diff --git a/flux/nn_prior/two_link_sys.jl b/flux/nn_prior/two_link_sys.jl
index 01646778a673974eb0e50aa1b64ac8b8be389095..6c1b8f5dd5d1d6c19d765bcd390f4d899f3b0363 100644
--- a/flux/nn_prior/two_link_sys.jl
+++ b/flux/nn_prior/two_link_sys.jl
@@ -15,7 +15,7 @@ Parameters.@with_kw struct TwoLinkSys <: AbstractSystem
end
-function generate_data(sys::TwoLinkSys, systype, seed, validation=false)
+function generate_data(sys::TwoLinkSys, systype, seed, validation=false; ufun=u->filt(ones(50),[50], 10u')')
Parameters.@unpack N, n, ns, h, σ0, sind, uind, s1ind = sys
srand(seed)
@@ -23,7 +23,7 @@ function generate_data(sys::TwoLinkSys, systype, seed, validation=false)
done = false
local x,u
while !done
- u = filt(ones(50),[50], 10randn(N+2,n))'
+ u = ufun(randn(n,N+2))
t = h:h:N*h+h
x0 = [-0.4,0,0,0]
prob = OrdinaryDiffEq.ODEProblem((t,x)->time_derivative(x, u[:,ceil(Int,t/h)]),x0,(t[[1,end]]...))
@@ -41,8 +41,7 @@ function generate_data(sys::TwoLinkSys, systype, seed, validation=false)
end
function true_jacobian(sys::TwoLinkSys, evalpoint, x, u)
- Jtrue = ForwardDiff.jacobian(x->time_derivative(x, u[:,evalpoint]), x[:,evalpoint])[:,1:4]
- Jtrue = [Jtrue ForwardDiff.jacobian(u->time_derivative(x[:,evalpoint], u), u[:,evalpoint])]
+ Jtrue = ReverseDiff.jacobian(x->time_derivative(x, u[:,evalpoint]), x[:,evalpoint])[:,1:4]
+ Jtrue = [Jtrue ReverseDiff.jacobian(u->time_derivative(x[:,evalpoint], u), u[:,evalpoint])]
Jtrue = expm([sys.h*Jtrue;zeros(2,6)])[1:4,:] # Discretize
- # Jtrue .-= [I zeros(4,2)] # Compensate for prediction of difference
end
diff --git a/flux/nn_prior/utilities.jl b/flux/nn_prior/utilities.jl
index cb3bb20676ae8e26d492ae630543477bc0b714d8..b6b6100f5108c47726a0281e598152e4d95fc399 100644
--- a/flux/nn_prior/utilities.jl
+++ b/flux/nn_prior/utilities.jl
@@ -293,37 +293,7 @@ end
get_res(res,n) = getindex.(res,n)
-function savefig2(filename, opts=[])
- fig = current()
- layout = size(fig.layout.grid)
- pgffig = fig.o
- pattern = "axis background/.style={fill={rgb,1:red,1.00000000;green,1.00000000;blue,1.00000000}}"
- foreach(f->(f.width="\\figurewidth"), pgffig)
- foreach(f->(f.height="\\figureheight"), pgffig)
- subfig = product(0:layout[2]-1, 0:layout[1]-1)
- for (f,sf) in zip(pgffig, subfig)
- for (i,ss) in enumerate(f.style)
- if !isempty(opts) && ss ∈ keys(opts)
- f.style[i] = opts[ss]
- end
- if contains(ss, pattern)
- f.style[i] = "xshift=$(1.05*sf[1])\\figurewidth, yshift=$(1.05*sf[2])\\figureheight"#replace(f.style[i], pattern, "")
- else
- f.style[i] *= "\n"
- end
- end
- end
- PGFPlots.save(filename, pgffig, include_preamble=false)
- temppath, temp = mktemp()
- for line in readlines(filename, chomp=true)
- # println(line)
- line = replace(line, "{unbounded coords", "unbounded coords",1)
- line = replace(line, "\\figureheight}, ymin", "\\figureheight, ymin",1)
- println(temp, line)
- end
- close(temp)
- mv(temppath, filename, remove_destination=true)
-end
+
try
foreach(treelike, [SystemD, DiffSystemD, VelSystemD, System, DiffSystem, VelSystem])
diff --git a/jump_lin_id/figs/ilc.tex b/jump_lin_id/figs/ilc.tex
index 5107fde33e992dbbeecd8e9625bfe4a78c457544..a2da6817fe4266c8f3c064fc018c5dbb9b1cd0c1 100644
--- a/jump_lin_id/figs/ilc.tex
+++ b/jump_lin_id/figs/ilc.tex
@@ -6,8 +6,8 @@
\definecolor{color2}{rgb}{0.242224297852199,0.64327509315763,0.304448651534115}
\begin{axis}[
-title={Median cost over 20 iterations},
-xlabel={Iteration},
+title={Median over 20 Monte-Carlo runs},
+xlabel={Learning iteration},
ylabel={Cost},
xmin=1, xmax=14,
ymin=1995.4298956103, ymax=10000,
diff --git a/jump_lin_id/figs/robot_train.tex b/jump_lin_id/figs/robot_train.tex
new file mode 100644
index 0000000000000000000000000000000000000000..e68d7bcf722fe831b88bf9b30891df0bbf500f75
--- /dev/null
+++ b/jump_lin_id/figs/robot_train.tex
@@ -0,0 +1,2840 @@
+\begin{tikzpicture}[]
+\begin{axis}[height = {\figureheight}, ylabel = {}, title = {$q_1$}, xmin = {1.0}, xmax = {200.0}, ymax = {0.238105426583605}, xlabel = {}, unbounded coords=jump
+, scaled x ticks = false
+, xticklabel style={rotate = 0}
+, xmajorgrids = false
+, xtick = {50.0,100.0,150.0,200.0}
+, xticklabels = {50,100,150,200}
+, xtick align = inside
+, axis lines* = left
+, scaled y ticks = false
+, yticklabel style={rotate = 0}
+, ymajorgrids = false
+, ytick = {-1.2000000000000002,-0.9000000000000001,-0.6000000000000001,-0.30000000000000004,0.0}
+, yticklabels = {-1.2,-0.9,-0.6,-0.3,0.0}
+, ytick align = inside
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index 0000000000000000000000000000000000000000..22586689b03f259cdb437ee38bf6fe2a296619ce
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+(199.0, -0.014175808066410056)
+(200.0, -0.010892783835507736)
+};
+\end{axis}
+
+\end{tikzpicture}
diff --git a/jump_lin_id/id_paper.tex b/jump_lin_id/id_paper.tex
index 1260a5c4ed45325c2ecb7c7a17c2dbc5b3324a5a..01a6325dbd4495d035cd50b1521ed103066f7de5 100644
--- a/jump_lin_id/id_paper.tex
+++ b/jump_lin_id/id_paper.tex
@@ -59,17 +59,12 @@
\definecolor{linecolor4}{rgb}{0.6,0,0.5}
%% Include comments
-\newcommand{\recmt}[1]{{\color{red}{\textbf{Reviewers comment:} #1}}}
-\newcommand{\cmt}[1]{{\color{yellow}{\textbf{Comment:} #1}}}
-%\newcommand{\rcmt}[1]{Rolfs comment: {\color{red}{ #1}}}
-%\newcommand{\revised}[3]{{\color{gray} #1}{\color{green} \textbf{Added:} #2}{\color{yellow}\textbf{Bagge's comment:} #3} }
-%\usepackage[utf8]{inputenc}
+% \newcommand{\recmt}[1]{{\color{red}{\textbf{Reviewers comment:} #1}}}
+% \newcommand{\cmt}[1]{{\color{yellow}{\textbf{Comment:} #1}}}
%% Exclude comments
-% \newcommand{\recmt}[1]{}
-%\newcommand{\cmt}[1]{}
-\newcommand{\rcmt}[1]{}
-\newcommand{\revised}[3]{#2}
+\newcommand{\recmt}[1]{}
+\newcommand{\cmt}[1]{}
\newcommand{\figurecaptionreduction}{\vspace{-2.5mm}}
\newcommand{\figurefontsize}{\footnotesize}
@@ -153,7 +148,7 @@ Identification of LTV Dynamical Models with\\ Smooth or Discontinuous Time Evolu
\pagestyle{empty}
\begin{abstract}
- We establish a connection between trend filtering and system identification which results in a family of new identification methods for linear, time-varying (LTV) dynamical models based on convex optimization. We demonstrate how the design of the cost function promotes either a continuous change in dynamics over time, or causes discontinuous changes in model coefficients occurring at a finite (sparse) set of time instances. We further discuss the introduction of priors on the model parameters for situations where excitation is insufficient for identification. The identification problems are cast as convex optimization problems and are applicable to, e.g., ARX-models and state-space models with time-varying parameters. We illustrate usage of the methods in simulations of a nonlinear robot arm with non-smooth friction and in model-based, trajectory centric reinforcement learning on a smooth nonlinear system.
+ We establish a connection between trend filtering and system identification which results in a family of new identification methods for linear, time-varying (LTV) dynamical models based on convex optimization. We demonstrate how the design of the cost function promotes either a continuous change in dynamics over time, or causes discontinuous changes in model coefficients occurring at a finite (sparse) set of time instances. We further discuss the introduction of priors on the model parameters for situations where excitation is insufficient for identification. The identification problems are cast as convex optimization problems and are applicable to, e.g., ARX-models and state-space models with time-varying parameters. We illustrate usage of the methods in simulations of jump-linear systems, a nonlinear robot arm with non-smooth friction and stiff contacts as well as in model-based, trajectory centric reinforcement learning on a smooth nonlinear system.
% \cmt{Rewrote abstract to make contribution stand out from previous work and highlight interesting nonlinear and non-smooth nature of examples.}
\end{abstract}
% \begin{keywords}
@@ -461,7 +456,7 @@ For the LTI problem to be well-posed, the system must be identifiable and the in
-\section{Example}
+\section{Example -- Jump-linear system}
\cmt{Add two link example where a stiff contact is established and there is a sign change in control signal and hence friction. Apply pwconstant method and maybe scrap linear system example. Fig 3 can be reproduced with analytical jacobian as comparison. Mention in abstract that we then evaluate on one non-linear but smooth system (pendcart) and one non-smooth system (twolink).}
\recmt{Only linear systems considered (pend-cart actually not linear.} \cmt{Highligt that pendcart is nonlinear or introduce simple robotic example. Mention early that an LTV system along a trajectory does not generalize far for a nonlinear system, hence the use of KL constraint.}
% \begin{figure}
@@ -500,17 +495,17 @@ to $$A_t = \left[
\right]
$$
occurred at $t=200$.
-The input was Gaussian noise of zero mean and unit variance, state transition noise and measuremet noise of zero mean and $\sigma = 0.2$ were added. The generated and filtered signals are shown in~\cref{fig:states}.
+The input was Gaussian noise of zero mean and unit variance, state transition noise and measuremet noise of zero mean and $\sigma = 0.2$ were added.
\Cref{fig:ss} depicts the estimated coefficients in the dynamics matrices for a value of $\lambda$ chosen using the L-curve method~\cite{hansen1994regularization}.
-\begin{figure}
- \centering
- \setlength{\figurewidth}{0.99\linewidth}
- \setlength{\figureheight }{5cm}
- \input{figs/states.tex}
- \figurecaptionreduction
- \caption{Piecewise constant state-space dynamics. True state values are shown with dashed lines. Gaussian state-transition and measurement noise with $\sigma = 0.2$ were added. \recmt{Hard to see and understand}}
- \label{fig:states}
-\end{figure}
+% \begin{figure}
+% \centering
+% \setlength{\figurewidth}{0.99\linewidth}
+% \setlength{\figureheight }{5cm}
+% \input{figs/states.tex}
+% \figurecaptionreduction
+% \caption{Piecewise constant state-space dynamics. True state values are shown with dashed lines. Gaussian state-transition and measurement noise with $\sigma = 0.2$ were added. \recmt{Hard to see and understand}}
+% \label{fig:states}
+% \end{figure}
\begin{figure}
\centering
\setlength{\figurewidth}{0.99\linewidth}
@@ -522,8 +517,29 @@ The input was Gaussian noise of zero mean and unit variance, state transition no
\end{figure}
+\section{Example -- Non-smooth robot arm with stiff contact}
+To illustrate the ability pf the proposed models to represent the non-smooth dynamics along a trajectory of a robot arm, we simulate a two-link robot with discontinuous Coulomb friction. Additionally, we let the robot establish a stiff contact with the environment to illustrate both strengths and weaknesses of the modeling approach.
-\section{Reinforcement learning}
+The state of the robot arm consists of two joint coordinates, $q$, and their time derivatives, $\dot q$. \Cref{fig:robot_train} illustrates the state trajectories, control torques and simulations of a model estimated with~\labelcref{eq:slow}. The figure clearly illustrates that the model is able to capture the dynamics both during the non-smooth sign change of the velocity, but also during establishment of the stiff contact. The learned dynamics of the contact is however time-dependent, which is illustrated in \Cref{fig:robot_val}, where the model is used on a validation trajectory where a different noise sequence was added to the control torque. Due to the novel input signal, the contact is established at a different time-instant and as a consequence, there is an error transient in the simulated data.
+\begin{figure*}[htp]
+ \centering
+ \setlength{\figurewidth}{0.5\linewidth}
+ \setlength{\figureheight }{4cm}
+ \input{figs/robot_train.tex}
+ % \includegraphics[width=\linewidth]{figs/robot_train}
+ \caption{Simulation of non-smooth robot dynamics with stiff contact -- training data vs. sample time index. The sign change in velocity, and hence a discontinuous change in friction torque, occurs in the time interval 50-100 and the contact is established in the time interval 100-150.}
+ \label{fig:robot_train}
+\end{figure*}
+\begin{figure*}[htp]
+ \centering
+ \setlength{\figurewidth}{0.5\linewidth}
+ \setlength{\figureheight }{4cm}
+ \input{figs/robot_val.tex}
+ \caption{Simulation of non-smooth robot dynamics with stiff contact -- validation data vs. sample time index. The dashed lines indicate the event times for the training data, highlighting that the model is able to deal effortless with the non-smooth friction, but inaccurately predicts the time evolution around the contact event.}
+ \label{fig:robot_train}
+\end{figure*}
+
+\section{Example -- Reinforcement learning}
\recmt{More details needed}
In this example, we use the proposed methods to identify LTV dynamics models for
reinforcement learning. The goal of the task is to dampen oscillations of a pendulum attached to a moving
@@ -627,6 +643,10 @@ can be generalized to $(\Delta k)\T \Lambda (\Delta k)$, where $\Lambda$ is an a
This allows incorporation of different scales for different variables with little added implementation complexity.
+\section{Conclusions}
+We have proposed a framework for identification of linear, time-varying models along trajectories of nonlinear dynamical systems using convex optimization. We showed how a Kalman smoother can be used to estimate the dynamics efficiently in a few special cases, and demonstrated the use of the proposed LTV models on three examples, highlighting their efficiency for trajectory-centric, model-based reinforcement learning, iterative learning control (ILC), and jump-linear system identification. We have also demonstrated the ability of the models to handle non-smooth friction dynamics as well as analyzed the identifiability of the models.
+
+
% The drawbacks of LTV models include situations where the observed time variation comes from non-linear effects. Non-linearities in the system may cause an apparent time-variation as the state varies. While an LTV-model can capture this variation, it will fail to generalize well to a subsequent experiment where the states take a significantly different trajectory. Many non-linear systems are, however, approximately \emph{locally} linear, such that they are well described by a linear model in a small neighborhood around the linearization/operating point. LTV models can thus be efficiently used for model-based reinforcement learning with variants of trust-region based optimization~\cite{schulman2015trust}.
diff --git a/jump_lin_id/two_link.jl b/jump_lin_id/two_link.jl
index 310ebbfb0f09f96afe3d2a69a0bb02118e54821d..9112d54b8712369686e187d186b88477f6e2a112 100644
--- a/jump_lin_id/two_link.jl
+++ b/jump_lin_id/two_link.jl
@@ -1,6 +1,7 @@
cd(@__DIR__)
# import Iterators
-using OrdinaryDiffEq, LTVModels
+using OrdinaryDiffEq
+@everywhere using LTVModels
using ValueHistories, IterTools, MLDataUtils, OrdinaryDiffEq, Parameters, InteractNext
@@ -30,15 +31,15 @@ u .+= 0.1randn(size(u))
const wall = -0.2
function affect!(i)
- # q = i.u[1:2]
- # qd = i.u[3:4]
- # p1,p2 = forward_kin(q)[2]
- # J = TwoLink.jacobian(q)
- # v1,v2 = J*qd
- # d = [0, wall-p2]
- # δq = J\d
- # i.u[3:4] = J\[v1, 0]
- # i.u[1:2] .+= δq
+ q = i.u[1:2]
+ qd = i.u[3:4]
+ p1,p2 = forward_kin(q)[2]
+ J = TwoLink.jacobian(q)
+ v1,v2 = J*qd
+ d = [0, wall-p2]
+ δq = J\d
+ i.u[3:4] = J\[v1, 0]
+ i.u[1:2] .+= δq
end
condition(x,t,i) = forward_kin(x)[2][2] > wall
cb = DiscreteCallback(condition,affect!)
@@ -51,59 +52,159 @@ x = hcat(sol(t)...)
u = u[:,1:N]
u = [u; ones(1,size(u,2))] # Extend to estimate friction
x,y = xy(System,x,N)
-plot(forward_kin(x[1:2,:])')
-plot!([1,N],[wall, wall], l=:dash, c=:black)
-error("True jacobian påverkas inte av väggen :/ Den syns bara i datan. Kolla om breakpoint identifieras när systemet träffar väggen")
+plot(forward_kin(x[1:2,:])', layout=2, subplot=1)
+plot!([1,N],[wall, wall], l=(:dash, :black), grid=false, subplot=1)
+vline!([136])
+plot!(u', subplot=2)
+vline!([62], subplot=2)
+# error("True jacobian påverkas inte av väggen :/ Den syns bara i datan. Kolla om breakpoint identifieras när systemet träffar väggen")
##
evalpoints = 1:sys.N
Jtrue = map(evalpoints) do evalpoint
- true_jacobian(sys, evalpoint, x, u)
+ true_jacobian(sys, evalpoint, x, u[1:2,:])
end
Jtrue = cat(3, Jtrue...)
-Jtrue = reshape(Jtrue, np, sys.N)
+Jtrue = reshape(Jtrue, 24, sys.N)
colors = [HSV(h,1,0.8) for h in linspace(0,200,5)]
-r2,p0 = 300,7
-model = LTVModels.fit_statespace_admm(x,u,1, extend=false)
-si = slider(1:np); sλ = slider(linspace(-3,3,50));
-function opt(λ)
- global n,m,np,x,u,N,model,lastλ
- j = 1
+sλ = linspace(-5,5,20)
+sr2 = linspace(-4,4,20)
+sp0 = linspace(-3,8,20)
+@everywhere using ParallelDataTransfer
+sendto(collect(2:5), x=x, u=u, n=n, m=m)
+admmvec = pmap(sλ) do λ
+ global x,u
+ model = LTVModels.fit_statespace_admm(x,u,10^λ, extend=false, tol=1e-8, iters=20000)
+end
+
+kalmanvec = pmap(IterTools.product(sr2,sp0)) do t
+ r2,p0 = t
+ global x,u,n,m
R1 = eye(n*(n+m))
- R2 = r2*eye(n)
+ R2 = 10^r2*eye(n)
P0 = 10^p0*R1
- model = LTVModels.fit_statespace_admm!(model,x,u,10^λ, extend=false, tol=1e-5)
- # model = LTVModels.fit_statespace_dp(x,u,M,extend=false)
- # model = LTVModels.fit_model(LTVModels.KalmanModel, x,u,R1,R2,P0, extend=true)
+ model = LTVModels.fit_model(LTVModels.KalmanModel, x,u,R1,R2,P0, extend=true)
end
-lastλ = 0.
-ui = @manipulate for i in si, λ in sλ
+kalmanvec = reshape(kalmanvec, 20,20)
+
+ui = @manipulate for λ in sλ, r2 in sr2, p0 in sp0, i in 1:24
global n,m,np,x,u,N,model,lastλ
j = 1
- if λ != lastλ
- J = opt(λ)
- lastλ = λ
- end
- J = reshape(cat(2, model.At, model.Bt), np, N)
- # plot(Jtrue[i,:],lab="True", l=:dash, c=:blue, layout=1, legend=false, size=(1920,1200), show=false)
- # ui = @manipulate for r2 in linspace(1,1000,20), p0 in linspace(1,7,50), i in 1:24
+ indad = findfirst(abs.(λ .- sλ) .< sqrt(eps()))
+ indk = (findfirst(abs.(r2 .- sr2) .< sqrt(eps())), findfirst(abs.(p0 .- sp0) .< sqrt(eps())))
+ Jadmm = reshape(cat(2, admmvec[indad].At, admmvec[indad].Bt), np, N)
+ Jkalman = reshape(cat(2, kalmanvec[indk...].At, kalmanvec[indk...].Bt), np, N)
+
+
+ plot(Jtrue[i,:],lab="True", l=:dash, c=:blue, layout=1, legend=false, show=false)
+ plot!(Jadmm[i,:], lab="admm", c=:red, subplot=1, show=false, show=false)
+ plot!(Jkalman[i,:], lab="kalman", c=:green, subplot=1, show=false, show=false)
+ # plot!(0.5maximum(abs.(Jadmm[i,:]))*sign.(qd)', linewidth=[3 1])
+ hline!([0], l=(:dash, :black), linewidth=2)
+end
+
+#=
+ui = @manipulate for r2 in linspace(-4,4,100), p0 in slider(linspace(-3,8,100)), i in 1:np
+ global n,m,np,x,u,N,model,lastλ
+
+ j = 1
+ R1 = eye(n*(n+m))
+ R2 = 10^r2*eye(n)
+ P0 = 10^p0*R1
# model = LTVModels.fit_statespace_dp(x,u,M,extend=false)
- # model = LTVModels.fit_model(LTVModels.KalmanModel, x,u,R1,R2,P0, extend=true)
+ model = LTVModels.fit_model(LTVModels.KalmanModel, x,u,R1,R2,P0, extend=true)
+
+ J = reshape(cat(2, model.At, model.Bt), np, N)
+ plot(Jtrue[i,:],lab="True", l=:dash, c=:blue, layout=1, legend=false, show=false)
- # yh = LTVModels.predict(model, x,u)
- # plot(y', lab="y", layout=6, show=false)
- # plot!(yh', lab="yh", subplot=1:4, show=false)
- # plot!(u', lab="u", subplot=5:6)
- plot(J[i,:], lab="Estimated", c=colors[j], subplot=1, show=false, show=false)
+ plot!(J[i,:], lab="Estimated", c=colors[j], subplot=1, show=false, show=false)
plot!(0.5maximum(abs.(J[i,:]))*sign.(qd)', linewidth=[3 1])
hline!([0], l=(:dash, :black), linewidth=2)
end
-# gui()
+=#
+
+@manipulate for ind = eachindex(sλ), normalize = [true,false]
+ # a = activation(admmvec[ind], normalize=normalize)
+ a = reshape(admmvec[ind].At, :, N)
+ a = sum(abs.(diff(a,2)), 1)
+ plot(a', show=false)
+ # plot(filt(ones(5),[5],abs.(diff(a))), show=false)
+ vline!([62,136])
+end
+
+@manipulate for ind1 = eachindex(sr2),ind2 = eachindex(sp0), normalize = [true,false]
+ a = activation(kalmanvec[ind1,ind2], normalize=normalize)
+ plot(filt(ones(5),[5],abs.(diff(a))), show=false)
+ vline!([62,136])
+end
+
+
+@manipulate for ind1 = eachindex(sr2),ind2 = eachindex(sp0)
+ m = kalmanvec[ind1,ind2]
+ plot(x', label="States", show=false, layout=(3,1), subplot=1, size=(800,1000), title="State trajectories", lab=["q1" "q2" "qd1" "qd2"], c=[:blue :blue :green :green])
+ plot!(LTVModels.simulate(m, x[:,1], u)[:,2:end]', label="Simulation", show=false, subplot=1, l=(:red, :dash))
+ vline!([62], show=false, subplot=1, text=["Velocity sign change"], l=(:black, :dash))
+ vline!([135], show=false, subplot=1, text=["Stiff contact"], l=(:black, :dash))
+ plot!(forward_kin(x[1:2,:])', subplot=2, title="End effector positions", lab=["x" "y"])
+ hline!([wall], l=(:dash, :black), grid=false, subplot=2, text=["Constraint"])
+ vline!([135], subplot=2, text=["Stiff contact"], l=(:black, :dash))
+ plot!(u', subplot=3, title="Control torques")
+ vline!([62], subplot=3, text=["Velocity sign change"], l=(:black, :dash))
+end
+
+
+pgfplots(grid=false, legend=:bottom)
+##
+
+
+r2,p0 = -3.,8.
+R1 = eye(n*(n+m))
+R2 = 10^r2*eye(n)
+P0 = 10^p0*R1
+model = LTVModels.fit_model(LTVModels.KalmanModel, x,u,R1,R2,P0, extend=true)
+
+
+signch = findfirst(u[2,:] .< 0)
+contact = findfirst(forward_kin(x[1:2,:])[2,:] .> wall)
+ind2 = length(sp0)
+fig=plot(x', label="States", show=false, layout=grid(3,2), subplot=1:4, size=(800,1000), title="State trajectories", title=["q1" "q2" "qd1" "qd2"], c=:blue, linewidth=4)
+vline!([signch].*ones(1,4), show=false, subplot=1:4, lab="Velocity sign change", l=(:black, :dash))
+vline!([contact].*ones(1,4), show=false, subplot=1:4, lab="Stiff contact", l=(:black, :dash))
+plot!(forward_kin(x[1:2,:])', subplot=5, title="End effector positions", lab=["x" "y"])
+hline!([wall], l=(:dash, :black), grid=false, subplot=5, lab="Constraint")
+vline!([contact], subplot=5, lab="Stiff contact", l=(:black, :dash))
+plot!(u[1:2,:]', subplot=6, title="Control torques", lab="")
+vline!([signch], subplot=6, lab="Velocity sign change", l=(:black, :dash))
+plot!(LTVModels.simulate(model, x[:,1], u)', label="Simulation", show=false, subplot=1:4, l=(:red,), legend=[false false true false])
+gui()
+# savefig2(@__DIR__()*"/figs/robot_train.tex", Dict("" => "legend style={at={(1.2,-0.2)}, anchor=north, legend columns=4},"))
+##
+
+# Validate
##
+let u, x, y
+u = torque(q,qd,qdd)
+u .+= 0.1randn(size(u))
+t = h:h:N*h+h
+q0 = [q[:,1]; qd[:,1]]
+prob = OrdinaryDiffEq.ODEProblem((x,p,t)->time_derivative(x, u[:,ceil(Int,t/h)]),q0,(t[[1,end]]...))
+sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8, callback=cb)
+x = hcat(sol(t)...)
+u = u[:,1:N]
+u = [u; ones(1,size(u,2))] # Extend to estimate friction
+x,y = xy(System,x,N)
+
+plot(x', label="States", show=false, layout=grid(2,2), subplot=1:4, size=(800,1000), title="State trajectories", title=["q1" "q2" "qd1" "qd2"], c=:blue, linewidth=4, reuse=false)
+vline!([signch].*ones(1,4), show=false, subplot=1:4, lab="Original Velocity sign change", l=(:black, :dash))
+vline!([contact].*ones(1,4), show=false, subplot=1:4, lab="Original Stiff contact", l=(:black, :dash))
-w = get_page()
-body!(w, ui)
+plot!(LTVModels.simulate(model, x[:,1], u)', label="Simulation", show=false, subplot=1:4, l=(:red,), legend=[false false true false])
+
+gui()
+# savefig2(@__DIR__()*"/figs/robot_val.tex", Dict("" => "legend style={at={(1.2,-0.2)}, anchor=north, legend columns=4},"))
+end
+##