \tau^\textit{TP},
	\end{align*}}%
	where $l_{t_i}$ (resp. $l_{g_{t_i}}$) denotes the location of the robot (resp. target $g_{t_i}$) at the current time $t_i$, $\tau_{t_{i}}$ denote the actual execution duration of $a_{t_i}$, $r_{arr}>0$, $r_{col} <0$, $\varepsilon_{a}>0$, $\varepsilon_{t}>0$, and $\varepsilon_{\tau}>0$ are hyperparameters.
	
	In particular, $r^{app}_{t_{i}}$ defines the penalty for departure from the target,
	$r^{arr}_{t_{i}}$ defines the reward for arrival at the target, $r_{col}$ defines the penalty for collisions, and $r^{step}$ defines a minor penalty for each action. $\varepsilon_{t}$ is set to penalize the execution time of the action, which encourages the robot to reach the goal faster.
	$\varepsilon_{\tau}$ is the coefficient of a fixed penalty for every action, which encourages the robot to reach the goal with fewer planning times.
	In our experiment, we set $r_{arr} = 500$, $r_{col} = -500$,  $\varepsilon_{a}=200$, $\varepsilon_{t}=12$, and $\varepsilon_{\tau} = 10$.
	
	
	
	
	\subsection{Network Architecture.}
	
	The architecture of our policy network~$\pi_\theta$ is shown in Figure~\ref{fig2}.
	Our value network has the same architecture as the policy network, except that the last layer is modified to only output the value function~$V_\phi$ without the Gaussian noise and the activation function.
	
	
	
	\subsection{Training Process.}
	
	We replace GAE with EGAE in DPPO. Then the objective function for the policy network $\pi_\theta$ is
	{\small
		\begin{align*}
			L^{\pi}_{\theta_{k}}(\theta) &=\mathbb{E}_{s, a \sim \theta_{k}} \Big[ \min \Big(\frac{\pi_{\theta}(a \mid s)}{\pi_{\theta_{k}}(a \mid s)} \hat{A}_\rho^\text{EGAE}(s, a), \\
			&\hspace{3.5cm} g(\epsilon,\, \hat{A}_\rho^\text{EGAE} (s, a))\Big)\Big], \\
			g(\epsilon, \hat{A}) &=\left\{\begin{array}{ll}
				(1+\epsilon) \hat{A} & \text{ if }\hat{A} \geqslant 0 \\
				(1-\epsilon) \hat{A} & \text{ otherwise,}
			\end{array}\right.
	\end{align*}}%
	where $\theta_{k}$ denotes the parameters of the policy at the $k$th epoch and $\epsilon$ is the clip ratio.
	
	The objective function for the value network $V_\phi$ is
	{\small
		\begin{align*}
			L^{V^{\pi}}_{\phi}(\phi) &=- \mathbb{E}_{s, \rho_{s} \sim \pi} \Big[ \Big( R(\rho_s)-V_{\phi}(s) \Big)^2 \Big],
	\end{align*}}%
	where $\rho_s$ denotes the episode starting from the state $s$ and $\phi$ denotes the parameters of the value function. $L^{\pi}_{\theta_{k}}(\theta)$ and $L^{V^\pi}_{\phi}(\phi)$ will both be estimated per iteration by all the data in the experience buffer.
	
	
	\begin{algorithm}[h]
		\caption{Adaptive Forward Simulation Time (AFST)}
		\label{alg:1}
		\begin{algorithmic}[1]
			\STATE Initialize the policy network $\pi_{\theta}$, the value network $V_\phi$.\
			\STATE Clear the experience buffer \emph{Buffer}.\
			\FOR{\emph{epoch} $k= 1,2,...,$ } 
			\STATE \emph{//collect data in parallel}
			\FOR{\emph{step} $i= 0,1,...,T_{ep}$ }
			\STATE $a_{t_i}^{raw}=\pi_{\theta}(s_{t_i}); V_{t_i}=V_\phi(s_{t_i})$
			\STATE Transfer $a_{t_i}^{raw}$ to $a_{t_i}$ according to Eq.~\eqref{activate}, Eq.~\eqref{2Dto3D}
			\STATE $s_{t_{i+1}}, r_{t_i} = \emph{step}(a_{t_i})$
			\STATE Add $(s_{t_i}, a_{t_i}^{raw}, r_{t_i}, V_{t_i})$ to \emph{Buffer}
			\STATE $s_{t_i} \gets s_{t_{i+1}}$
			\IF{$\emph{trajectory length}>T_m$ or \emph{arrive} or \emph{collide}}
			\STATE Finish the current episode $\rho$ by $V_{t_{i+1}}=0$
			\STATE Estimate $\hat{A}_{\rho}^\text{EGAE}(s_{t_j}, a_{t_j})$ for all $(s_{t_j}, a_{t_j})$ in $\rho$
			\STATE Add each $\hat{A}_{\rho}^\text{EGAE}(s_{t_j},a_{t_j})$ to \emph{Buffer}
			\STATE Add each $R(\rho_{s_{t_j}})$ to \emph{Buffer}
			\STATE $s_{t_i}=\emph{reset()}$
			\ENDIF
			\ENDFOR
			\STATE Update $\pi_{\theta}$ with $L^{\pi}_{\theta_{k}}(\theta)$ and $lr_\theta$ for $E_\pi$ iterations
			\STATE Update $V_\phi$ with $L^{V^\pi}_{\phi}(\phi)$ and $lr_\phi$ for $E_v$ iterations
			\STATE Clear \emph{Buffer}
			\ENDFOR
		\end{algorithmic}
	\end{algorithm}
	
	As illustrated in Figure~\ref{fig3:subfig:a},
	we train the networks in environments that are constructed by a customized simulator based on OpenCV with Algorithm~\ref{alg:1}.
	In particular, the simulator can load an environment as a gray image, where obstacles and robots are specified as corresponding pixels in the image. 
	In DPPO, the algorithm collects experiences in a distributed setting from multiple environments where robots share the same navigation policy~$\pi_\theta$.
	
	\section{Experiments}
	
	In this section, we present the simulated and physical evaluation methods. Their results both validate our hypothesis that AFST can achieve better performance in complex unknown environments. In our experiments, we
	use a turtlebot2, a differential drive, a circular robot with a radius of $0.17m$, the maximum velocities of which are $v_m=0.6$ m/s and $\omega_{m}=0.9$ rad/s. The local grid maps are $6m \times 6m$. The robot is in the middle of the local map and can only perceive the information around $3m$ thereby cannot plan too far ahead.
	These settings are the same in both simulation and real-world experiments.

	
	\begin{table}[b]
		\begin{center} 
			\caption{Hyperparameters}  
			\label{table1}  
			\begin{tabular}{ll}
				\hline\noalign{\smallskip}  
				Hyperparameter & Value \\  
				\hline\noalign{\smallskip}  
				Time scale $\tau^\textit{TP}$  & $0.4s$ \\
				Learning rate $lr_{\theta}$   & $ 3\times10^{-4} $ \\  
				Learning rate $lr_{\phi}$  & $ 1\times10^{-3} $ \\  
				$\gamma$ in EGAE & $ 0.975 $\\  
				$\lambda$ in EGAE & $0.95$ \\
				Clip ratio $\epsilon$ & $ 0.2 $ \\
			
				Steps per epoch $T_{ep}$ & $ 2000 $ \\
				Pixels of a local map & $ 48\times48 $ \\
				Episode length $T_{m}$ & $200$ \\
				Training iterations per epoch $E_\pi$ & 80 \\
				Training iterations per epoch $E_v$  & 80 \\
				\hline\noalign{\smallskip} 
			\end{tabular}
		\end{center} 
	\end{table}
	
	\subsection{Implementation}
	We specify two modes to implement the navigation strategy:
	\begin{itemize}
		\item {\bf \emph{Mode1}}: the \emph{control interval} is equal to FST, which means executing the action as long as the algorithm plan ahead.
		\item {\bf \emph{Mode2}}: the \emph{control interval} is set to a fixed small value, which means that no matter how long the FST is planned, the robot will replan the next action after executing the current action for the fixed small interval.
	\end{itemize}
	In most cases, we use \emph{Mode1}, such as simulation training, simulation testing, and physical experiments in static scenarios. \emph{Mode2} is used in physical experiments with dynamic obstacles for better safety. We set the fixed \emph{control interval} of \emph{Mode2} to $0.1s$.
	
	We train the AFST with hyperparameters listed in Table~\ref{table1}. 
	In the experiments, we find out that nearly every value of $\tau^\textit{TP}$ results in a similar success rate, which is higher than the other baselines.\footnote{Detailed evaluations can be found in the appendix.}
	Among the values of $\tau^\textit{TP}$, $0.4s$ is slightly better.
	Both the policy and value networks are implemented in TensorFlow and trained with the Adam optimizer on a computer with i7-9900 CPU and Nvidia Titan RTX GPU. 
	It takes around 10 hours to run about 1100 epochs in DPPO when the networks converge in training environments. 
	
	\subsection{Simulated Experiments}
	\subsubsection{Training and Testing Scenarios}
	
	
	\begin{figure}[tp]
		\centering
		\subfigure[Sparse]{
			\label{fig3:subfig:a} 
			\includegraphics[width=0.16\linewidth]{figure/5-1.png}
		}
		\subfigure[Dense]{
			\label{fig3:subfig:b}
			\includegraphics[width=0.16\linewidth]{figure/5-2.png}
		}
		\subfigure[Spiral]{
			\label{fig3:subfig:c} 
			\includegraphics[width=0.16\linewidth]{figure/5-3.png}
		}
		\subfigure[Zigzag]{
			\label{fig3:subfig:d}
			\includegraphics[width=0.16\linewidth]{figure/5-4.png}
		}
		\subfigure[Hybrid]{
			\label{fig3:subfig:e}
			\includegraphics[width=0.16\linewidth]{figure/5-5.png}
		}
		\caption{We consider five scenarios in our experiments and their global top views are shown as the above, where yellow curves denote the navigation paths of the robot, the yellow dots denotes the planning moment, the green boxes denote the random 
		region of the starting position, and the blue boxes denote the random region of the target position.  
		We can see in the red boxes that our robot can adjust its forward simulation time to adapt to multiple spatial features and obstacle densities. More importantly, our robot successfully escape the local minimums marked with the red circles. }
		\label{fig3}
	\end{figure}
	
	
	Figure~\ref{fig3} illustrates five different scenarios that are considered in our experiments. 
	Please note that we only consider environments in the \emph{sparse} scenario as training environments and evaluate the performance in these five scenarios:
	
	\begin{itemize}
		\item {\bf \emph{Sparse}} scenarios are $10m \times 10m$ and randomly choose locations for 6 big obstacles. The starting and the target positions of the robot are randomly chosen.
		Their random regions are shown in the Fig.~\ref{fig3}
		\item {\bf \emph{Dense}} scenarios are $10m \times 10m$ and randomly choose locations for 32 small obstacles. The starting and the target positions are also randomly chosen.
		\item {\bf \emph{Spiral}} scenarios are $6m \times 6m$, using a spiral map with 5 randomly placed obstacles, the starting and the target positions are fixed.
		\item {\bf \emph{Zigzag}} scenarios are $6m \times 6m$, using a zigzag map with 5 randomly placed obstacles, where the starting and target positions are fixed.
		\item {\bf \emph{Hybrid}} scenarios are $10m \times 10m$, combining the characteristics of \emph{Dense} and \emph{Zigzag} scenarios, where the locations of 32 obstacles, the starting and the target position are randomly chosen.
	\end{itemize}  
	
	
	\subsubsection{Comparison Baselines and Evaluation Metrics}
	
	We compare AFST with four baseline methods, i.e., 
	\begin{itemize}
		\item {\bf \emph{DWA}}: Dynamic Window Approach~\cite{dwa}, an elegant and classical local planner,  with a fixed forward simulation time, which is set to 3s in our experiments.
		\item {\bf \emph{CAMDRL}}: Collision Avoidance via Map-Based Deep Reinforcement Learning (CAMDRL)~\cite{chen2020distributed}, a map-based robot navigation method with fixed action duration. Its inputs of the network are local maps generated by the data of laser. The original GAE is applied in it.
		\item {\bf \emph{GO-DWA}}: Goal-Oriented Dynamic Window Approach (GO-DWA), a hierarchical planning approach that selects subgoals as actions and reaches the subgoals with DWA. We refer to the scheme and the action space of GO-MPC~\cite{brito2021go} and replace its MPC with DWA.
		\item {\bf \emph{SDDQN}}: Semi-Markov DoubleDQN~\cite{schmoll2021semi}, a DRL-based method to solve SMDP problems with discrete action space. We set the range of the translational velocity $v \in \{0, 0.3, 0.6\}$, the rotational velocity $\omega \in \{-0.9, 0, 0.9\}$, the planning duration  $d \in \{0.4, 0.8, 1.2, 1.6, 2.0\}$. 
		
	\end{itemize}
	For a fair comparison, all methods share the same laser data as the input, all DRL-based methods use the same reward function as AFST. The action duration in CAMDRL is set to a fixed value of 0.4s as~\cite{chen2020distributed} does. We train these models in the same set of environments in the \emph{sparse} scenario and test their performance in the same set of environments for all four scenarios. 
	
	We use \emph{success rate} (SR) and \emph{reach time} (RT) to evaluate the performance. The \emph{success rate} is the ratio of tests that the robot reaches its target within 200 steps without any collision. The \emph{reach time} is the average time taken by the robot to reach the target. The evaluation results of trajectory length are presented in the appendix.
	
	
	
	
	\begin{table}[bp]
		\centering
		\setlength\tabcolsep{3.5 pt}
		\caption{Evaluation results of different methods}
		\begin{tabular}{rlrrrrrr}
			\toprule
			\multicolumn{1}{l}{Metric} & Method & \multicolumn{2}{l}{\#scenario} &       &       &       & \multirow{2}[4]{*}{Average } \\
			\cmidrule{3-7}          &       & Sparse & Dense & Spiral & Zigzag & Hybrid &  \\
			\midrule
			\multicolumn{1}{l}{Success} & DWA   & 0.616 & 0.618 & 0     & 0     & 0.002 & 0.248 \\
			\multicolumn{1}{l}{rate} & CAMDRL & 0.782 & 0.760 & 0     & 0     & 0.542 & 0.418 \\
			\multicolumn{1}{l}{(SR)}& GO-DWA & 0.886 & 0.700 & 0.610 & 0.082 & 0.206 & 0.498 \\
			& SDDQN & 0.868 & 0.854 & 0.826 & 0.078 & 0.624 & 0.650 \\
			& AFST  & \textbf{0.946} & \textbf{0.910} & \textbf{1.00} & \textbf{0.904} & \textbf{0.778} & \textbf{0.908} \\
			\midrule
			\multicolumn{1}{l}{Reach} & DWA   & 39.8  & 20.4  & /     & /     & 37.6  & / \\
			\multicolumn{1}{l}{time} & CAMDRL & 19.2  & 11.2  & /     & /     & \textbf{28.3} & / \\
			\multicolumn{1}{l}{(RT)}& GO-DWA & 35.0  & 22.5  & 31.0  & \textbf{23.5} & 38.0  & 30.0 \\
			& SDDQN & 26.0  & 14.6  & 30.9  & 127   & 46.3  & 49.0 \\
			& AFST  & \textbf{16.9} & \textbf{9.61} & \textbf{23.1} & 39.5  & 34.2  & \textbf{24.7} \\
			\bottomrule
		\end{tabular}%
		\label{table2}%
	\end{table}%
	
	
	
	
	\subsubsection{Testing Results}
	
	Table~\ref{table2} summarizes the test results using \emph{Mode1}. Under every scenario, we performed 500 tests for each model. We detail our observations below:
	
	{\bf \emph{The performance of DWA}}:
	DWA has inherent safety rules in that the robot is only allowed to take collision-free actions (assuming obstacles are static).
	However, DWA cannot achieve $100\%$ SR as the robot may get trapped in the local minimum. This problem causes the SR of DWA to be decreased to zero in some scenarios and its RT to be increased in others.

	
	{\bf \emph{The performance of CAMDRL}}:
	As a method with fixed action duration, CAMDRL has a tolerable SR in the \emph{sparse} and \emph{dense} scenarios as local minima areas don't occur very often with randomly located obstacles. However, it is entirely trapped in the local minimums of the \emph{spiral} and the \emph{zigzag} scenarios, the detailed analysis of this phenomenon is presented in Sec.~\ref{why3D}. This phenomenon is mitigated in \emph{hybrid} scenarios as the sizes of their obstacles are smaller than the ones in  \emph{zigzag} scenarios.
	
	{\bf \emph{The performance of GO-DWA}}:
	GO-DWA achieves a high SR in the \emph{sparse} scenario. However, we find that it is difficult for GO-DWA to select an accurate local goal between dense obstacles. Sometimes, the local goal will overlap with obstacles, which leads to the limited generalization of GO-DWA.
	
	{\bf \emph{The performance of SDDQN}}:
	SDDQN's performance is the closest to AFST, which indicates the advantage of using SMDP models. But its deterministic policy is not as good at handling aliased states as stochastic policy as mentioned in Sec.~\ref{approach}. Besides, SDDQN's discrete action space limits its generalization. 
	Therefore, SDDQN has a lower SR and longer RT than AFST in testing scenarios.
	
	{\bf \emph{The performance of AFST}}:
	AFST achieves the highest SR in all testing scenarios. The high SR and the trajectories in Fig.~\ref{fig3} prove it can adjust the FST to a proper small value when running through the narrow gaps (Fig.~\ref{fig3:subfig:a}), and can adjust the FST to a proper big value when escaping the local minimums(Fig.~\ref{fig3:subfig:d}). Although the SR of AFST in the training scenario is not much higher than that of other methods, its SR in unseen testing scenarios is distinctly higher, which indicates its excellent generalization. Moreover, AFST has the shortest mean RT as it always moves with the maximum speed according to Eq.~\eqref{2Dto3D} and can escape from local minimums with fewer failed attempts.
	
	
	\begin{table}[bp]
		\centering
		\setlength\tabcolsep{3.5 pt}
		\caption{Results of an ablation study}
	
		\begin{tabular}{rlrrrrrr}
			\toprule
			\multicolumn{1}{l}{Metric} & Method & \multicolumn{2}{l}{\#scenario} &       &       &       & \multirow{2}[4]{*}{Average } \\
			\cmidrule{3-7}          &       & Sparse & Dense & Spiral & Zigzag & Hybrid &  \\
			\midrule
			\multicolumn{1}{l}{Success} & -ELU  & 0.936 & 0.840 & 0.990 & 0.532 & 0.556 & 0.771 \\
			\multicolumn{1}{l}{rate} & -2D-to-3D & 0.860 & 0.850 & 0.104 & 0     & 0.588 & 0.481 \\
			\multicolumn{1}{l}{(SR)}& -EGAE & 0.760 & 0.646 & 0.228 & 0     & 0.436 & 0.414 \\
			& -SMDP & 0.944 & 0.906 & \textbf{1.00} & 0.804 & 0.686 & 0.868 \\
			& AFST  & \textbf{0.946} & \textbf{0.910} & \textbf{1.00} & \textbf{0.904} & \textbf{0.778} & \textbf{0.908} \\
			\midrule
			\multicolumn{1}{l}{Reach} & -ELU  & 17.2  & \textbf{9.03} & 27.0  & 51.1  & \textbf{26.1} & 26.1 \\
			\multicolumn{1}{l}{time} & -2D-to-3D & 17.4 & 9.78  & 79.4  & /     & 34.4  & / \\
			\multicolumn{1}{l}{(RT)}& -EGAE & 17.5  & 9.16  & 33.4  & /     & 31.2  & / \\
			& -SMDP & 17.0  & 11.0  & 23.7  & \textbf{37.5} & 27.9  & \textbf{23.4} \\
			& AFST  & \textbf{16.9}  & 9.61  & \textbf{23.1} & 39.5  & 34.2  & 24.7 \\
			\bottomrule
		\end{tabular}%
		\label{table3}%
	\end{table}%
	
	
	

	\begin{comment}
	\begin{table}[bp]
	\centering
	\caption{Average wall time (WT) and planning time (PT) of methods in real-world scenarios}
	\begin{tabular}{rlrrrr}
	\toprule
	\multicolumn{1}{l}{Method} & Metrics & \multicolumn{3}{c}{Scenario} \\
	\cmidrule{3-5}          &       & \multicolumn{1}{l}{Scatter} & \multicolumn{1}{l}{Spiral} & \multicolumn{1}{l}{Zigzag} \\
	\midrule
	\multicolumn{1}{l}{Chen et al.} & WT & 20.20   & fail     & fail \\
	& PT & 9.79 & fail     & fail \\
	\multicolumn{1}{l}{AFST} & WT & 19.20  & 24.06    & 28.17 \\
	& PT & 1.30 & 4.14  & 1.24 \\
	\bottomrule
	\end{tabular}%
	\label{table4}%
	\end{table}%
	\end{comment}
	
	\subsubsection{Ablation Studies}
	
	We also conduct ablation studies to investigate the utility of several modifications we propose in our scheme. Specifically,
	`-ELU' means replacing the modified ELU with the ReLu~\cite{relu}, `-2D-to-3D' means directly outputting 3D actions without the 2D-to-3D conversion, `-EGAE' means replacing EGAE with TD(0) in the settings of the SMDP, 
	and `-SMDP' means formulating the problem as a lifted MDP\footnote{We can also model the problem with different action duration as a MDP. Because the wall time between successive time steps is not forced to be uniform in MDPs. The fixed discount rate $\gamma$ in `-SMDP' is $0.99$.} and using GAE to evaluate the advantage function.	
	The results are shown in Table~\ref{table3}. In general, we find the above modifications nearly all lead to performance improvements. 
	
	ELU has the characteristic of soft saturation, which improves the robustness to noise. Therefore, `-ELU' is less robust for long-distance navigation tasks and can almost only achieve short distance tasks. This is the reason why `-ELU' has lower SR but shorter RT compared with AFST in \emph{Dense} and \emph{Hybrid} scenarios.
	Without the 2D-to-3D conversion, the action space would be much larger, which makes it much harder to sample proper actions and learn successful policies. 
	Hence, the SR of `-2D-to-3D' is low, which indicates the effectiveness of the conversion.
	TD(0) estimates the advantage function with a low variance but high bias. Thus the performance of ‘-EGAE’ is poor.
	For the detailed implementation, the only difference between `-SMDP' and AFST is the power of the discount rate. The experimental results show that `-SMDP' achieves similar performance to AFST. In some scenarios, it is even hard to distinguish their effectiveness.
	However, MDPs cannot evaluate the value function on an equitable time scale as SMDPs, the rewards gained from different execution duration will be multiplied by the same discount rate, which encourages the long duration actions. Thus, the policy of `-SMDP' is a little more aggressive than the one of AFST. Then, the average SR of `-SMDP' is lower than AFST and the average RT is shorter.
	The results of the ablation study show that each of our components plays a considerable role in the algorithm.
	

	
	
	\begin{figure}[tp]
		\centerline{\includegraphics[width=1.0\linewidth]{figure/real.png}}
		\caption{We also conduct real-world experiments on a TurtleBot 2 with a Kobuki base using a Hokuyo UTM-30LX 2D LiDAR and a NVIDIA Jetson TX2 (shown in (a)). We consider three scenarios: Spiral (b), Zigzag (c), and scattered (d) (with scattered obstacles and pedestrians). }
		\label{fig5}
	\end{figure}
	
	
	\subsection{Physical Experiments}
	
	To further verify the generalization and effectiveness of our learning policy, we deploy the trained model of AFST and CAMDRL to an actual differential-drive robot in the real world as illustrated in Figure~\ref{fig5}. 
	
	
	We place paper boxes as obstacles to set up \emph{spiral} and \emph{zigzag} environments.
	In these static scenarios, we use \emph{Mode1} to implement AFST.
	Then, AFST can reach the target with fewer planning steps than CAMDRL as AFST usually plans for longer execution duration per step. Moreover, CAMDRL cannot get out of the local minimums in some difficult scenarios. Hence, AFST can not only tackle more difficult navigation tasks in unknown scenarios but also has the potential to navigate faster with less computational overhead.
	The testing of the planning time is presented in the demonstration video at \url{https://youtu.be/8Gx-MMLqGmA}.
	
	In addition, we also set up environments with scattered obstacles and pedestrians. 
	In these dynamic scenarios, we use \emph{Mode2} to implement AFST.
	Although we did not train our robot with dynamic obstacles, it is still able to avoid dynamic obstacles successfully.
	
	\begin{figure}[tp]
		\centerline{\includegraphics[width=1.0\linewidth]{figure/analysis.png}}
		\caption{We analyze the effect of dynamic action repetition in a gridworld environment. (a) The robot is facing a task to reach the goal marked by a red pentacle.  (b) The blue cells are those the robot can reach in one action from $A_{fix}^1$. (c) The blue cells are those the robot can reach in one action from $A_{dyn}^3$. The cells marked by orange circles are those the robot can reach in one action from $A_{fix}^3$.}
		\label{fig6}
	\end{figure}
	
	
	\section{Discussions}\label{conc}
	\subsection{Analysis}\label{why3D}
	{\bf \emph{Why dynamic action repetition is more efficient in our scenarios?}}
	Roughly speaking, we believe that dynamic action repetition reduces the sample complexity\footnote{The sample complexity is the number of training-samples that we need to supply to the algorithm} and makes it easier to explore under certain circumstances.
	
	We present a simple example to analyze its effect. 
	Considering an gridworld environment in Fig.~\ref{fig6}, the cells of the grid correspond to the state of the environment. From any state the robot can perform one of four actions per second, \emph{up}, \emph{down}, \emph{left} and \emph{right}. The robot can only get a positive reward when it reaches the target cell, otherwise, the reward is zero.
	
	Assume that the robot is initialized with a stochastic policy, which chooses each action with equal probability. Then, the robot needs to explore the goal at least once before start learning effective policy. In the first episode, we define the probability of reaching the goal (with the minimum number of actions) as $p_{reach}(A)$, where $A$ denotes action space. As shown in Fig.~\ref{fig6} (b) and (c), $A_{fix}^1$ means the action space with fixed duration ($1s$), $A_{dyn}^3$ means the action space with dynamic duration (the maximum duration is $3s$). In the task of Fig.~\ref{fig6} (a), $p_{reach}(A_{fix}^1)=\frac{1}{4^3}<p_{reach}(A_{dyn}^3)=\frac{1}{12}$, which means the sample complexity of using $A_{fix}^1$ is higher than $A_{dyn}^3$ in such empty scenarios. 
	When the target is further away from the starting position, $p_{reach}(A_{fix}^1)$ will tend to zero faster, which makes it more difficult to learn an effective navigation policy with $A_{fix}^1$. 
	
	Some readers may wonder what will happen if increasing the fixed duration. As shown in Fig.~\ref{fig6} (c), the robot cannot reach certain cells with $A_{fix}^3$. Of course, if we allow the robot to move half (or less) a cell per second, it will be able to reach every cell with $A_{fix}^3$. But it may need more time to reach the goal. In addition, dynamic action repetition is not always beneficial. It may increase the sample complexity when the maximum duration is too long\footnote{AFST doesn't need to set maximum duration as its initial policy is Gaussian random.} or the obstacles in the environment are too dense. In our training scenarios, we need dynamic action repetition to reduce the sample complexity to learn the capability of overcoming the local minimums. 

	
	
	
	\subsection{Limitations and Future Works}
	\subsubsection{\bf \emph{Dynamic collision avoidance}}
	In this paper, AFST only uses one frame of the local map for collision avoidance thereby cannot extract any dynamic information, which makes it inefficient for AFST to avoid dynamic obstacles. Besides, it is hard for the robot to react to dynamic obstacles timely only training with \emph{Mode1}. We will further develop the potential of \emph{Mode2} in dynamic scenarios.
	\subsubsection{\bf \emph{Robustness and safety}}
	Similar to many learning-based navigation methods, AFST is not as robust as optimization-based methods. The inherent exploration capabilities of AFST help itself get out of local minimums, but it also increases the uncertainty of obstacle avoidance. Therefore, more safety constraints should be added in the future.
	\subsubsection{\bf \emph{Kinematical constraints and smoothness}}
	The transformation in TP-space assumes that the robot would move with the maximum translational velocity or rotate with the maximum rotational velocity in the environment, which is a common simplification for low-speed robots. 
	However, this assumption is unrealistic for high-speed robots. 
	One of our future works is to extend AFST for considering the kinematical constraints of the robot and the smoothness of its trajectories.


	\subsection{Conclusion}
	In this paper, we focus on the navigation problem in unknown environments. We introduce adaptive FST to alleviate the local minimum problem. To better adapt the FST, we propose the first DRL-based navigation method modeled by a SMDP with continuous action space, AFST. We show that AFST outperforms several baseline schemes under multiple scenarios both in simulation and the real world.
	We also conduct ablation experiments to indicate the effectiveness of our components. 
	
	
	
	\bibliographystyle{IEEEtran}
	