diff --git a/flux/nn_prior/two_link_sys.jl b/flux/nn_prior/two_link_sys.jl
index 6c1b8f5dd5d1d6c19d765bcd390f4d899f3b0363..ce9ca9b32cad95104ce6594b9029d37ce7edef14 100644
--- a/flux/nn_prior/two_link_sys.jl
+++ b/flux/nn_prior/two_link_sys.jl
@@ -26,7 +26,7 @@ function generate_data(sys::TwoLinkSys, systype, seed, validation=false; ufun=u-
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]]...))
+ prob = OrdinaryDiffEq.ODEProblem((x,p,t)->time_derivative(x, u[:,ceil(Int,t/h)]),x0,(t[[1,end]]...))
sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8)
x = hcat(sol(t)...)
done = all(abs.(x[1:2,:]) .< 0.9π)
diff --git a/jump_lin_id/figs/robot_train.tex b/jump_lin_id/figs/robot_train.tex
index e68d7bcf722fe831b88bf9b30891df0bbf500f75..d0374008c91b57bf3eb955f5c1574dc0b4ecdcbb 100644
--- a/jump_lin_id/figs/robot_train.tex
+++ b/jump_lin_id/figs/robot_train.tex
@@ -1,22 +1,22 @@
\begin{tikzpicture}[]
-\begin{axis}[height = {\figureheight}, ylabel = {}, title = {$q_1$}, xmin = {1.0}, xmax = {200.0}, ymax = {0.238105426583605}, xlabel = {}, unbounded coords=jump
+\begin{axis}[height = {\figureheight}, ylabel = {}, title = {$q_1$}, xmin = {1.0}, xmax = {200.0}, ymax = {1.2041925960958817}, xlabel = {}, unbounded coords=jump
, scaled x ticks = false
, xticklabel style={rotate = 0}
-, xmajorgrids = false
+, xmajorgrids = true
, 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}
+, ymajorgrids = true
+, ytick = {-1.0,-0.5,0.0,0.5,1.0}
+, yticklabels = {-1.0,-0.5,0.0,0.5,1.0}
, ytick align = inside
, axis lines* = left
-, xshift=0.0\figurewidth, yshift=0.0\figureheight, legend style={at={(1.2,-0.2)}, anchor=north, legend columns=4}, ymin = {-1.3180282010216717}, width = {\figurewidth}]\addplot+ [color = {rgb,1:red,0.00000000;green,0.00000000;blue,1.00000000},
+, xshift=0.0\figurewidth, yshift=0.0\figureheight, legend style={at={(1.2,-0.2)}, anchor=north, legend columns=4}, ymin = {-1.4928056891063268}, width = {\figurewidth}]\addplot+ [color = {rgb,1:red,0.00000000;green,0.00000000;blue,1.00000000},
draw opacity=1.0,
-line width=4,
+line width=2,
solid,mark = none,
mark size = 2.0,
mark options = {
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solid
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diff --git a/jump_lin_id/figs/robot_val.tex b/jump_lin_id/figs/robot_val.tex
index 22586689b03f259cdb437ee38bf6fe2a296619ce..f847755d9558c48b851693233c23bb4700ae4ceb 100644
--- a/jump_lin_id/figs/robot_val.tex
+++ b/jump_lin_id/figs/robot_val.tex
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+(176.0, -0.4881786226847886)
+(177.0, -0.504793897644223)
+(178.0, -0.37607868293011476)
+(179.0, -0.44265135407749046)
+(180.0, -0.3256337554222763)
+(181.0, -0.2434783848052363)
+(182.0, -0.16048316965740023)
+(183.0, -0.0535728748128485)
+(184.0, -0.08376184278162085)
+(185.0, 0.03276789588947382)
+(186.0, 0.19667388293619287)
+(187.0, 0.23313051998729345)
+(188.0, 0.2888401925740635)
+(189.0, 0.2710298241026224)
+(190.0, 0.2653751582306831)
+(191.0, 0.36716581321485464)
+(192.0, 0.508160579354618)
+(193.0, 0.6172010019590102)
+(194.0, 0.751840230851124)
+(195.0, 0.6897624946589895)
+(196.0, 0.6802511267640267)
+(197.0, 0.6171927821498235)
+(198.0, 0.6839852503196648)
+(199.0, 0.7326319951443234)
+(200.0, 0.6749949042970869)
};
\end{axis}
diff --git a/jump_lin_id/id_paper.tex b/jump_lin_id/id_paper.tex
index 01a6325dbd4495d035cd50b1521ed103066f7de5..793ce2fccd75f7acea3a8ff4c59d7e5a462fca0a 100644
--- a/jump_lin_id/id_paper.tex
+++ b/jump_lin_id/id_paper.tex
@@ -161,13 +161,13 @@ Identification of LTV Dynamical Models with\\ Smooth or Discontinuous Time Evolu
% ---------------------------------------------------------
\section{Introduction}\label{sec:introduction}
-The difficulty of the task of identifying time-varying dynamical models of systems varies greatly with the model considered and the availability of measurements of the state sequence. For smoothly changing, linear dynamics,\footnote{Linear dynamics in the sense of models that are linear in the parameters.} the recursive least-squares algorithm with exponential forgetting (RLS$\lambda$) is a common option. If a Gaussian random-walk model for the parameters is assumed, a Kalman filtering/smoothing algorithm \cite{rauch1965maximum} gives the filtering/smoothing densities of the parameters in closed form. The assumption of smoothly (Gaussian) varying dynamics is often restrictive. Discontinuous dynamics changes occur, for instance, when an external controller changes operation mode, when a sudden contact between a robot and its environment is established, an unmodeled disturbance enters the system or when a system is suddenly damaged.
+The difficulty of the task of identifying time-varying dynamical models of systems varies greatly with the model considered and the availability of measurements of the state sequence. For smoothly changing dynamics, linear in the parameters, the recursive least-squares algorithm with exponential forgetting (RLS$\lambda$) is a common option. If a Gaussian random-walk model for the parameters is assumed, a Kalman filtering/smoothing algorithm \cite{rauch1965maximum} gives the filtering/smoothing densities of the parameters in closed form. The assumption of smoothly (Gaussian) varying dynamics is often restrictive. Discontinuous dynamics changes occur, for instance, when an external controller changes operation mode, when a sudden contact between a robot and its environment is established, an unmodeled disturbance enters the system or when a system is suddenly damaged.
Identification of systems with non-smooth dynamics evolution has been studied extensively. The book~\cite{costa2006discrete} treats the case where the dynamics are known, but the state sequence unknown, i.e., state estimation. In~\cite{nagarajaiah2004time}, the authors examine the residuals from an initial constant dynamics fit to determine regions in time where improved fit is needed by the introduction of additional constant dynamics models. Results on identifiability and observability in jump-linear systems in the non-controlled (autonomous) setting are available in~\cite{vidal2002observability}. The main result on identifiability in \cite{vidal2002observability} was a rank condition on a Hankel matrix constructed from the collected output data, similar to classical results on the least-squares identification of ARX-models which appears as rank constraints on the, typically Toeplitz or block-Toeplitz, regressor matrix. Identifiability of the methods proposed in this article are discussed in~\cref{sec:identifiability}.
-An LTV model can be seen as a first order approximation of the dynamics of a nonlinear system around a trajectory. We emphasize that such an approximation will in general fail to generalize far from the this trajectory, but many methods in reinforcement learning and control make efficient use of the linearized dynamics for optimization, while ensuring validity of the approximation by constraints or penalty terms. An example provided towards the end of this work highlights such a method.
+An LTV model can be seen as a first order approximation of the dynamics of a nonlinear system around a trajectory. We emphasize that such an approximation will in general fail to generalize far from the this trajectory, but many methods in reinforcement learning and control make efficient use of the linearized dynamics for optimization, while ensuring validity of the approximation by constraints or penalty terms. An example provided in \cref{sec:rl} highlights such a method.
-An important class of identification methods that have been popularized lately are \emph{trend filtering} methods~\cite{kim2009ell_1, tibshirani2014adaptive}. Trend filtering methods work by specifying a \emph{fitness criterion} that determines the goodness of fit, as well as a \emph{regularization} term, often chosen with sparsity promoting qualities. As a simple example, consider the reconstruction $\hat y$ of a noisy signal $y = \left\{y_t\inspace{}\right\}_{t=1}^T$ with piecewise constant segments. To this end, we might formulate and solve the convex optimization problem
+An important class of identification methods that has been popularized lately is \emph{trend filtering} methods~\cite{kim2009ell_1, tibshirani2014adaptive}. Trend filtering methods work by specifying a \emph{fitness criterion} that determines the goodness of fit, as well as a \emph{regularization} term, often chosen with sparsity promoting qualities. As a simple example, consider the reconstruction $\hat y$ of a noisy signal $y = \left\{y_t\inspace{}\right\}_{t=1}^T$ with piecewise constant segments. To this end, we might formulate and solve the convex optimization problem
\begin{equation} \label{eq:tf}
\minimize{\hat{y}} \normt{y-\hat{y}}^2 + \lambda\sum_t |\hat{y}_{t+1} - \hat{y}_t|
\end{equation}
@@ -205,7 +205,7 @@ where $\otimes$ denotes the Kronecker product and $K=n^2+nm$ is the number of mo
\section{Time-varying dynamics}
-We now move on to the contribution of this work, and extend out view to systems where the dynamics change with time. We limit the scope of this article to models on the form
+We now move on to the contribution of this work, and extend our view to systems where the dynamics change with time. We limit the scope of this article to models on the form
\begin{equation}
\label{eq:tvk}
\begin{split}
@@ -231,7 +231,7 @@ The following sections will introduce a number of optimization problems with dif
A slowly varying signal is characterized by \emph{small first-order time differences}.
To identify slowly varying dynamics parameters, we thus penalize the squared 2-norm of the first-order time difference of the model parameters, and solve the optimization problem
\begin{equation} \label{eq:slow}
- \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda\sum_t \normt{\w_{t+1} - \w_{t}}^2
+ \minimize{\w} \normt{\y-\hat{\y}}^2 + \lambda^2\sum_t \normt{\w_{t+1} - \w_{t}}^2
\end{equation}
where $\sum_t$ denotes the sum over relevant indices $t$, in this case $t\in [1,T-1]$.
This optimization problem has a closed form solution given by
@@ -430,12 +430,19 @@ the effective number of free parameters increases until the
problem gets ill-posed for $\lambda = 0$.
We formalize the above arguments as
\begin{proposition}
- Optimization problems \labelcref{eq:slow,eq:smooth,eq:pwconstant,eq:pwlinear} have unique global minima for $\lambda > 0$ if and only if the corresponding LTI optimization problem has a unique solution.
+ Optimization problems \labelcref{eq:slow,eq:pwconstant} have unique global minima for $\lambda > 0$ if and only if the corresponding LTI optimization problem has a unique solution.
\end{proposition}
\begin{proof}
- The cost function is a sum of two convex terms. For the global minimum to be non-unique, the Hessians of the two terms must share nullspace.
+ The cost function is a sum of two convex terms. For a global minimum to be non-unique, the Hessians of the two terms must share nullspace.
In the limit $\lambda \rightarrow \infty$ the problem reduces to the LTI problem. The nullspace of the regularization Hessian, which is invariant to $\lambda$, does thus not share any directions with the nullspace of $\tilde{\A}\T \tilde{\A}$ which establishes the equivalence of identifiability between the LTI problem and the LTV problems.
\end{proof}
+\begin{proposition}
+ Optimization problems \labelcref{eq:smooth,eq:pwlinear} have unique global minima for $\lambda > 0$ if and only if the corresponding LTI optimization problem has a unique solution and the Jacobian of the generating system is bounded along parts of the trajectory.
+\end{proposition}
+\begin{proof}
+ Again, the cost function is a sum of two convex terms and for a global minimum to be non-unique, the Hessians of the two terms must share nullspace.
+ In the limit $\lambda \rightarrow \infty$ the regularization term reduces to a linear constraint set, allowing only parameter vectors that lie along a line through time. Let $v \neq 0$ be such a line and let $L$ be an upper bound on the Frobenius norm of the Jacobian. $v \in \operatorname{null}{(\tilde{\A}\T \tilde{\A})}$ implies $\exists \alpha : \norm{k+\alpha v} > L$ which establishes the equivalence of identifiability between the LTI problem and the LTV problems.
+\end{proof}
For the LTI problem to be well-posed, the system must be identifiable and the input $u$ must be persistently exciting of sufficient order~\cite{johansson1993system}.
@@ -520,14 +527,14 @@ The input was Gaussian noise of zero mean and unit variance, state transition no
\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.
-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.
+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:pwconstant}. 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.}
+ \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. For numerical stability, all time-series are normalized to zero mean and unit variance, hence, the velocity zero crossing is explicitly marked with a dashed line. The control signal plot clearly indicates the discontinuity in torque around the unnormalized zero crossing of $\dot{q}_2$.}
\label{fig:robot_train}
\end{figure*}
\begin{figure*}[htp]
@@ -536,10 +543,10 @@ The state of the robot arm consists of two joint coordinates, $q$, and their tim
\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}
+ \label{fig:robot_val}
\end{figure*}
-\section{Example -- Reinforcement learning}
+\section{Example -- Reinforcement learning} \label{sec:rl}
\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
@@ -644,7 +651,7 @@ This allows incorporation of different scales for different variables with littl
\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.
+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.
diff --git a/jump_lin_id/two_link.jl b/jump_lin_id/two_link.jl
index 77022fd363f361539af3d99aa59b847b7b9ff341..3137c90877471df28639de6e992a2059f964f61e 100644
--- a/jump_lin_id/two_link.jl
+++ b/jump_lin_id/two_link.jl
@@ -14,6 +14,7 @@ if !isdefined(:AbstractEnsembleSystem)
@everywhere include("../flux/nn_prior/linear_sys.jl")
end
+##
n,m,N,h,σ0 = 4,3,200,0.02,0.0
np = n*(n+m)
sys = TwoLinkSys(N=N, h=h, σ0 = σ0)#; warn("Added noise")
@@ -25,38 +26,40 @@ q = inverse_kin(traj[1],:up)
p = forward_kin(q)
qd = centraldiff(q')'/sys.h
qdd = centraldiff(qd')'/sys.h
-u = torque(q,qd,qdd)
-u .+= 0.2randn(size(u))
+u0 = torque(q,qd,qdd)
+u0 .+= 0.1randn(size(u0))
# x,y,u = generate_data(sys, System, seed; ufun=x->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!)
-t = h:h:N*h+h
+t = 0:h:N*h
q0 = [q[:,1]; qd[:,1]]
-prob = OrdinaryDiffEq.ODEProblem((x,p,t)->time_derivative(x, u[:,ceil(Int,t/h)]),q0,(t[[1,end]]...))
+prob = OrdinaryDiffEq.ODEProblem((x,p,t)->time_derivative(x, u0[:,floor(Int,t/h)+1]),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
+u = u0[:,1:N]
+u = [u; ones(1,size(u,2))]
x,y = xy(System,x,N)
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
@@ -68,17 +71,19 @@ Jtrue = reshape(Jtrue, 24, sys.N)
colors = [HSV(h,1,0.8) for h in linspace(0,200,5)]
-sλ = linspace(-5,5,5)
+sλ = linspace(-3,2,16)
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)
+ model = LTVModels.fit_statespace_admm(x,u,10^λ, extend=false, tol=1e-8, iters=20000, D=2)
end
-kalmanvec = pmap(IterTools.product(sr2,sp0)) do t
+kalmanvec = map(IterTools.product(sr2,sp0)) do t
r2,p0 = t
global x,u,n,m
R1 = eye(n*(n+m))
@@ -94,13 +99,13 @@ ui = @manipulate for λ in sλ, r2 in sr2, p0 in sp0, 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)
+ Jadmm = flatten(admmvec[indad...])
+ Jkalman = flatten(kalmanvec[indk...])
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!(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
@@ -145,7 +150,7 @@ 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(x', label="States", show=false, layout=(3,1), subplot=1, size=(800,1000), title="State trajectories", lab=["\$q_1\$" "\$q_2\$" "\$\\dot{q}_1\$" "\$\\dot{q}_2\$"], 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))
@@ -161,36 +166,37 @@ pgfplots(grid=false, legend=:bottom)
##
-p0 = 1.
-R1 = eye(np)
-R2 = 10^(1.)*eye(n)
-P0 = 10^p0*R1
+p0 = 10.
+R1 = 10^(0.)*eye(np)
+R2 = 10^(-0.)*eye(n)
+P0 = 10^p0*eye(np)
sx = std(x,2)
su = std(u[1:2,:])
mx, mu = mean(x,2), mean(u[1:2,:],2)
xt = (x.-mx) ./ sx
ut = copy(u)
-ut[1:2,:] = (u[1:2,:].-mu) ./ su
+ut[1:2,:] = (ut[1:2,:].-mu) ./ su
-Pt = cat(3,[diagm([fill(10^4,n^2);fill(10^10,n*m)]) for i=1:N]...)
+Pt = cat(3,[diagm([fill(10^10,n^2);fill(10^10.,n*m)]) for i=1:N]...)
model = KalmanModel(zeros(n,n,N),zeros(n,m,N),zeros(1,1,N),false)
prior = KalmanModel(zeros(n,n,N),zeros(n,m,N),Pt,false)
model = LTVModels.fit_model!(model,prior, xt,ut,R1,R2,P0, extend=true)
-
+plot(.√[maximum(eigvals(Symmetric(model.Pt[:,:,i])) )for i = 1:N])
+model = LTVModels.fit_statespace_admm(xt,ut,10^-0.5, extend=false, tol=1e-8, iters=20000, D=1, zeroinit=false)
signch = findfirst(u[2,:] .< 0)
contact = findfirst(forward_kin(x[1:2,:])[2,:] .> wall)
ind2 = length(sp0)
-fig=plot(xt', label="States", show=false, layout=grid(3,2), subplot=1:4, size=(1800,1000), title="State trajectories", title=["q1" "q2" "qd1" "qd2"], c=:blue, linewidth=4)
+fig=plot(xt', label="States", show=false, layout=grid(3,2), subplot=1:4, size=(1800,1000), title="State trajectories", title=["\$q_1\$" "\$q_2\$" "\$\\dot{q}_1\$" "\$\\dot{q}_2\$"], c=:blue, linewidth=2)
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"])
+plot!(forward_kin(x[1:2,:])', subplot=5, title="End effector positions", lab=["x" "y"], c=[:blue :red])
hline!([wall], l=(:dash, :black), grid=false, subplot=5, lab="Constraint")
vline!([contact], subplot=5, lab="Stiff contact", l=(:black, :dash))
-plot!(ut[1:2,:]', subplot=6, title="Control torques", lab="")
+plot!(ut[1:2,:]', subplot=6, title="Control torques", lab="", c=[:blue :red])
vline!([signch], subplot=6, lab="Velocity sign change", l=(:black, :dash))
-plot!(LTVModels.simulate(model, xt[:,1], ut)', label="Simulation", show=false, subplot=1:4, l=(:red,), legend=[false false true false])
+plot!(LTVModels.simulate(model, xt[:,1], ut)', label="Simulation", show=false, subplot=1:4, l=(:red,), legend=[false false :left false])
gui()
# savefig2(@__DIR__()*"/figs/robot_train.tex", Dict("" => "legend style={at={(1.2,-0.2)}, anchor=north, legend columns=4},"))
##
@@ -199,26 +205,26 @@ gui()
##
# @views model.At[:,:,:] .*= 0.99
-uv = torque(q,qd,qdd)
-uv .+= 0.0randn(size(uv))
-t = h:h:N*h+h
-q0 = [q[:,1]; qd[:,1]]
-prob = OrdinaryDiffEq.ODEProblem((x,p,t)->time_derivative(x, uv[:,ceil(Int,t/h)]),q0,(t[[1,end]]...))
+uv = 1.0torque(q,qd,qdd)
+uv .+= 0.1randn(size(uv))
+prob = OrdinaryDiffEq.ODEProblem((x,p,t)->time_derivative(x, uv[:,floor(Int,t/h)+1]),q0,(t[[1,end]]...))
sol = solve(prob,Tsit5(),reltol=1e-8,abstol=1e-8, callback=cb)
xv = hcat(sol(t)...)
uv = uv[:,1:N]
uv = [uv; ones(1,size(uv,2))] # Extend to estimate friction
xv,yv = xy(System,xv,N)
-xt = (xv.-mx) ./ sx
-ut = copy(uv)
-ut[1:2,:] = (uv[1:2,:].-mu) ./ su
-plot(xt', label="States", show=false, layout=grid(2,2), subplot=1:4, size=(1800,1000), title="State trajectories", title=["q1" "q2" "qd1" "qd2"], c=:blue, linewidth=4, reuse=false)
+
+xt2 = (xv.-mx) ./ sx
+ut2 = copy(uv)
+ut2[1:2,:] = (uv[1:2,:].-mu) ./ su
+
+plot(xt2', label="States", show=false, layout=grid(2,2), subplot=1:4, size=(1800,1000), title="State trajectories", title=["\$q_1\$" "\$q_2\$" "\$\\dot{q}_1\$" "\$\\dot{q}_2\$"], c=:blue, linewidth=2, reuse=false, grid=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))
-plot!(LTVModels.simulate(model, xt[:,1], ut)', label="Simulation", show=false, subplot=1:4, l=(:red,), legend=[false false true false])
+plot!(LTVModels.simulate(model, xt2[:,1], ut2)', label="Simulation", show=false, subplot=1:4, l=(:red,), legend=[false false :left false])
# plot!(ut', subplot=5:6)
gui()