prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What algorithm is engaged in the PLMS-PPIC method?

Read the following text and answer briefly.

Context 1: \section{Introduction}\label{S1}

The multiple access interferences (MAI) is the root of user
limitation in CDMA systems \cite{R1,R3}  The parallel least mean
square-partial parallel interference cancelation (PLMS-PPIC) method
is a multiuser detector for code division multiple access (CDMA)
receivers which reduces the effect of MAI in bit detection  In this
method and similar to its former versions like LMS-PPIC \cite{R5}
(see also \cite{RR5}), a weighted value of the MAI of other users is
subtracted before making the decision for a specific user in
different stages \cite{cohpaper}  In both of these methods, the
normalized least mean square (NLMS) algorithm is engaged
\cite{Haykin96}  The $m^{\rm th}$ element of the weight vector in
each stage is the true transmitted binary value of the $m^{\rm th}$
user divided by its hard estimate value from the previous stage  The
magnitude of all weight elements in all stages are equal to unity 
Unlike the LMS-PPIC, the PLMS-PPIC method tries to keep this
property in each iteration by using a set of NLMS algorithms with
different step-sizes instead of one NLMS algorithm used in LMS-PPIC 
In each iteration, the parameter estimate of the NLMS algorithm is
chosen whose element magnitudes of cancelation weight estimate have
the best match with unity  In PLMS-PPIC implementation it is assumed
that the receiver knows the phases of all user channels  However in
practice, these phases are not known and should be estimated  In
this paper we improve the PLMS-PPIC procedure \cite{cohpaper} in
such a way that when there is only a partial information of the
channel phases, this modified version simultaneously estimates the
phases and the cancelation weights  The partial information is the
quarter of each channel phase in $(0,2\pi)$ 

The rest of the paper is organized as follows: In section \ref{S4}
the modified version of PLMS-PPIC with capability of channel phase
estimation is introduced  In section \ref{S5} some simulation
examples illustrate the results of the proposed method  Finally the
paper is concluded in section \ref{S6} 

\section{Multistage Parallel Interference Cancelation: Modified PLMS-PPIC Method}\label{S4}

We assume $M$ users synchronously send their symbols
$\alpha_1,\alpha_2,\cdots,\alpha_M$ via a base-band CDMA
transmission system where $\alpha_m\in\{-1,1\}$  The $m^{th}$ user
has its own code $p_m( )$ of length $N$, where $p_m(n)\in \{-1,1\}$,
for all $n$  It means that for each symbol $N$ bits are transmitted
by each user and the processing gain is equal to $N$  At the
receiver we assume that perfect power control scheme is applied 
Without loss of generality, we also assume that the power gains of
all channels are equal to unity and users' channels do not change
during each symbol transmission (it can change from one symbol
transmission to the next one) and the channel phase $\phi_m$ of
$m^{th}$ user is unknown for all $m=1,2,\cdots,M$ (see
\cite{cohpaper} for coherent transmission)  According to the above
assumptions the received signal is
\begin{equation}
\label{e1} r(n)=\sum\limits_{m=1}^{M}\alpha_m
e^{j\phi_m}p_m(n)+v(n),~~~~n=1,2,\cdots,N,
\end{equation}
where $v(n)$ is the additive white Gaussian noise with zero mean and
variance $\sigma^2$  Multistage parallel interference cancelation
method uses $\alpha^{s-1}_1,\alpha^{s-1}_2,\cdots,\alpha^{s-1}_M$,
the bit estimates outputs of the previous stage, $s-1$, to estimate
the related MAI of each user  It then subtracts it from the received
signal $r(n)$ and makes a new decision on each user variable
individually to make a new variable set
$\alpha^{s}_1,\alpha^{s}_2,\cdots,\alpha^{s}_M$ for the current
stage $s$  Usually the variable set of the first stage (stage $0$)
is the output of a conventional detector  The output of the last
stage is considered as the final estimate of transmitted bits  In
the following we explain the structure of a modified version of the
PLMS-PIC method \cite{cohpaper} with simultaneous capability of
estimating the cancelation weights and the channel phases 

Assume $\alpha_m^{(s-1)}\in\{-1,1\}$ is a given estimate of
$\alpha_m$ from stage $s-1$  Define
\begin{equation}
\label{e6} w^s_{m}=\frac{\alpha_m}{\alpha_m^{(s-1)}}e^{j\phi_m} 
\end{equation}
From (\ref{e1}) and (\ref{e6}) we have
\begin{equation}
\label{e7} r(n)=\sum\limits_{m=1}^{M}w^s_m\alpha^{(s-1)}_m
p_m(n)+v(n) 
\end{equation}
Define
\begin{subequations}
\begin{eqnarray}
\label{e8} W^s&=&[w^s_{1},w^s_{2},\cdots,w^s_{M}]^T,\\
\label{e9}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!X^{s}(n)\!\!\!&=&\!\!\![\alpha^{(s-1)}_1p_1(n),\alpha^{(s-1)}_2p_2(n),\cdots,\alpha^{(s-1)}_Mp_M(n)]^T 
\end{eqnarray}
\end{subequations}
where $T$ stands for transposition  From equations (\ref{e7}),
(\ref{e8}) and (\ref{e9}), we have
\begin{equation}
\label{e10} r(n)=W^{s^T}X^{s}(n)+v(n) 
\end{equation}
Given the observations $\{r(n),X^{s}(n)\}^{N}_{n=1}$, in modified
PLMS-PPIC, like the PLMS-PPIC \cite{cohpaper}, a set of NLMS
adaptive algorithm are used to compute
\begin{equation}
\label{te1} W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T,
\end{equation}
which is an estimate of $W^s$ after iteration $N$  To do so, from
(\ref{e6}), we have
\begin{equation}
\label{e13} |w^s_{m}|=1 ~~~m=1,2,\cdots,M,
\end{equation}
which is equivalent to
\begin{equation}
\label{e14} \sum\limits_{m=1}^{M}||w^s_{m}|-1|=0 
\end{equation}
We divide $\Psi=\left(0,1-\sqrt{\frac{M-1}{M}}\right]$, a sharp
range for $\mu$ (the step-size of the NLMS algorithm) given in
\cite{sg2005}, into $L$ subintervals and consider $L$ individual
step-sizes $\Theta=\{\mu_1,\mu_2,\cdots,\mu_L\}$, where
$\mu_1=\frac{1-\sqrt{\frac{M-1}{M}}}{L}, \mu_2=2\mu_1,\cdots$, and
$\mu_L=L\mu_1$  In each stage, $L$ individual NLMS algorithms are
executed ($\mu_l$ is the step-size of the $l^{th}$ algorithm)  In
stage $s$ and at iteration $n$, if
$W^{s}_k(n)=[w^s_{1,k},\cdots,w^s_{M,k}]^T$, the parameter estimate
of the $k^{\rm th}$ algorithm, minimizes our criteria, then it is
considered as the parameter estimate at time iteration $n$  In other
words if the next equation holds
\begin{equation}
\label{e17} W^s_k(n)=\arg\min\limits_{W^s_l(n)\in I_{W^s}
}\left\{\sum\limits_{m=1}^{M}||w^s_{m,l}(n)|-1|\right\},
\end{equation}
where $W^{s}_l(n)=W^{s}(n-1)+\mu_l \frac{X^s(n)}{\|X^s(n)\|^2}e(n),
~~~ l=1,2,\cdots,k,\cdots,L-1,L$ and
$I_{W^s}=\{W^s_1(n),\cdots,W^s_L(n)\}$, then we have
$W^s(n)=W^s_k(n)$, and therefore all other algorithms replace their
weight estimate by $W^{s}_k(n)$  At time instant $n=N$, this
procedure gives $W^s(N)$, the final estimate of $W^s$, as the true
parameter of stage $s$ 

Now consider $R=(0,2\pi)$ and divide it into four equal parts
$R_1=(0,\frac{\pi}{2})$, $R_2=(\frac{\pi}{2},\pi)$,
$R_3=(\pi,\frac{3\pi}{2})$ and $R_4=(\frac{3\pi}{2},2\pi)$  The
partial information of channel phases (given by the receiver) is in
a way that it shows each $\phi_m$ ($m=1,2,\cdots,M$) belongs to
which one of the four quarters $R_i,~i=1,2,3,4$  Assume
$W^{s}(N)=[w^{s}_1(N),w^{s}_2(N),\cdots,w^{s}_M(N)]^T$ is the weight
estimate of the modified algorithm PLMS-PPIC at time instant $N$ of
the stage $s$  From equation (\ref{e6}) we have
\begin{equation}
\label{tt3}
\phi_m=\angle({\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m}) 
\end{equation}
We estimate $\phi_m$ by $\hat{\phi}^s_m$, where
\begin{equation}
\label{ee3}
\hat{\phi}^s_m=\angle{(\frac{\alpha^{(s-1)}_m}{\alpha_m}w^s_m(N))} 
\end{equation}
Because $\frac{\alpha^{(s-1)}_m}{\alpha_m}=1$ or $-1$, we have
\begin{eqnarray}
\hat{\phi}^s_m=\left\{\begin{array}{ll} \angle{w^s_m(N)} &
\mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=1\\
\pm\pi+\angle{w^s_m(N)} & \mbox{if}~
\frac{\alpha^{(s-1)}_m}{\alpha_m}=-1\end{array}\right 
\end{eqnarray}
Hence $\hat{\phi}^s_m\in P^s=\{\angle{w^s_m(N)},
\angle{w^s_m(N)+\pi, \angle{w^s_m(N)}-\pi}\}$  If $w^s_m(N)$
sufficiently converges to its true value $w^s_m$, the same region
for $\hat{\phi}^s_m$ and $\phi_m$ is expected  In this case only one
of the three members of $P^s$ has the same region as $\phi_m$  For
example if $\phi_m \in (0,\frac{\pi}{2})$, then $\hat{\phi}^s_m \in
(0,\frac{\pi}{2})$ and therefore only $\angle{w^s_m(N)}$ or
$\angle{w^s_m(N)}+\pi$ or $\angle{w^s_m(N)}-\pi$ belongs to
$(0,\frac{\pi}{2})$  If, for example, $\angle{w^s_m(N)}+\pi$ is such
a member between all three members of $P^s$, it is the best
candidate for phase estimation  In other words,
\[\phi_m\approx\hat{\phi}^s_m=\angle{w^s_m(N)}+\pi \]
We admit that when there is a member of $P^s$ in the quarter of
$\phi_m$, then $w^s_m(N)$ converges  What would happen when non of
the members of $P^s$ has the same quarter as $\phi_m$? This
situation will happen when the absolute difference between $\angle
w^s_m(N)$ and $\phi_m$ is greater than $\pi$  It means that
$w^s_m(N)$ has not converged yet  In this case where we can not
count on $w^s_m(N)$, the expected value is the optimum choice for
the channel phase estimation, e g  if $\phi_m \in (0,\frac{\pi}{2})$
then $\frac{\pi}{4}$ is the estimation of the channel phase
$\phi_m$, or if $\phi_m \in (\frac{\pi}{2},\pi)$ then
$\frac{3\pi}{4}$ is the estimation of the channel phase $\phi_m$ 
The results of the above discussion are summarized in the next
equation
\begin{eqnarray}
\nonumber \hat{\phi}^s_m = \left\{\begin{array}{llll} \angle
{w^s_m(N)} & \mbox{if}~
\angle{w^s_m(N)}, \phi_m\in R_i,~~i=1,2,3,4\\
\angle{w^s_m(N)}+\pi & \mbox{if}~ \angle{w^s_m(N)}+\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\angle{w^n_m(N)}-\pi & \mbox{if}~ \angle{w^s_m(N)}-\pi, \phi_m\in
R_i,~~i=1,2,3,4\\
\frac{(i-1)\pi+i\pi}{4} & \mbox{if}~ \phi_m\in
R_i,~~\angle{w^s_m(N)},\angle
{w^s_m(N)}\pm\pi\notin R_i,~~i=1,2,3,4 \\
\end{array}\right 
\end{eqnarray}
Having an estimation of the channel phases, the rest of the proposed
method is given by estimating $\alpha^{s}_m$ as follows:
\begin{equation}
\label{tt4}
\alpha^{s}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
q^s_m(n)e^{-j\hat{\phi}^s_m}p_m(n)\right\}\right\},
\end{equation}
where
\begin{equation} \label{tt5}
q^{s}_{m}(n)=r(n)-\sum\limits_{m^{'}=1,m^{'}\ne
m}^{M}w^{s}_{m^{'}}(N)\alpha^{(s-1)}_{m^{'}} p_{m^{'}}(n) 
\end{equation}
The inputs of the first stage $\{\alpha^{0}_m\}_{m=1}^M$ (needed for
computing $X^1(n)$) are given by
\begin{equation}
\label{qte5}
\alpha^{0}_m=\mbox{sign}\left\{\mbox{real}\left\{\sum\limits_{n=1}^{N}
r(n)e^{-j\hat{\phi}^0_m}p_m(n)\right\}\right\} 
\end{equation}
Assuming $\phi_m\in R_i$, then
\begin{equation}
\label{qqpp} \hat{\phi}^0_m =\frac{(i-1)\pi+i\pi}{4} 
\end{equation}
Table \ref{tab4} shows the structure of the modified PLMS-PPIC
method  It is to be notified that
\begin{itemize}
\item Equation (\ref{qte5}) shows the conventional bit detection
method when the receiver only knows the quarter of channel phase in
$(0,2\pi)$  \item With $L=1$ (i e  only one NLMS algorithm), the
modified PLMS-PPIC can be thought as a modified version of the
LMS-PPIC method 
\end{itemize}

In the following section some examples are given to illustrate the
effectiveness of the proposed method 

\section{Simulations}\label{S5}

In this section we have considered some simulation examples 
Examples \ref{ex2}-\ref{ex4} compare the conventional, the modified
LMS-PPIC and the modified PLMS-PPIC methods in three cases: balanced
channels, unbalanced channels and time varying channels  In all
examples, the receivers have only the quarter of each channel phase 
Example \ref{ex2} is given to compare the modified LMS-PPIC and the
PLMS-PPIC in the case of balanced channels 

\begin{example}{\it Balanced channels}:
\label{ex2}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex2})} \label{tabex5} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s = 2 &  $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{3 24\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 18\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s = 2 &  $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{3-5} & & s = 3 & $\hat{\phi}^s_m=\frac{2 85\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 88\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider the system model (\ref{e7}) in which $M$ users
synchronously send their bits to the receiver through their
channels  It is assumed that each user's information consists of
codes of length $N$  It is also assumd that the signal to noise
ratio (SNR) is 0dB  In this example there is no power-unbalanced or
channel loss is assumed  The step-size of the NLMS algorithm in
modified LMS-PPIC method is $\mu=0 1(1-\sqrt{\frac{M-1}{M}})$ and
the set of step-sizes of the parallel NLMS algorithms in modified
PLMS-PPIC method are
$\Theta=\{0 01,0 05,0 1,0 2,\cdots,1\}(1-\sqrt{\frac{M-1}{M}})$,
i e  $\mu_1=0 01(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_4=0 2(1-\sqrt{\frac{M-1}{M}}),\cdots,
\mu_{12}=(1-\sqrt{\frac{M-1}{M}})$  Figure~\ref{Figexp1NonCoh}
illustrates the bit error rate (BER) for the case of two stages and
for $N=64$ and $N=256$  Simulations also show that there is no
remarkable difference between results in two stage and three stage
scenarios  Table~\ref{tabex5} compares the average channel phase
estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and PLMS-PPIC, when the the number of users is
$M=15$ 
\end{example}

Although LMS-PPIC and PLMS-PPIC, as well as their modified versions,
are structured based on the assumption of no near-far problem
(examples \ref{ex3} and \ref{ex4}), these methods and especially the
second one have remarkable performance in the cases of unbalanced
and/or time varying channels 

\begin{example}{\it Unbalanced channels}:
\label{ex3}
\begin{table}
\caption{Channel phase estimate of the first user (example
\ref{ex3})} \label{tabex6} \centerline{{
\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{6}{*}{\rotatebox{90}{$\phi_m=\frac{3\pi}{8},M=15~~$}} & N(Iteration) & Stage Number& NLMS & PNLMS  \\
&&&&\\
\cline{2-5} & \multirow{2}{*}{64}& s=2 &  $\hat{\phi}^s_m=\frac{2 45\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 36\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 71\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 80\pi}{8}$ \\
\cline{2-5} & \multirow{2}{*}{256}& s=2 &  $\hat{\phi}^s_m=\frac{3 09\pi}{8}$ & $\hat{\phi}^s_m=\frac{2 86\pi}{8}$ \\
\cline{3-5} & & s=3 & $\hat{\phi}^s_m=\frac{2 93\pi}{8}$ & $\hat{\phi}^s_m=\frac{3 01\pi}{8}$ \\
\cline{2-5} \hline
\end{tabular} }}
\end{table}
Consider example \ref{ex2} with power unbalanced and/or channel loss
in transmission system, i e  the true model at stage $s$ is
\begin{equation}
\label{ve7} r(n)=\sum\limits_{m=1}^{M}\beta_m
w^s_m\alpha^{(s-1)}_m c_m(n)+v(n),
\end{equation}
where $0<\beta_m\leq 1$ for all $1\leq m \leq M$  Both the LMS-PPIC
and the PLMS-PPIC methods assume the model (\ref{e7}), and their
estimations are based on observations $\{r(n),X^s(n)\}$, instead of
$\{r(n),\mathbf{G}X^s(n)\}$, where the channel gain matrix is
$\mathbf{G}=\mbox{diag}(\beta_1,\beta_2,\cdots,\beta_m)$  In this
case we repeat example \ref{ex2}  We randomly get each element of
$G$ from $[0,0 3]$  Figure~\ref{Figexp2NonCoh} illustrates the BER
versus the number of users  Table~\ref{tabex6} compares the channel
phase estimate of the first user in each stage and over $10$ runs of
modified LMS-PPIC and modified PLMS-PPIC for $M=15$ 
\end{example}

\begin{example}
\label{ex4} {\it Time varying channels}: Consider example \ref{ex2}
with time varying Rayleigh fading channels  In this case we assume
the maximum Doppler shift of $40$HZ, the three-tap
frequency-selective channel with delay vector of $\{2\times
10^{-6},2 5\times 10^{-6},3\times 10^{-6}\}$sec and gain vector of
$\{-5,-3,-10\}$dB  Figure~\ref{Figexp3NonCoh} shows the average BER
over all users versus $M$ and using two stages 
\end{example}


\section{Conclusion}\label{S6}

In this paper, parallel interference cancelation using adaptive
multistage structure and employing a set of NLMS algorithms with
different step-sizes is proposed, when just the quarter of the
channel phase of each user is known  In fact, the algorithm has been
proposed for coherent transmission with full information on channel
phases in \cite{cohpaper}  This paper is a modification on the
previously proposed algorithm  Simulation results show that the new
method has a remarkable performance for different scenarios
including Rayleigh fading channels even if the channel is
unbalanced 





ground_truth: The normalized least mean square (NLMS) algorithm.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the effect of accounting for path preference on the robot's belief update?

Read the following text and answer briefly.

Context 2: Paper Info

Title: Incorporating Human Path Preferences in Robot Navigation with Minimal Interventions
Publish Date: 16 Mar 2023
Author List: Oriana Peltzer, Dylan Asmar, Mac Schwager, Mykel Kochenderfer

Figure

Hyperplane arrangement of a twodimensional space containing two obstacles (colored in gray) The robot is located inside the pink polytope, surrounded by three adjacent obstacle-free polytopes Each hyperplane on the boundary of the robot's polytope corresponds to one of the nonredundant constraints in eq (4) (b)Graph derived from the hyperplane arrangement The nodes on the graph designate polytopes, and edges designate transitions to adjacent polytopes To estimate the human's preference, the robot updates a posterior over the goal and over which of the graph transitions φ 1 , φ 2 and φ 3 is preferred by the human (c)Example preference defined over the graph The location of the goal is indicated in yellow in the lower right polytope For each node, the outgoing pink arrow designates the edge on the graph corresponding to the preferred transition between polytopes 
Simple, 10 × 10, 8 polytopes (b) Map 2: Office, 10 × 10, 56 polytopes (c) Map 3: Classroom, 20 × 20, 73 polytopes (d) Sampled observations and robot's executed trajectories 
Fig 5: Maps used for simulating the robot navigation problem with path preferences In (d), the heading angles observed are indicated with arrows The goal is indicated with a pink circle, and the orange robot corresponds to the starting location The blue robot follows a policy that accounts for path preference, while the green robot does not The opacity of the robots increases with time 
Map 1 problem setup and example realizations for goal-only (green) and path preference (blue) solution methods The robot starts at the lower left corner of the environment, and the goal of the task (pink circle) is in the upper left area The robot does not know which goal, among 10 options (shown in light blue squares), is the correct goal The human provides noisy observations, indicated by arrows, at each iteration The green robot selects actions according to the goal-only baseline, and the blue robot uses our proposed method to infer path preferences The polytopes composing G are drawn in blue Probability of correct goal WLPHVWHS +J (c) Entropy of goal distribution g 
Fig  6: Probability of the correct goal, fig 6b, and entropy of the goal belief distribution P (g), fig 6c, for the same problem setup, fig 6a In this problem instance, the human's preference is to go to the goal by passing on the right side of the obstacle Results are averaged over 50 runs and the area filled represents one standard deviation above and below the mean value The goal-only baseline shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference 
Success rates in the simple environment (Map 1) The results are averaged over 6 randomly sampled problem instances (start location, goal location, and goal possibilities), and over 50 runs per problem instance ∆T is the number of time steps separating two consecutive human inputs The robot's mission time is Tmax = 30 time steps We selected γ h = 1 5, corresponding to relatively noisy human inputs and making the problem more difficult to solve for the robot 
Computation times for Goal Only and Path Preference methods on Map 1 (fig 5a),Map 2 (fig 5b), and Map 3 (fig 5c),averaged over 100 runs with randomly sampled problem instances The 95 % confidence interval is provided with the mean We evaluate computation time at the first iteration of each run (where the search depth takes on its highest value Tmax) 

abstract

Robots that can effectively understand human intentions from actions are crucial for successful human-robot collaboration  In this work, we address the challenge of a robot navigating towards an unknown goal while also accounting for a human's preference for a particular path in the presence of obstacles 
This problem is particularly challenging when both the goal and path preference are unknown a priori  To overcome this challenge, we propose a method for encoding and inferring path preference online using a partitioning of the space into polytopes  Our approach enables joint inference over the goal and path preference using a stochastic observation model for the human 
We evaluate our method on an unknown-goal navigation problem with sparse human interventions, and find that it outperforms baseline approaches as the human's inputs become increasingly sparse  We find that the time required to update the robot's belief does not increase with the complexity of the environment, which makes our method suitable for online applications 

INTRODUCTION

Collaboration between humans and robots has become increasingly important and one key aspect of this collaboration is the ability for robots to adapt to human decisions  In many scenarios, such as a robot navigating through a busy room to deliver an item, it is important for the robot to take into account human preferences 
For instance, humans may prefer a specific path that would allow their colleagues to notice the item being delivered, but this preference may change dynamically based on various factors such as changes in the environment or unforeseen circumstances  While some preferences can be incorporated into the path-planning process, accommodating dynamic user preferences in real-time remains challenging 
In this paper, we propose a way to enable robots to adapt to human preferences dynamically by leveraging real-time feedback to inform decision-making  In this work, we tackle the problem of robot navigation in which the robot cannot observe the goal or the preferred path to the goal, but must make navigation decisions that are influenced by humans through recommended actions 
Prior work has explored how to adapt to a human's preference through feedback, but such approaches often require a high level of intervention, which can be time-consuming and impractical in real-world scenarios  To optimize the use of human input and quickly infer the human's preference, Fig  : An autonomous robot navigates in a simulated classroom towards a goal location (pink circle) 
At the start of its mission, it receives direction indications (arrows) from a human that indicate which path it should take to get to the goal  In this scenario, the human wants the robot to go around the desks on the right side of the classroom  A robot that does not reason over path preferences (green) will take the shortest path to the goal regardless of the human's input 
Our method (blue) infers the human's path preference from these indications and adapts to their recommendations  we propose an approach that leverages probabilistic representations of human preference and incorporates real-time feedback  Previous research by Bajcsy et al  considered an online adaptation problem in a manipulation task, where the person can apply forces to the robot to indicate their preferences 
By allowing the robot to continue its task while taking into account a probabilistic representation of human preference, their approach does not require frequent inputs  Building on this idea, we adopt a similar approach to adapt to a human's preference in the context of a robot autonomously navigating through a known environment, such as a cluttered office space 
Specifically, we focus on allowing the human to influence the robot's trajectory with respect to obstacles, by providing guidance on preferred routes or paths, while the robot continues to execute its task  Paths can be represented using homotopy classes   However, homotopies can pose computational challenges when used to encode and infer human preferences 
When the robot maintains a belief over homotopy classes, the inference problem can become exponentially complex with the number of obstacles in the space  Additionally, when the goal is unknown, the number of variables increases with the number of candidate destinations  This complexity can render the decision-making problem intractable 
Our solution is to encode path preference based on a partitioning of the environment into polytopes   This representation allows path preferences to be expressed as sets of preferred transitions between adjacent polytopes  Paths belonging to different homotopy classes correspond to different sequences of transitions 
By leveraging conditional independence assumptions, we can make the Bayesian inference problem tractable  These assumptions exploit the fact that human actions provide information about the path in a piece-wise manner  For example, indicating a preference for navigating around a particular obstacle only provides information about the local area and not the entire path 
Finally, after updating its belief representation over the human's preference, the robot can adapt to indications by replanning online  Our contributions are as follows  • We formulate the human-robot collaboration problem as a Partially Observable Markov Decision Process (POMDP) where both the goal of the task and the human's path preference are unknown random variables 
• We propose an encoding of a human's path preference using a partitioning of the environment into polytopes, along with conditional independence assumptions that make the Bayesian inference problem tractable to infer the task goal and path preference online  • Through simulations in two environments of different sizes and complexity, we show that our method is effective for solving problems where the robot must reach a goal that is unknown a-priori while simultaneously adapting to a human's indications 
Our method shows higher success rates compared to baseline approaches when the human inputs are sparse  Our approach enables a robot to make effective navigation decisions in collaboration with a human, even when the goal and path preference are not known in advance, and with minimal human input  In recent years, there has been a growing interest in shared autonomy and interactive systems, where humans and robots work together to accomplish tasks 
Several approaches have been proposed to address the challenge of enabling effective collaboration between human and robot agents while still achieving high task performance  Losey et al  and Jeon, Losey, and Sadigh propose a framework where a human operator is given control of a task-relevant latent action space while an autonomous system handles the rest 
Dragan and Srinivasa present a formalism for arbitrating between a user's input and a robot's policy when both human and robot share control of the same action space  Cognetti et al  [7] provide a method for real-time modifications of a path,       Fig  : We model the intent inference problem with the above diagram 
At each step in time, the robot receives an observation ot from the human conditioned on its current location st, the intended goal g, and the human's path preference θ  The robot updates its belief over g and θ and transitions to a next location st+1  while Hagenow et al  present a method that allows an outside agent to modify key robot state variables and blends the changes with the original control 
However, a common challenge of these approaches is the high level of intervention required from humans  Best and Fitch propose a method for predicting an agent's intended trajectory from observations  Rather than maintaining a belief over the agent's future path, they infer the agent's intended goal among a set of candidate locations at the boundary of the space 
This approach provides information on where the agent is heading and generates a distribution of candidate future trajectories for the agent  Inferring the goal of the task among a discrete set of candidates is also relevant to the area of shared autonomy  Javdani, Srinivasa, and Bagnell propose a formalism for shared control of a robotic arm, where the robot must assist the human in picking up an object but needs to infer which object the human has chosen from joystick inputs 
Planning with homotopy class constraints is useful in problems where the robot's requirements are given with respect to obstacles, and Yi, Goodrich, and Seppi consider topological constraints provided by human operators  Bhattacharya propose an efficient algorithm for solving pathplanning problems under homotopic constraints 
However, the number of homotopy classes for a given problem can be infinite, and as the robot changes location and updates its representation of the world, carrying out inference over homotopy classes in a dynamic environment requires recomputing the set of homotopies at every iteration, making the belief update challenging 
Prior work has addressed the challenge of shared autonomy by considering how robots can infer a human's intended goal, or how they can infer the preferred path to a goal  However, we argue that inferring the goal and the path as separate problems can lead to over-confidence in incorrect beliefs about the user's preferences 
To illustrate this point, consider the following scenario: a robot and a human are collaborating to move an object from one end of a room to Fig  : Using the hyperplanes composing the H-representation of each obstacle, we construct a hyperplane arrangement of the obstacle-free space (a)  We define the human's preference for the robot's one step action choices as the posterior distribution (given all human input up to that point) over transitions from the current to the neighboring polytopes, i e  edges on the graph 
Each time the robot transitions to a new polytope, the set of neighbor polytopes and the distribution over human preferences are updated  another, but there is an obstacle in the way  The human would like the robot to take a path around the obstacle on the left, even though the goal is on the right  If the robot only infers the goal from the human's inputs, it may incorrectly assume that the goal is on the right, and become over-confident in this belief 
On the other hand, if the robot only infers the preferred path, it may mistakenly assume that the goal is on the left, leading to a failure in completing the task  To overcome these challenges, our work proposes a joint inference approach that considers both the human's intended goal and their preferred path to that goal 
Specifically, we model the human's preference over different homotopy classes and leverage a conditional independence assumption to provide a tractable solution  In our approach, we assume that the human's inputs are noisily rational conditioned on both the goal and the preference  By jointly inferring the goal and path preference, we can avoid over-confidence in incorrect beliefs about the user's preferences, leading to improved system performance 
We consider the problem of robot navigation in a known environment to an unknown destination, where a human can intervene and provide a heading direction to the robot using a joystick or force cues  The human also has a preference on which path the robot should take with respect to obstacles, and our objective is for the robot to understand the human's intentions and execute the task with minimal interventions 
Let g be a discrete random variable denoting the goal of the task, belonging to a set of candidates Ω g , and let θ be a discrete-valued random variable representing the human's path preference, belonging to a set of possible preferences Θ  The physical location of the robot at time index t is denoted by s t ∈ R 2 , and the robot's action at time index t, belonging to some action space A, is denoted by a t  
The transition model T (s t+1 | s t , a t ) is deterministic, meaning the robot has full control over its future location  At any time step, the human may provide an observation to the robot  When the human intervenes, the robot receives a direction (heading angle) that can be mapped to a future location in space 
More specifically, we map the direction to an intended location, which is the resulting robot location after advancing in the indicated direction for one time step  For simplicity, we consider that the robot directly makes an observation o t of the location indicated by the human  We assume that the robot has a stochastic observation model for the human P (o t | s t , g, θ) that is conditioned on both the goal of the task g and the human's preferred path θ 
We further assume that having chosen a goal and path preference, the human takes actions to noisily minimize a cost function C g,θ that measures the cost of moving from the robot's current location to the goal along the preferred path  For example, C g,θ (s t , o t ) can be the length of the shortest path from location s t to the goal g after taking a first step to o t , and constrained by path preference θ 
We use C g,θ to induce a probability distribution over observations, given by: where γ h is a hyperparameter that designates the rationality coefficient  This model assumes the human will pick the lowest cost action with the highest probability and the likelihood of an action decreases exponentially with the increase in cost  
Our inclusion of the path preference θ sets our approach apart from   The model is shown in fig  represented as a Bayesian Network 

Inference

At each time step where the human provides an observation, the posterior P (g, θ) is given through the Bayesian update We note that the number of Bayesian updates required at each iteration to update the belief is equal to the cardinality of Ω g × Θ  In addition, each Bayesian update involves computing C g,θ (  
,   ) in eq  ( ), which involves solving an optimization problem (such as a shortest path problem)  In section IV, we propose a specific encoding of preference θ for resolving eq  ( ), while ensuring the number of computations of the cost C g,θ ( ,  ) per update does not grow exponentially with the number of obstacles 

Decision Making

We consider a navigation problem where the robot receives reward according to the model R(s t , g, θ, a t )  We wish to find the optimal policy π that maximizes the expected discounted sum of future rewards, with discount factor γ  The above problem is a Partially Observable Markov Decision Process (POMDP)  
In this section, we propose an encoding of human's path preference θ for computing the posterior in eq  ( )  Devifrom the concept of homotopy classes, we define the preference according to a partitioning of the environment into polytopes, as shown in fig  , creating a hyperplane arrangement of the space 
Hyperplane arrangements have been used by Vincent and Schwager in the context of Neural Network verification  In our setting, we leverage this representation to define path preferences as preferred transitions between adjacent regions of the space 

Hyperplane Arrangement

We assume a two-dimensional environment composed of m polytopic obstacles, each defined by their half-space representation (H-representation) where A i ∈ R di×2 and b i ∈ R di , and where d i is the number of edges (hyperplanes) composing polytope i  Let n = i d i be the total number of hyperplanes  We leverage each obstacle's H-representation to construct a hyperplane arrangement of the environment as shown in fig 
 e  a partitioning of the space into polytopes  More specifically, each location in space belongs to a polytope j for which we can write an H-representation of the form where α j i ∈ {−1, 1} di is a vector specific to polytope j and obstacle i corresponding to the relative position of any point in the set with respect to each hyperplane in O i  
Fig  : Intent inference model in a hyperplane arrangement of the obstacle free space  We spatially decompose the preference θ into a set of preferred neighboring polytopes per region of the space  Within each polytope j, the human preference pj is a discrete distribution over the preferred neighbor in N (j) 
We assume that for a location st belonging to polytope j, and given goal g and preference pj, the observation ot and any other preference p i,i =j are conditionally independent  Concatenating elements from each obstacle's Hrepresentation, we can write polytope j's H-representation as where Some of the constraints in eq  ( ) (corresponding to rows of A, b and α j ) are redundant, i e  the set P j does not change upon their removal 
We can further reduce the Hrepresentation of a polytope to include only non-redundant constraints  By removing the rows corresponding to redundant constraints, we obtain new matrices A j e , b j e and α j e such that we can write the polytope's reduced H-representation as The non-redundant constraints correspond to edges of the polytope 
In other words, as the robot continually moves in space, the first hyperplane that it will cross upon exiting the polytope will correspond to one of the polytope's nonredundant constraints  Vincent and Schwager outline an iterative method for removing redundant constraints by solving n linear programs 
We use this method in practice for computing α j e for each polytope  We can now characterize each polytope by a vector α j e ∈ {−1, 1} n j e , where n j e ≤ n is the number of essential constraints of the polytope  The polytopes P j partition the environment into a hyperplane arrangement 

Path Preference

In this section, we provide a definition of preference θ according to a graphical representation of the environment based on the hyperplane arrangement  Under this representation, a path preference corresponds to a set of preferred transitions  In other words, for each polytope in the space, the human will have a preference to which neighboring polytope they wish to transition 
Let G := (V, E) be an undirected graph, where vertices are obstacle-free polytopes, and edges connect two adjacent polytopes  Each polytope is described by a unique vector α j as defined in eq  ( )  Two polytopes are adjacent if they share non-redundant constraints (rows in eq  ( )) corresponding to the same hyperplane (i e  they are on opposite sides of the hyperplane) 
Let N (v) be the set of neighbors of a vertex v  For each vertex, we denote p v the discrete-valued random variable describing which edge in N (v) the human intends to transition to  Using this formalism, we define a path preference as the set of preferred transitions over all nodes in the graph, Let m θ = v∈V |N (v)| be the cardinality of Θ, and m g = |Ω g | the number of possible goals 
A priori, the number of Bayesian updates required to update the belief at every iteration should be m θ × m g   Now, let us assume the conditional independence relationships described by the new problem diagram in fig    More specifically, we introduce the assumption that conditioned on a robot location s t , the goal g, and the preference for the corresponding vertex p v in the graph, the observation o t and the preference for any other vertex are conditionally independent 
In other words, the observations the human provides can be defined conditioned only on the robot location, the goal, and the human's preference for its current vertex p v   By introducing this assumption, each update step only requires updating the joint (p v , g), reducing the number of cost computations to |N (v)| × m g  
We can notice that by introducing this assumption, we removed the direct relationship between the number of polytopes in the environment and the complexity of the Bayesian update in eq  ( )  In practice, components of θ are not mutually independent  For example, if the human preference at a vertex v 1 is
, it is unlikely that the human will also prefer p v2 = (v 2 , v 1 ) (turning back)  We can improve our model by assuming a dependent relationship between preferences for adjacent edges, which does not significantly increase the complexity of the inference problem  An interesting property of our encoding is that any two paths that belong to different homotopy classes will cross different sequences of polytopes, i e  they correspond to a different sequence of edges on G 
This can be proved by contradiction  Let us suppose that two continuous trajectories ξ 1 and ξ 2 , with the same start and end points and that do not intersect any obstacle, traverse the same regions in G in the same order  From the construction of the hyperplane arrangement, each polytope that the paths traverse through is obstacle-free 
Therefore, within each polytope, there is no obstacle in the area located in between the portions of ξ 1 and ξ 2 that belong to the region  A smooth transformation of ξ 1 into ξ 2 can be obtained by transforming each portion of ξ 1 belonging to the polytopes it intersects into the corresponding portion of ξ 2 for the same polytopes, where the extremities of the trajectory portions are connected to one another along the polytope's edges (where the same edge is crossed by both paths) 
Along this transformation, the paths do not intersect any obstacle, and therefore ξ 1 and ξ 2 belong to the same homotopy class 

EXPERIMENTS

We evaluate our model on a simulated navigation task where the robot must reach a goal that is unknown a priori while respecting the path preferences indicated by a human  The robot navigates in a grid world containing obstacles  The transition model is deterministic: the robot selects an adjacent location on the grid to reach at the next time step 
The robot is also allowed to take diagonal actions  Each location s t in the map can be mapped to a vertex v t ∈ G  Therefore, the actions leading to locations mapped to different vertices correspond to edges on the graph  We note f (s t , a t ) the edge crossed by taking action a t from location s t  
The robot is given a mission time limit T max for reaching the goal  In this problem, we assume that the human selects actions to noisily minimize a cost function C g,θ , where θ is defined as per eq  ( ), corresponding to the length of the shortest path to the goal constrained by the preference (where the robot is only allowed to make transitions on G along preferred edges) 
More specifically, where δ(s t , g | o t , p vt ) designates the length of the shortest path from s t to g passing by o t and constrained by preference p vt   This is a slight variant of the cost function proposed by Best and Fitch , where we add in a conditioning on the path preference  We compute costs by running the A path planning algorithm on the environment maps (grid worlds with diagonal actions) and impose preference constraints by pruning invalid transitions from the search tree 
Reward model  At each step in time, the robot receives a reward which is a sum of three components: a goal-specific reward a preference-specific reward or penalty We compute solutions to the POMDP defined in section III-B with the online solver POMCP , and with the particularity that within the rollouts, the robot does not expect to collect human inputs 
Each time a solution is computed, the robot takes an action and may receive an observation  If it does, it updates its belief distribution over the unknown problem variables and resolves the POMDP over a receding horizon 

Baselines

• Goal only  The robot solves the POMDP while ignoring the effects of path preference  Similarly to , we assume the human is taking action to minimize a goaldependent cost C g (s t , o t ) = δ(s t , g | o t ), where the conditioning on the preference is removed  We also omit the path preference's contribution to the reward R pref  
• Compliant  The robot complies with the human input, but does not take an initiative  If the user stops providing information, the robot continues in the last direction indicated for 5 time steps (conserving its momentum), then stops  • Blended  We designed an arbitration function to decide between our proposed policy (accounting for path preferences) and the user's recommendation when the robot receives inputs 
Our metric to evaluate confidence in the robot's prediction for the purpose of arbitration is the entropy of the intention distribution H(g, p i ), where p i denotes the preferred neighbor for the current region  Because our representation of the world is discrete, the arbitration is given by a step function 
Denoted by U , the action corresponding to the human's input, and P , the robot's prediction for the optimal action, we write the policy where we chose h = 1 6 as the confidence threshold 

Results

When evaluating the algorithm, we consider that a run is successful if the robot reached the goal within its allocated mission time T max and only made transitions between graph vertices corresponding to the human's preferences  We vary the time delay between human inputs, from constant guidance (∆ T = 1) to only a single observation (∆ T ≥ T max ) 
Success rates  Table I reports the success rates for experiments conducted over six randomly sampled problem instances and 50 runs per instance in Map 1 (fig  )  When the human provides inputs at every iteration, the compliant policy shows the highest success rates  However, as ∆ T increases, the compliant robot is not able to accomplish the task within the allotted time as it does not receive sufficient inputs to do so, and performance decreases compared to the autonomous baselines 
We find that in these runs, accounting for path preference consistently improves performance compared with the goal-only baseline  Results also show that blending the user's input with the robot's policy (Path Preference + Blend) when the human provides information leads to improved performance  Belief entropy 
Figure shows a challenging problem instance where the directions the human provides do not align directly with the shortest path to the goal  By ignoring the effects of preferences in the problem model (goal only), the robot quickly infers from observations that the upper left goal is less likely than others (P (g) drops) 
The strong decrease in entropy shows that the robot becomes overconfident in this prediction  Overconfidence in an incorrect goal will prevent the agent from finding the correct goal once the human's indications directly align with it, as it needs to correct for the wrong predictions, as shown in the path realization (fig 
)  In this realization, the goal-only method (green robot) fails to search the upper left area within the allotted time  By accounting for path preferences in its model, the blue robot's entropy over the goal distribution decreases more steadily, allowing for it to leverage the human's latest observations and reach the goal successfully 
shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference  Computation time  In table II we provide the time required to solve the POMDP, and the time required to update the robot's belief as it receives new observations 
We compute solutions on three maps: a simple 10 × 10 grid world with 8 polytopes (fig  ), a 10 × 10 grid world with 56 polytopes (fig  ), and a 20×20 grid world with 73 polytopes (fig  )  The latter environment being larger, we increase the mission time and the depth of the search tree in POMCP from T max = 30 (Map 1 and Map 2) to T max = 60 (Map 3) 
We do not notice an increase in the time required to update the robot's belief with an increase in problem complexity, which is consistent with our observation that the complexity of the Bayesian update should not increase with the number of obstacles or polytopes  On the contrary, the belief update time on Map 2 and Map 3, containing more obstacles, is reduced compared to the first map 
More obstacles result in fewer iterations when solving the constrained shortest path problem with A   Adding constraints due to the obstacles and polytopes reduces the size of the A search tree  C  Limitations Simulation environments  In our simulations, we hardcoded the preference policy over the maps (e g  in Map 1, go around the table counter-clockwise) 
We randomly sampled problem instances (start and goal locations, and goal options) to reduce the bias introduced by these preference choices  To best evaluate and compare the different approaches, it would be best to sample preferences among a distribution of preferences chosen by a human (for example, from benchmarks resulting from a collection of data) 
Creating such a benchmark is an interesting direction for future work  Hyperplane arrangement construction  The main limitation of our approach is that the size and geometry of each polytope depends strongly on the geometry of the obstacles, as seen in fig    Because of this, the robot can make predictions over preferences that are too refined compared with the topology of the environment 
A direct consequence is that when the size of the polytopes is small, the information provided by the human can be incorrectly interpreted as a preference on the robot's immediate action  Our method can be improved by changing the structure of the hyperplane arrangement so that it relies on the topology of the environment, but does not vary strongly with the geometry of the features in the environment 
For this purpose, topometric maps and region construction algorithms are promising directions  We presented an approach for encoding and inferring a human's path preference in an environment with obstacles  By leveraging a partitioning of the space into polytopes and a stochastic observation model, our method allows for joint inference over the goal and path preference even when both are unknown a-priori 
Our experiments on an unknown-goal navigation problem with sparse human interventions demonstrate the effectiveness of our approach and its suitability for online applications  The time required to update the robot's belief does not increase with the complexity of the environment, which further highlights the practicality of our method 



ground_truth: The belief entropy decreases more steadily.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the problem encountered when building the fuselage sides?

Read the following text and answer briefly.

Context 3: Probably one of the most frustrating things about building experimental aircraft, especially when starting with a minimum of pre-fabricated parts, is to start building and ending up with an unexpected result  Every builder starts a new project by wanting it to go "perfectly " So when things aren't going well, especially at the beginning, the frustration can lead to an unfinished airplane 
This is the first article in a series dedicated to helping builders of the Rand Robinson KR series planes build a straight and true fuselage -- the first part of the construction process  Borrowing from modern boatbuliding techniques, focus will be on the KR-2S, but the principles apply to the entire lineup of KR-1 & KR-2 series planes 
While building the KR-2(s) a common surprise is encountered by builders when the completed fuselage sides are laid into position to form the fuselage box section  With many hours spent building the sides flat, finding the once straight longerons that now bow up from the building surface, form a most dissatisfying "banana" shape  Especially when using the preformed fiberglass parts, this curve in the top longeron is not acceptable  The builder is left wondering what went wrong and no amount of clamping or brute force forming will solve the problem to any degree of satisfaction  The problem is not the builder's fault  The solution starts by understanding the three dimensional relationship of the assembled parts being built 
First understand that the plans show the finished form of the plane  They show the "projected" form as you would expect to see it if viewing an actual plane from the top, ends and from the side  Since the sides are sloped (flared) outward, looking from the side, the distances given by measuring the profile drawing are "foreshortened" and don't give the proper shape for building the fuselage with a flat top longeron  What needs to be done is to "develop" the "true" distances and shape of the flat panel so that when it is curved into position, the longerons lay flat 
Second, understand that the dimensions called for in the plans put a twist in the sides that tends to work the panel in two directions of curvature  This twist makes the panel "undevelopable" meaning that that shape cannot be unrolled into an equivalent flat shape  This is important when laying out the side and bottom panels onto flat plywood  To illustrate this, try forming a piece of paper around a soda can  The paper can be formed flat around the can either straight or at a diagonal to it's length  It has only one direction of curvature and is by definition "developable"  Now try to form the same piece of paper around a baseball  It won't lie flat on the surface without some deformation (folding, wrinkling or tearing) of the paper  The ball has curvature in more that one direction and is a "compounded" shape  Paper (or plywood) can only be readily formed in developable shapes as opposed to aluminum or other metal which can accept in plane deformation  A developable surface is needed to lay out a curved surface when the materials used can't be deformed with any degree of in-plane strain 
Initially, the fuselage sides are laid out flat with reference to the top longeron measured to a straight chalk line  The bowing problem starts when the side panels are bent and sloped to form the fuselage box section  If the sides were not sloped (tumbled home), the section formed would be cylindrical and the longerons would lie flat  Since the sides are tumbled home, the section formed is now conical  When a conical shape is cut with a plane (building surface) not perpendicular to it's axis, the shape formed is elliptical -- exactly what happens with the top longeron  When it's built flat, bent to form a cylindrical section, and sloped to form a conical section, it takes on an elliptical shape firewall to tailstock 
This method borrows heavily from proven techniques used in the marine trades  It should be stressed at this point that although the layout procedure is not complicated, it is important to take your time  If the layout is not going well initially, start over! Better to erase layout errors now than to have them built it and cause surprises later 
Layout to ensure a fair and true fuselage starts by drawing a reference line (baseline) on the building surface  Refer to figures 2 & 3 and use a wire guide to draw a very straight baseline  About 500 lbs  Of tension should be adequate  One could use a chalk line, but we're talking airplanes here, not house framing 
The main layout difference is that the baseline isn't used as a reference for the top longeron  The baseline references the mid point of the firewall for the developed (and true dimensioned) side panel  Although the baseline will still be the reference, the top and bottom longerons will be laid separately 
Layout differences don't end there  Each of the stations (vertical members) will be laid out with a calculated separation so that when the panels are formed into position, they land on the spacing called for in the plans  Another major difference is that the bottom & side panels are applied after forming the fuselage box section  This is mainly to obtain the ability to "fair" the side and bottom surfaces and insure a straight and true shape 
Refer to figure 1 for the layout of the new developed side panel  The firewall (station a) is layed out perpendicular to the baseline  Longitudinal (station) measurements are given along the length of the baseline from the firewall  Vertical dimensions are given to reference the angle and breadths of the station at the baseline 
Notice that the top longeron is bowed outward and that the stations are spaced slightly greater than called out in the plans  When the panels are formed into the box frame section ,they will work into the dimensions specified in the plans 
Strike a centerline, longer than is needed on the building surface using a wire guide  Draw off the firewall line perpendicular to the centerline at one end 
Using the distances listed in the balloons, mark them off on the centerline  Distances are measured to the nearest sixteenth of an inch  Take time to mark them off carefully  Don't mark off the distances in a cumulative fashion  Use the firewall as a common reference 
Using the angles listed at each station, mark off a station line longer than is needed  The angles are measured to the nearest hundredth of a degree  Take time to mark them off carefully 
At each station, start by marking off each short (bottom longeron) line distance from the centerline  Use your set of trammels or beam compass for doing this  Mark the intersection of the short line with the station line 
At each station, mark off each long (top longeron) line distance from the intersection of the short line distance and the station line  Again the trammels or beam compass is best for completing this step  Mark the intersection of the long line distance with the station line 
Using the longeron as a batten, trace out the inside and outside curves of the longeron  After the batten is secure, in between each station, fasten a keeper block inside and outside to preserve the shape of the longeron taking care to avoid potential future interference with the diagonal members to be installed later  The fairing blocks can be removed or left in place if they won't interfere with building  The vertical station members and their diagonals can now be measured and positioned  Remember to refer to the plans for the material thickness direction 
After vertical and diagonal members are cut and fitted, take time to draw their outlines on the building surface to cut down on time and confusion when laying out the opposite side 
Finishing the side panel is accomplished in a manner similar to that called for in the handbook with the exception that the side and bottom skin panels will be attached later 
The next article in the series will discuss jigging and building techniques to ensure alignment and straightness of the flat built side panels  Also covered will be building a "strongback" jig to assure alignment of the side panels when they are formed into their final shape 
Part 3 in the series will cover assembly of the side panels using the jigs  Some joint details will be discussed that will ensure a stronger and more fair fuselage assembly  Also covered will be the layout & attachment of the side and bottom ply skins 
U S  Mail: Densmore Associates, inc 
ANSI "D" size, computer generated plots of all the layout drawings in this series are available from the author for $30 plus postage & handling  Full (true size) scale plots may be made available depending on demand 
"Scarfing" is the practice of splicing plywood so that short pieces of plywood can be used to span long distances  On the KR, it is required on both the fuselage skins and spar webs  The angle of the splice should be 10 to 12 degrees to maintain strength across the joint  Also, joints should coincide with structural members, such as spar webs or fuselage truss members 
This scarfer is made by mating a regular plunge router (this one costs about $50) to a table saw  Obviously, you really only need a table saw to cut the chamfer, but it does make a nice heavy table for scarfing  You could just as easily use a large work table as the base First, set the table saw for a 5 5 degree cut (for a 1:12 joint, or 6 5 degree cut for a 10:1 joint), and run a 1 x 6 through on edge to chamfer a corner on the board  Then drill the board for three router mounting holes (two are countersunk) and connect the assembly to the table saw with two 1/4 inch bolts  Use a long (2-3 inch) straight cutting bit to do the cutting  Adjust the bit so it doesn't interfere with your table top, and go to town  Keep pressure on the plywood to ensure contact with the table while you're scarfing  Make sure you feed your material from the same end as you would if you were sawing, or the router will take your plywood away from you and put a big dent in your garage door 
In the late 60's Ken Rand and Stuart Robinson were working as flight system engineers for Douglas Avionics  Ken was working as an electrical engineer, having previously worked for Sperry as an autopilots project engineer, while Stu's degree was in aeronautical engineering from Northrop University  They were two of the guys at the end of the DC-8,9, and 10 assembly lines responsible for correcting some of the nits and picks in various systems before delivery to the customer 
They both wanted to build a fast, inexpensive airplane which was also economical to maintain  Several designs were considered, and plans were bought first for the Jeanie's Teenie and then the Taylor Monoplane  The Monoplane was more to their liking, but would require some modification to fit their needs  A cooperative redesign effort ensued, with virtually no dimensions left untouched  Only the basic fuselage structure, airfoil, and powerplant were retained  The tail shape was Stu's, and came directly from the big DC-8s parked on the ramp outside his office window  The landing gear was designed by Ken, after seeing the gear on a Dewey Bird at Santa Paula airport 
Ken was killed in his KR2 a short time later while flying over Cajon Pass in what was apparently a bad weather / low fuel accident  Ken's wife Jeanette became owner of RR overnight, and stepped up to keep the plans and parts coming  Much of the engineering needs are handled by Bill Marcy of Denver, who's been helping out since early '79 
To date, almost 6000 KR1, 9200 KR2, and 760 KR2S plan sets have been sold  1200 KR2s are estimated to be flying, with 5 KR2Ss now in the air  Much of the development work done on KR's is now done by the builders themselves  KR builders tend to be innovative, which leads to some interesting modifications  Some of the mods that work eventually creep into the plans  The KR2S is a case in point  Many builders who'd heard of the pitch sensitivity and tight cabin of the KR2 began to build an enlarged version, with the length determined by the most commonly available longeron material  The result is a KR2 that is stretched 2" between firewall and main spar, and 14" behind the main spar  Higher gross weights dictated more wing area, with the new standard becoming the Diehl wing skin  Those who plan to carry passengers commonly stretch the cabin width a few inches, although 1 5 inches is the limit if you still want to use RR's premolded parts 
Mike Stearns addresses the KR Forum crowd 
This year's KR Forum featured guest speakers Mike Stearns, Steve Trentman, and Bill Marcey  Mike Stearns spoke on several topics, including the many sources for KR and homebuilding information available on the Internet  He also mentioned KRNet, the list server devoted entirely to KR aircraft, as well as several notable World Wide Web home pages  He also brought a sample of the new Rand Robinson wing skins with him, and discussed their high temperature core prepreg construction  His KR2S will receive the first set, which is currently being installed at Hinson Composites 
Steve Trentman spoke on his turbine installation  It uses a turbine engine which saw duty as an A7 attack jet starter engine  Total weight is about 85 pounds, while putting out around 90 horsepower  There is a small stockpile of these engines available from government surplus  sources  This engine can only be throttled back to 52% power, which leads to some pretty interesting landings  One inflight failure has been logged so far, with very little damage to the aircraft  More on this exciting development in next month's issue of KROnline 
Les Palmer's KR2 N202LP won Best KR2, Best Engine Installation, and People's Choice awards at the 1995 KR Gathering at Columbia, TN  After researching the KR series, and reading Neil Bingham's "A Critical Analysis of the KR2" (Jan 88 Sport Aviation), Les decided to build his as a single seater, stretched 24" in the tail, while maintaining a stock width firewall  His fuselage is made from Douglas fir, which weighs in at 4 lbs heavier than if constructed from spruce  It is skinned with 1/8" birch plywood  Spars are covered with plywoood on both fore and aft sides, ala KR2S  Diehl wing skins provide the lift  Horizontal stabilizer and elevator were stretched 7" longer on each side, while the vertical stabilizer and rudder were stretched 8" taller    The fuselage to cowling junction was made more graceful by adding 1 5 inches to the height of the firewall end of the fuselage sides 
Les's canopy is a Dragonfly, using a four linkage system to swing forward when opening  The canopy frame fits snugly into a recess in the foward deck, providing an excellent wind and water seal  The fiberglass work is exemplary 
Seating is luxurious for one 
The cowling is also a work of art, and uses NACA ducts for efficiency  Female molds were made for all the fiberglass parts on Les's plane, so he could proabably be persuaded to make more, if demand dictates  Les also machines a multitude of KR aluminum and steel parts which he now offers for sale 
The firewall was reinforced with aluminum brackets and angles bolted between the longerons in anticipation of the 200 lb Subaru EA-81 engine installation  His 100 HP Asian version is outfitted with an American Holley 5200 caburetor and manifold  It uses a PSRU of Les's own design, featuring two spur gears with a 1 69:1 reduction ratio and a toothed belt  Other than tapping the crank for larger bolts to mount the redrive, no other engine modifications were required  Also, this is probably the only air conditioned KR2 on the planet  The prop is a 60/63 Hegy 
Originally built as a taildragger, the fixed gear is made from 4130 steel tubing  Custom cast 6 00x6 aluminum wheels and steel rotors are mated with 6" Cleveland calipers for braking  An early taxi test accident damaged the main gear, and prompted Les to change to tricycle gear  Again, he designed his own fiberglass main gear, and uses a Diehl nose wheel fork with a 4130 strut and 6" wheel up front 
Early tests revealed cooling problems, which prompted a radiator move from the firewall to a lower cowling location 
The first flight was almost a disaster, as test pilot Randy Smith lost power right after takeoff  He managed a 180 with a safe downwind landing with only minor nosewheel pant damage  The culprit proved to be a spark plug with too much reach, which was quickly remedied  Subsequent flights have shown water temp to be about 210 degrees, oil temp is 220-230, and airspeed is about 180 mph 
Shopping for the Partially Built KR 
This story starts about twenty years ago when I first started looking at the KR-2 as the plane I'd like to build  The only problem at that time was a lack of money, lack of knowledge, and a lack of job stability  I liked the design, except for the low ground clearance of the retractable gear and that a KR was going to be a tight fit for me to fly 
Over the past twenty years I've owned a number of planes, but still always wanted to build my own  I needed one that would fit me, my budget requirements, and have the speed and performance that I wanted  When "KITPLANES" published the article featuring Roy Marsh's new KR-2S, it was the first I had heard of any major modifications or improvements to the same old KR design  I believe that article and Roy Marsh's workmanship have probably been the greatest boon to Rand Robinson (RR) in the last twenty years  It certainly caught my eye! Here was the same design I had decided I wanted to build twenty years ago, with all of the improvements I wanted  It was sitting on fixed gear with some reasonable ground clearance  It had the capability to be built large enough to accommodate me  It has enough prefab parts available that it didn't have to be 100% scratch built if I decided to hurry the project along  And it had the speed I wanted  I knew that Roy's published speeds were probably not realistic expectations for the average KR, but after knocking around for the last three years in my Champ, anything over 90 mph seems pretty fast to me 
After purchasing the info kit and the sales video from Rand Robinson, the next step after deciding for sure to build this plane was to order the KR-2 plans and the KR-2S addendum  I finally got my plans and was putting together my first order to start the plane, when my partner in the Champ pointed out that there was a partially completed KR-2S for sale in Trade-a-plane  My initial answer was "No, I don't even want to look at it  I want to build my own from scratch " My partner insisted that for the advertised price and the fact that it wasn't too far away, I ought to at least give the guy a call and investigate it  "No, I don't think I want to buy someone else's problems," I persisted  That night I went home and crunched up some numbers on the calculator and finally came to the conclusion that for the sake of my budget for the next several years, I really should give this guy a call 
Three days later, I flew to his place about 400 miles away to take a look at his project  At this point I should probably mention that I consider myself to be fairly knowledgeable about airplane construction, although the vast majority of my experience is with tube and fabric  The rest of this article deals with what I looked for and more importantly what I missed and have had to repair in the last year since I purchased the project 
When we went to the seller's house, I found that the left wing was built using the Dan Diehl wing skins and the right wing skins were leaning against the wall inside the house  Also the canopy was in the house with the canopy covered with paper and tape  I wanted to inspect the fuselage first, so off we went to the shop 
There I found a fuselage sitting on it's gear painted in primer gray  The first step was to inspect the quality of workmanship of what could be seen as it sat  The interior of the fuselage looked as if it had been built with a great deal of care  The fit and finish of all of the interior wood was very nice  Even the gussets looked like they had been painstakingly perfectly fitted  The glass work on the turtle back also looked very precise and clean  It was evenly faired into the vertical and horizontal stabs  The tail also appeared to be well built with the exception of a depression directly over the front and rear spars in the horizontal stabs  He explained that when he moved recently, that he had shot the plane with gray primer to protect it from the weather since he wouldn't have ready access to a shop to put it in right away  It ended up sitting out in the hot south Texas summer sun for a few weeks before he got a shop rented to work in  That caused the glass (or possibly the foam inside the horizontal stab) to swell, except that it held onto the spar, so it was slightly ballooned in front of and behind the spars  His recommendation was to fill it back smooth with micro 
I also found a small linear crack in the lower left wing spar cap on the left wing stub  It appeared to be from over tightening the rear spar wing attach fitting bolts  His explanation was that the crack wasn't important because the rear spars only job is to keep the wings from folding back  I also noticed that the holes for attaching the outer wing to the wing stub were badly rounded out on the rear spar  He explained that the Diehl wing skins require the rear spar to be swept slightly more forward than the stock wings  This won't allow you to use the rear spar attach fittings from RR and that I would need to fabricate a new set of rear spar attach fittings 
I also found that the aileron bellcranks were not built or installed as per plans, but found that they looked professional  I couldn't check for function since the right bellcrank and sheeve wasn't installed, the left wing also wasn't installed, and the right wing didn't exist yet 
Next we pulled the inspection panels off of the fuselage and tail and looked at everything I could see with a good flashlight  I didn't find anything else that might be questionable about the fuselage except for a cracked elevator trim tab that was damaged when it fell off it's hanging place on the wall 
Next we spent some time going over his builders log and builders photo album  I still hadn't seen anything that would dissuade me from buying this project 
At this point it was starting to get late and my ride down needed to get airborne for the flight home  I needed to make a decision about whether I wanted this project or not, but I hadn't inspected the wings and canopy yet  I took a cursory look at the left wing and saw lots on micro built up on it and some bubbles in the leading edge, but nothing that looked seriously wrong to my amateur eye  The right wing was only a set of spars in the shop and the Diehl wing skins in the house, so there wasn't much to look at there  The canopy was wrapped in paper and tape, so there wasn't much to look at there either  I decided that even if there were serious problems in the wing that was built, I would be money ahead to go ahead and buy the project  For the advertised price, I could build a new set of wings and still be way ahead financially  We negotiated a final price, shook hands, took my ride to the airport, and started off in search of a U-haul to haul the project home 
Now, at this point, some of you are thinking about what I surely must have forgotten to inspect and why didn't I take a local A & P or EAA member along for the ride  First of all, I don't know any mechanics locally that have any experience with glass and our EAA chapter of which I am VP is woefully lacking in fiberglass knowledge  Secondly, as you will see, I missed plenty  Some by ignorance, some by just not looking close enough 
Now for a list of the problems that I found over the last year and a few of the fixes that I came up with 
I found that the lower set of rear spar attach fittings on the left rear spar were installed backwards with the longer spaced hole towards the fuselage  Since this is the same place that also had the cracked spar cap, it required a major change  Also in the same area he had drilled through the rear spar with a hole saw to create a place for the aileron cable to pass through and managed to cut out the second from the outside vertical brace in the spar  Then he chose to install the aileron bellcranks in front of the rear spar, and cut another hole through the rear spar for the aileron push rod  He also managed to cut out the outside vertical brace in the spar  Since the holes were already drilled through the spar, the choices were to either cut out that section of spar cap and scarf a new piece in, cut the whole rear spar carrythrough out of the fuselage including ruining the left lower wing skin, or do something else creative to reinforce the spar cap and install a custom built set of attach fittings 
I also found that after I built and installed the right side wing stub ribs and skin that the aileron bellcrank setup would not work as installed  The cable that crosses between the two bellcranks had a sharp uphill from the sheeve to the bellcrank in the last 12 inches on either side  This combined with the radius that the bellcranks turn caused the cross cable to pull up tight when the ailerons were pushed to either end of their travel, but allowed the cables to go very slack when the ailerons were centered  Also the Aileron pushrods needed to pass directly through the lower set of rear wing attach fittings to attach to the aileron  This whole rear spar and aileron bellcrank setup was going to either have to be redesigned or cut out and built to plans  The bottom line is that the problems I observed when I inspected this part were much more serious than expected when I had to fix it 
I decided that I had to remove the rear fittings from the left wing to be replaced with the new set that my neighborhood machinist was cutting out for me  When I put the wing on the work bench to start removing the rear fittings, I thought I had better take a closer look at the bubbles in the leading edge  I found that as I pushed on the leading edge, it delaminated between the glass lay-up on top and the upper and lower wing skin edges that were floxed together underneath  I concluded that that area had to come apart and took a belt sander to the leading edge  What I found was that the leading edge had been floxed together and glassed over, but the mold release had never been scrubbed off the leading edge of the wing  It peeled apart for rebuild quite easily 
When I got back to removing the rear spar attach fittings, I noticed that the woodwork inside the wing looked awfully dull  The reason was that the wing had been closed up without varnishing any of the woodwork  This was rectified with a small hole saw, a number of extensions and a modified undercoating sprayer 
I also found that the aluminum drain fitting in the bottom of the left wing tank had been glassed into place upside down  The tapered pipe threads were tapered the wrong way to install the draincock into the tank  Retapping the fitting the right direction seemed to be a good fix for that problem 
When I finally got around to attaching the wing to the fuselage, I found that the front spar attach fittings were badly misaligned  Although they could be forced into alignment, I didn't think I needed that kind of preload on the main spar fittings  This problem was fixed by calling on my local neighborhood machinist to build me an aligning fixture and reaming the attach holes to the next larger size and ordering the new sized bolts 
On the fuselage I found that although it had new Cleveland wheels and brakes on it, one of the brakes had a severe wobble to it  I must complement the manufacturers for taking care of that problem  One call to the Cleveland factory and they shipped me a new set of wheels and brakes even though the receipt for this set was over four years old and in the original builders name  Their only concern was that this set had never been placed in service yet 
I chose to sand the load of micro off the left wing to see what it was covering  When I got down to the glass, I found that there was no glass for the aft inch and a half of the underside of the wing in front of the aileron hinge  With the Diehl wing skins, you build the wings, then cut the ailerons out of trailing edge of the wing  He had mismeasured and cut too much material off the bottom side of the trailing edge in front of the aileron  It was filled by floxing a piece of spruce into the gap to fill the space between the back edge of the fiberglass and the aileron mount  I chose to wrap the trailing edge of that wing, and the other wing to match with a couple of lay-ups of glass 
When I sanded the primer off the aforementioned damaged trim tab, I found that the hinge was floxed to the leading edge of the foam insides of the tab, but not the glass  I also chose to wrap the front of the trim tab with a lay-up of glass 
I decided to pull the paper off the canopy and take a look at it before I'm ready to bolt it on and fly  The original builder had blown his own canopy and after some of the previous problems, I was beginning to have some concerns about not having looked it over closely enough  The canopy turned out to have been blow a little too large  It ended up with a little larger bubble for headroom, which I didn't object to  However, it had more headroom on the right side than the left  Yes, it was just a little bit lopsided  The main problem was that the canopy is stretched thin enough that it can be easily pushed in with one hand when the weather is warm   My fear was that this is just thin enough that it may decide to lay on my head or in my lap when flying on a warm day  It will have to be replaced 
I'm sure that many that are reading this could see several of the potential problems before I mentioned them, but some others may not have and I'm sure that there could have been many other problems that didn't but could have existed on this project  This is also not intended to be critical of the gentleman that started this project as many parts of it, especially the wood work are better than I could have done and much of his work is outstanding  I prefer to think that I'll end up with a better plane with his woodwork combined with my glasswork  This article is intended to feature some of the problems that you may run into in buying someone else's project 
The final question is, knowing what I have found over the past year, would I have still purchased this project  The answer is yes, but primarily because the price was right in that I am still money and work ahead of where I would be if I had started the project from scratch  There are a few things that I would have done differently, but nothing that I can't live with  Although I won't be able to say that I built it all from scratch, I have built and rebuild enough of the plane that I should have no problem qualifying under the 51% rule 
You can send comments directly to the author via e-mail at "jscott@LANL GOV" 
Here is an brief explanation of how I built my turtledecks  The jig was constructed from scrap plywood and a few 1x4s that I ripped into stringers  I made two temporary bulkheads from the plywood, one for each end  Remember the forward bulkhead needs to be shaped in a way that will closely match the aft end of your canopy frame  Make an aft bulkhead by placing a straight edge at the top of your forward bulkhead and the trailing edge of your horizontal stabilizer  This will give you an idea of how tall your aft bulkhead needs to be  As far as location, I placed my aft bulkhead just forward of the lower/front of my vertical fin  I constructed the jig on the fuselage, it is glued together with automotive bondo 
After the bulkheads were bondoed to the fuselage I used the stringers that I ripped from the 1x4s and bondoed them to the bulkheads  This gave me a male form to cover with thin plastic or posterboard  I stapled two layers of posterboard to the jig(thin plastic would work better)  The posterboard wraps down two inches onto the fuselage  After I was satisfied with the way it looked, I then covered the entire thing with duct tape (fiberglass will not stick to duct tape) On top of this I wetout one layer of tri-ply cloth (22oz) that I had left over from an earlier project, and one layer of 8oz  bid  Remember to mask off your fuselage so you don't get epoxy on it  If you are not familiar with composite lay-ups, you should plan on razor cutting your lay-ups 4 to 6 hours after wetout while the lay-up is still soft enough to cut with a razorblade 
After the lay-up cured (2 or 3 days) it was removed from the jig, and the jig was removed from the fuselage and discarded  (be careful, the bondo sticks very well to the spruce, you could splinter your wood during removal) I now have a fiberglass skin that tends to hold the shape of the jig but is still flexible enough to work with  I made two bulkheads out of 1/4 last-a-foam (AS&S) using the plywood formers from the jig as a guide  I covered these foam bulkheads with one 8oz layer of glass on each side, with a glass to glass edge on the bottom  After cure these bulkheads were bondoed into place (to the fuselage)and the fiberglass skin was pulled down tight and floxed to the bulkheads  When the flox cured the bondo joints were broken, again being careful not to harm the wood  The turtledeck was removed from the fuselage and 2 inch tapes added to the bulkheads inside and out 
At this point the turtledeck looked great and only weighed about 5lbs  but I noticed you could deform the skin by pushing hard on the outside  So I flipped the turtledeck over and from 1/4 inch last-a-foam, I cut two inch wide strips that would run the entire length, forward and aft inside the turtledeck  In effect these would act as composite stringers, I made enough of these two inch wide strips to make up three stringers  One down the center (sort of a backbone) and one on each side of the "backbone" half the distance to the edge of the turtledeck  I sanded the edge of the foam so that when covered with a layer of bid @ 45degrees there would be a nice transition from the turtledeck skin up onto the foam and then back onto the turtledeck I scuff sanded and glued the foam stringers in with micro  I covered the foam stringers with one layer of 8oz bid @ 45degrees 



ground_truth: The longerons bow up from the building surface, forming a "banana" shape.