# Time-Symmetric Boundary Conditions and Quantum Foundations

Ken Wharton  Department of Physics and Astronomy, San José State University,
San José, CA 95192-0106

###### Abstract

Despite the widely-held premise that initial boundary conditions (BCs)
corresponding to measurements/interactions can fully specify a physical
subsystem, a literal reading of Hamilton’s principle would imply that both
initial and final BCs are required (or more generally, a BC on a closed
hypersurface in spacetime). Such a time-symmetric perspective of BCs, as
applied to classical fields, leads to interesting parallels with quantum
theory. This paper will map out some of the consequences of this counter-
intuitive premise, as applied to covariant classical fields. The most notable
result is the contextuality of fields constrained in this manner, naturally
bypassing the usual arguments against so-called “realistic” interpretations of
quantum phenomena.

##  I Introduction

Without measurement theory, theoretical physics would be a purely mathematical
endeavor; it is the link between mathematical expressions and physical events
that provides a testable connection between equations and reality. But despite
the temporal symmetries evident in our most fundamental physics equations,
such symmetries are rarely applied to measurement theory itself. Given a
physical subsystem described by some equation, it is usually assumed that to
find the solution to the equation one must impose initial boundary conditions
(IBCs), but not final boundary conditions (FBCs). This intuitive assumption
leads to an asymmetric map between physical measurements and mathematical
boundary conditions (BCs), where the map from the initial measurement to the
IBC is treated differently than the map from the equation solution to the
final measurement. This asymmetry is especially evident in the realm of
quantum theory.

It is notable that outside of measurement theory there exists a well-known,
time-symmetrical application of mathematical boundary conditions: the use of
Hamilton’s principle to construct equations of motion from a Lagrangian
Lanczos . In this case, one mathematically fixes both IBCs and FBCs (in
coordinate space), and then determines the equations of motion for which the
action is extremized. Despite this symmetry, when using the resulting Euler-
Lagrange equations to describe physical systems, one typically reverts to the
usual procedure of imposing only IBCs (determined via
measurements/interactions), ignoring the FBCs that generated the equations in
the first place. Presumably, in the minds of many physicists, there is an
important distinction between mathematically imposing an FBC for the purpose
of Hamilton’s principle and actually imposing an FBC via a physical
measurement. However, there seems to be little distinction between these two
where IBCs are concerned.

The purpose of this paper is to motivate and explore the consequences of
treating measurement-BCs in the same symmetrical manner as they are treated in
Hamilton’s principle. External measurements (both before and after the
subsystem in question) will be treated as physical constraints, imposed on the
subsystem in exactly the same way that boundaries are imposed when using
Hamilton’s principle. This interpretation of measurements as BCs on the prior
system (as opposed to merely revealing the status of events determined by
IBCs) is counter-intuitive, but there is further motivation for such a
framework (as seen in the next section). The subsequent analysis indicates
that classical fields constrained by FBCs exhibit many of the characteristics
of quantum systems. These results point to a natural resolution of several
foundational issues in quantum mechanics, presented in a classical context.

##  II Motivating FBCs

Variational principles notwithstanding, most physicists have a strong
intuition that any given subsystem in the universe should be determined by
IBCs alone. Such an intuition is bolstered by the time-asymmetric second law
of thermodynamics, which of course can only be derived using time-asymmetric
assumptions. The most natural such assumption is that of correlated IBCs (and
no corresponding FBCs), from which coarse-grained entropy can be statistically
shown to increase Schulman . Still, far from “proving” that nature is
fundamentally time-asymmetric, all this demonstrates is that our observable
universe happens to have an IBC which is highly correlated on a coarse-grained
level (the low-entropy Big Bang).

Furthermore, when one considers single- or few-particle phenomena, it is not
obvious that lessons from statistical regimes should apply in this more
fundamental context. Indeed, if the quantum level of description is considered
to be fundamental (fine-grained, with no underlying level on which statistics
can be applied), then one would not expect any entropy-like asymmetry to arise
from the use of time-symmetric equations. As Eddington pointed out long ago,
“When [Dirac’s theory] is applied to four particles alone in the universe, the
analysis very properly brings out the fact that in such a system there could
be no steady one way direction of time, and vagaries would occur which are
guarded against in our actual universe . . . of about $10^{79}$
particles.”Eddington  Recovering a distinction between IBCs and FBCs on this
fundamental level therefore requires some new, time-asymmetric addition to the
equations (as Eddington also advocated). But time-symmetry is an important
principle in physics, so violating it in any way should not be done lightly –
particularly when the phenomena being described have no apparent time-
asymmetry to explain.

Although time-symmetry and Hamilton’s principle are two reasons for
considering FBCs on the same footing as IBCs, another compelling reason is the
experimental fact of the Heisenberg uncertainty principle. In the context that
Heisenberg first envisioned UP , this is not necessarily an indication of some
difference between quantum and classical systems, but rather a strict limit on
how accurately one can measure any system. Simply put, this principle makes
IBCs based on actual initial measurements inherently insufficient to describe
the evolution of classical systems. Without such a restriction, imposing FBCs
would be trivial: together the IBCs and FBCs would overconstrain the system,
and FBCs would not “determine” anything at all; they would merely reveal what
was already deducible from the IBCs. But there is, in fact, no experimental
system with sufficient IBCs to make FBCs redundant. Viewed in this light, the
uncertainty principle seems to discourage total reliance on IBCs (although
quantum theory has resisted this conclusion, as discussed in the next
section).

The uncertainty principle does not merely imply that IBCs contain insufficient
data; it also generally indicates that IBCs will be insufficient by a factor
of two (in full concordance with both time-symmetry and Hamilton’s principle).
In the simplest possible case, a single particle in a known potential requires
an initial position and an initial velocity; the uncertainty principle allows
exactly half of this information to be measured and imposed as an IBC. More
generally, quantum measurement theory for a single spinless particle permits
measurements that constrain the initial values of some complex scalar function
$\psi(\bm{x},t_{0})$ (to within a global phase). Such an IBC is sufficient to
solve the Schrödinger equation (SE), as the SE is a first-order differential
equation in time. But the corresponding classical system, a complex scalar
field $\phi(\bm{x},t)$, is governed by the second-order-in-time Klein-Gordon
equation Goldstein , and requires double the initial data:
$\phi(\bm{x},t_{0})$ and $\dot{\phi}(\bm{x},t_{0})$. Again, one is limited to
half of the IBCs needed for a solution to the corresponding classical problem
Wharton .

Even more generally, the IBCs and FBCs can together be treated as a single
closed-hypersurface boundary condition in spacetime, as is generally done in
covariant extensions of Hamilton’s principle. In this case, while one can no
longer pick out a well-defined “IBC” (portions of the boundaries can be time-
like, and therefore neither IBCs nor FBCs), the same half-data restriction
still seems to generally hold locally on the boundary surface. For example,
the time-like boundary of a perfect conductor in classical electromagnetism
can constrain 3 of the 6 components of the E and B fields on the boundary, but
leaves the other 3 components fully unconstrained. Further evidence that
nature has provided us with systems underconstrained by their IBCs comes from
the “$\psi$-epistemic” view of quantum states espoused by Spekkens Spekkens .
In a simple toy theory, Spekkens demonstrates that restricting ones initial
knowledge of a classical system (by exactly a factor of two!) leads to a vast
array of consequences normally associated with quantum phenomena, including
interference, quantum teleportation, and an apparent “collapse” upon
measurement. The primary quantum phenomena not recoverable from such a picture
are Bell-inequality violations.

But even this exception implies that one should seriously consider FBCs,
something Spekkens’s model does not do. In his model, while the known IBCs are
insufficient, the unconstrained parameters are assumed to be determined by
some additional, random IBC, yielding a well-defined probability distribution
for future measurements of the corresponding unknown parameters. This, of
course, is the type of “hidden variable” that was considered by Bell in the
proof of his inequality Bell , so it is no surprise that Spekkens’s toy theory
is also constrained in a similar manner. Both Spekkens and Bell assumed that
any future measurement on the system would merely reveal what was already
determined by (known or unknown) IBCs.

And yet experiments tell us that nature does not respect Bell’s inequality,
leading many physicists to consider dramatic changes to our best models of
spacetime, introducing non-locality and/or preferred reference frames. An
alternate approach, much less popular but still scattered throughout the
literature OCB ; CWR ; Cramer ; Sutherland1 ; Price ; Miller ; Wharton1 , is
to permit hidden variables that are somehow correlated with the future
measurement settings, in which case violation of Bell-type inequalities
becomes trivial. The question of how such retrocausal correlations might arise
is not always specified, but the above analysis should point to an obvious
option: perhaps the future measurement is not merely revealing information,
but actually a constraint on the prior development of the system. Systems
constrained by FBCs are naturally “contextual”, in that their internal
parameters depend upon that future measurement/interaction. One expects Bell-
inequality violations in such a system. Perhaps the strongest reason to
consider FBCs is that it might provide a way to explain such violations in a
way compatible with the local nature of spacetime assumed by general
relativity.

##  III The Case for Classical Fields

The previous sections motivate an equivalence between external measurements on
a spacetime subsystem and the mathematical boundary conditions on the
underlying equations that describe that system. Note that any connected region
of spacetime only has one boundary on which external measurements can be
imposed; if this region is effectively infinite in spatial extent, then the
boundary is often considered to be formed by two spacelike surfaces. So for
any given region, three or more complete external measurements on the same
subsystem are not logically possible. (Measurement devices take up a spacetime
volume themselves.) This approach assumes that measurements made by physical
devices are both constraints on spacetime-adjacent subsystems as well as being
part of larger subsystems that are themselves constrained by other
measurements/BCs. In this view the ultimate boundary conditions are
cosmological.

Given this framework where all measurements are BCs on some subsystem, there
are three general categories of systems/equations that one might consider;
classical particles, classical fields, and quantum fields. Previous
applications of FBCs to physical systems have primarily focused on classical
particles Schulman  and quantum fields Schulman ; Miller1 ; AV ; Oeckl .
(Sutherland recently pursued this idea in the context of Bohmian quantum
mechanics, spanning both classical particles and quantum fields Sutherland .)
Apart from my own research program (as well as recent work by Dolce Dolce ),
there seems to be no one pursuing the application of FBCs onto classical
fields.

Classical fields may seem like a curious middle ground between the other two
options, more abstract than particles but not as foundational as quantum
systems. Still, there is a case to be made that classical fields constrained
by closed hypersurface BCs might be an alternative to quantum systems
constrained by only IBCs. First I will address the particle/field issue, and
then tackle the classical/quantum question.

The strongest case for considering fields over particles is to look at our
most successful fundamental physical theories, general relativity (GR) and
quantum field theory (QFT). GR is most naturally expressed in terms of fields
(the Einstein equation is a field equation, derivable from a continuous
Lagrangian density), and although it might be argued that this beautiful
formalism is only an approximation of underlying discrete events, efforts to
discretize spacetime itself have so far been unsuccessful. QFT has even
clearer field-based underpinnings, and analysis of inequivalent Fock-space
representations in accelerating reference frames has demonstrated that
“particle number” is not a generally covariant concept NoPs . This strongly
implies that particles are not even “real”, let alone fundamental.

Apart from these appeals to GR and QFT, it is notable that various efforts to
describe quantum phenomena using a particle ontology (deBroglie-Bohm-style
interpretations, stochastic electrodynamics, etc.) all are forced to introduce
fields into the ontology as well as particles. This is because it is extremely
difficult to extract wave-like behavior from particles alone; conversely, it
is trivial to describe a highly-localized field, at least at some particular
instant. For all of these reasons, it seems evident that fields should at
least be given equal status to particles when asking foundational questions,
if not strict priority.

When searching for answers to foundational questions, one may be tempted to
not consider classical systems at all. After all, it is widely known that
classical physics fails in many instances, apparently forcing us into a
quantum framework. But there is a key flaw in this argument. What is known to
fail are classical systems constrained by IBCs alone. Such systems cannot
possibly violate Bell-type inequalities, for example, so this implies is that
some change must be made to the “classical + IBC-only” framework. Modern
physics has already made such a change, jumping to the “quantum + IBC-only”
framework that we have found so difficult to reconcile with GR. While the
above arguments may tempt one to go straight to a “quantum + IBC + FBC”
framework (as is being pursured by Oeckl and others Oeckl ), such a path would
bypass the mostly unexplored “classical + IBC + FBC” framework. If this latter
framework could address the experiments that motivate the classical $\to$
quantum transition, then there would be no need to ever leave the classical
framework to begin with. This would provide a possibility to explain the
origin of some of the curious postulates of quantum theory, rather than just
assuming them from the outset.

Indeed, the traditional classical $\to$ quantum transition has effectively
bypassed some of the arguments for FBCs, making the addition of FBCs to the
quantum framework an attempt to solve the same problem in two different ways.
The uncertainty principle, for example, is no longer seen as statement of
unknown parameters that might be supplied by an FBC, but typically interpreted
to mean that the system can still be perfectly well-defined using only half of
the classical parameters. Quantum theory accomplishes this halving of the
parameter space by dropping from classical second-order (in time) partial
differential equations to the first-order equations described by the SE and
its natural relativistic extensions (e.g. the Wheeler-deWitt equation). It is
notable that this “halving” introduces severe problems in curved space-time,
where there is no covariant way to separate out the underlying second-order
equations into two first-order equations DeWitt . Viewed in this light, the
entire structure of quantum theory is built around the premise that one should
be able to do physics with IBCs only; adding FBCs to this framework is
therefore quite unnatural.

This conclusion is evident in the results of the few researchers who have
explored the “quantum + IBC + FBC” framework. Schulman has found that imposing
both IBCs and FBCs to the SE is a strong overconstraint, and has had to resort
to finding approximate solutions that are necessarily inexact Schulman .
Aharanov and Vaidman’s two-state formalism AV  has overcome this problem by
re-doubling the size of the quantum state to permit the addition of FBCs; a
similar treatment can be found in other two-time boundary approaches
Sutherland ; Wharton1 ; Miller1 . (Miller’s approach Miller1  is arguably more
similar to Schulman’s, in that it resorts to non-unitary evolution to get from
the IBC to the FBC.) I have argued Wharton  that it is far more natural to
start with a second-order partial differential equation (like the Klein-Gordon
equation) and apply both IBCs and FBCs rather than to halve-and-then-double
the parameter space to accomplish effectively the same thing – especially
given that this “halving” procedure fails in curved spacetime DeWitt .

In this perspective, the “classical field + IBC + FBC” framework should be
viewed as a competitor to the “quantum + IBC” framework; both are attempting
to deal with limited IBCs in a completely different manner. While it is
certainly reasonable to have doubts that such a classical framework might
explain the same experiments as quantum theory, it remains a fact that this
framework has been largely unexplored. The below analysis of the first-order
consequences of the “classical field + IBC + FBC” framework does not encounter
any major problems, and even supports its general plausibility. Section 5 then
discusses how many of the difficult interpretational questions of quantum
theory would vanish under such a classical alternative.

##  IV First-Order Consequences: Quantization and Contextuality

For those unfamiliar with two-time boundary problems, the simplest and best
analogy is that of two-spatial boundary problems. Such systems are widely
familiar in contexts ranging from high-Q laser cavities to a quantum particle
in an infinite square well. In both of these examples, the enclosed field is
found to have a quantized wavelength (and therefore a quantized wavenumber)
$k_{n}=n\pi/L$, where $L$ is the distance between the boundaries. Notice that
quantization is a natural consequence of two-boundary problems, even for
classical systems (such as electromagnetic fields in laser cavities).

Applying this same two-boundary logic to a field subject to both an IBC (at
time $t=t_{0}$) and an FBC (at time $t=t_{f}$), leads to a similar
quantization of the “temporal wavelength”, aka the period. The corresponding
“wavenumber” in this case is just the angular frequency $\omega$, which should
therefore be quantized in the same manner:

$\omega_{n}=\frac{n\pi}{\Delta t}$ (1)

where $\Delta t=t_{f}-t_{0}$. This is a novel “frequency quantization” which
would seem to necessarily occur given both an IBC and FBC.

Sequential measurements of fields does not usually reveal the quantization
from Eqn. (1), and this may seem to be reason to reject this approach. But
consider that for most systems it is impossible to externally constrain
$\Delta t$ in the same way that one can externally constrain $L$. In fact, the
only experimental set-up where this is generally possible is that of a laser
cavity, where the length is directly related to the time between interactions
via $L=c\Delta t$. In this special case, the above “temporal quantization”
leads to $\omega_{n}=n\pi c/L$, which indeed is the quantization that is
actually observed in this instance. Of course, one traditionally comes to this
conclusion by using the relation between the frequency and the quantized
wavenumber $\omega_{n}=ck_{n}$, so this known result is consistent with the
use of FBCs, but not dependent upon it.

For fields that do not propagate at some known speed, it is quite difficult to
physically constrain $\Delta t$ to the accuracy where one might see the
effects of (1). For example, consider a classical scalar field $\phi$ governed
by the generally covariant Lagrangian density

${\cal{L}}=\frac{1}{2}\left(g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi-\frac{m^{2}c^{2}}{\hbar^{2}}\phi^{2}\right)$
(2)

Here $\nabla$ is the covariant 4-derivative, summation is implied, and $g$ is
the metric. The parameters $m$ and $\hbar$ are chosen such that the
corresponding Euler-Lagrange equation in flat Minkowski spacetime

$\left(\frac{\partial^{2}}{\partial
t^{2}}-c^{2}\nabla^{2}+\frac{m^{2}c^{4}}{\hbar^{2}}\right)\phi=0$ (3)

has the dispersion relation
$\omega(\bm{k})=\sqrt{c^{2}k^{2}+m^{2}c^{4}/\hbar^{2}}$. This matches the
energy-momentum relationship for a relativistic particle with mass $m$,
subject to the deBroglie relations $E=\hbar\omega$, $\bm{p}=\hbar\bm{k}$.
Equation (3) is the Klein-Gordon equation (KGE), and of course is a second-
order partial differential equation (in time).

Comparing the dispersion relation to (1), one finds

$\frac{n\pi}{\Delta t}=\omega_{n}\geq\frac{mc^{2}}{\hbar}$ (4) $\Delta t\leq
n\pi\frac{\hbar}{mc^{2}}$ (5)

From this one can see see that this quantization requires the time between the
IBC and FBC to be constrained to an accuracy less than the Compton period
$h/(mc^{2})$. As this value is less than $10^{-20}$ seconds for electrons, and
even smaller for more massive particles, it is unsurprising that the frequency
quantization implied by (1) is not observed (outside of laser cavities).

However, our current inability to make measurements at well-defined sub-
attosectond intervals does not mean that this framework has no consequences.
For example, consider two consecutive measurements made on this scalar field
with the additional constraint of a 1D infinite-potential square well of
length L. Now there are two, independent, quantization conditions: the usual
wavenumber quantization $k=n_{x}\pi/L$ and the new frequency quantization
$\omega=n_{t}\pi/\Delta t$. Inserting these expressions into the dispersion
relation yields the constraint

$n_{t}^{2}L^{2}+n_{x}^{2}c^{2}(\Delta
t)^{2}=\frac{m^{2}c^{4}}{\pi^{2}\hbar^{2}}L^{2}(\Delta t)^{2}$ (6)

The fact that $n_{x}$ and $n_{t}$ are both integers is now a strong constraint
on $\Delta t$ (given a well-defined value for L). Most values of $\Delta t$
will have no solution at all, and for those values that do have solutions, the
vast majority will now have only one integer pair ($n_{x}$,$n_{t}$) that
satisfies (6). Therefore, given two consecutive measurements on a classical
field in a 1D cavity, one expects to typically find well-defined single values
of $k$ and $\omega$; not the superposition of modes that one sees in a laser
cavity where $n_{x}=n_{t}$. In other words, particle-like measurement results
have emerged from simply constraining classical fields with FBCs. (Note that
if no final measurement was made, there would have been no such constraint,
and a generic superposition of solutions to the KGE would still be permitted.)
It seems plausible that this general result would continue hold for any
potential well, not merely the infinite square well.

Another interesting result arises from the previous example: The time between
measurements $\Delta t$ is no longer a completely arbitrary external
parameter. There are values of $\Delta t$ with no acceptable solution, and
therefore measurements may not occur at certain times. This is consistent with
the probabilistic treatment of measurement time in Wharton , but is certainly
contrary to how most physicists think about this parameter. The use of FBCs
has now dictated which hypersurfaces may correspond to physical measurements,
and which ones may not. Taking this logic to its conclusion, one finds that
such constraints on classical field measurements (as dictated by global
solutions to an action-extremization problem) naturally leads to exactly the
sort of quantization observed in experiments, including the quantization of
angular momentum in units of $\hbar/2$ Wharton3 .

This relates to a technical issue concerning the boundaries in covariant
formulations of Hamilton’s principle. Action-extremization only generally
leads to the Euler-Lagrange equations if the boundaries constrain the value of
the field in coordinate space Lanczos ; Wharton3 . If these boundaries are now
to be interpreted as physical measurements, one encounters a problem:
constraints on boundaries in coordinate space necessarily correspond to
measurements of time-even quantities (parameters that are identical under
time-reversal). For example, the $j$-momentum density of a classical scalar
field, as defined by the stress-energy tensor component $T^{0j}$, is
proportional to $\dot{\phi}\,\partial\phi/\partial x^{j}$, a time-odd
quantity. Such a measurement cannot be imposed as a BC on $\phi$ in coordinate
space, because $\dot{\phi}$ is independent from $\phi$ on any space-like
hypersurface. If one assumed complete experimental control over when and where
measurements could be made, this would imply that Hamilton’s principle would
fail any time a time-odd quantity was measured.

Fortunately, the analysis in Wharton3  demonstrates that there is an
interesting “loophole” that can save Hamilton’s principle for time-odd
measurements. It turns out that boundary constraints on $\dot{\phi}$ can still
permit an extremized action (leading to the same Euler-Lagrange equations),
provided that the hypersurfaces corresponding to those boundaries have a
particular geometrical structure. (It is the constrained geometry of the
hypersurfaces that leads to angular momentum quantization in Wharton3 .) As
above, this implies that our assumed freedom of when and where we can make
measurements may be illusory, and indeed this lack of freedom seems to lead to
an apparent measurement-induced quantization.

But the most important consequence of the above example is simply that the
global solution to the equations in a space-time subsystem depend upon the
next measurement on that subsystem. Even if we cannot control $\Delta t$ on a
Compton period time-scale, we can certainly control it at much grosser levels,
and in Eqn (6) such external control will lead to different values of $n_{t}$
throughout the spacetime volume bounded by the two sequential measurements. In
other words, if we are to treat $\phi$ as a “real” field, our future
measurement decisions at $t\approx t_{f}$ will affect the reality of the field
at times $t<<t_{f}$. Such influence is usually termed “retrocausality”.

Remarkably, the retrocausality that emerges from such a picture is of the
least objectionable sort, as it only affects the so-called “inaccessible past”
Price , hidden parameters that cannot be determined until their “future cause”
at $t_{f}$. (If one was to make a measurement between $t_{0}$ and $t_{f}$,
then that intermediate measurement would still be at $t_{f}$ by definition.)
Therefore no paradoxes can be constructed. Furthermore, these are precisely
the sort of counter-intutive hidden parameters needed to violate Bell’s
theorem in a natural way. Such contextuality is not an added, ad hoc feature
of this framework, but an inevitable result of treating IBCs and FBCs on the
same conceptual footing.

##  V Lessons for Quantum Foundations

Quantum measurement theory typically comes with a built-in temporal asymmetry:
complete initial measurements always correspond to a pure-state IBC, while
complete final measurements are often made on mixed states and are not imposed
as FBCs. The (generalized) Born rule that gives the conditional probabilities
of a final measurement outcome (conditioned upon the IBC and subsequent time
evolution) is typically used in a time-asymmetric manner, in the sense that it
is always conditioning on the past. Although one might choose to condition on
the future instead of the past, and use the Born rule in reverse, one would
find that the FBC becomes a pure state and the intermediate quantum state is
no longer the same as it was in the past-conditioned case. The Born rule
therefore seems inconsistent with Hamilton’s Principle, as the latter imposes
both IBCs and FBCs on the same system in a time-neutral manner.

In both the forward- and backward- application of Born’s rule, despite the
different intermediate mathematics, the joint probability between the past and
future outcomes remains unchanged ABL ; it is this joint probability that
therefore appears more fundamental than conditional probabilities when
approaching such time-symmetric problems. For a complex classical field
constrained by both an IBC and a FBC, I have shown that a invariant joint
probability distribution can be constructed from the solution space of the
intermediate field (given the boundary constraints) Wharton . From this one
can construct the usual conditional probabilities in the appropriate limit,
conditioning on any chosen portion of the boundary. In this perspective,
probability is not some mysterious attribute of quantum systems, but instead
has the same source as it does in classical physics: ignorance. Knowing the
IBCs is not enough to define the fields, and therefore one retreats to a
probabilistic framework. Once the IBCs and FBCs are known, however, one can
“retrodict” the field values in the intermediate region between the two
boundaries if desired.

The so-called “quantum collapse” disappears in such a framework, in the same
manner as envisioned by the “$\psi$-epistemic” proponents Spekkens . In this
classical-field context, the solution to the SE ($\psi$) is merely our best
guess of what the field looks like given only the IBC, but does not perfectly
correspond to the actual field $\phi$ (which obeys the KGE, constrained by
IBCs and FBCs). Once the final measurement result (the FBC) becomes known, one
updates ones knowledge of $\psi$ in a discontinuous manner. Still, there is no
discontinuity in reality, for the field $\phi$ is perfectly consistent with
this final measurement result (as it is pre-constrained by the eventual FBC).

Another difference between $\psi$ and $\phi$ lies in the multi-particle
sector. To encode the probabilities of all possible measurements in a single
function $\psi$, one is forced to expand the dimensionality of $\psi$ into
multi-particle configuration space. But in this framework the actual
probabilities are not encoded by $\phi$ or $\psi$ themselves, but rather the
solution space of $\phi$ given any particular future measurement set-up. There
is no need for $\phi$ to encode the results of possible measurements which
will not actually occur, so there is no need to expand $\phi$ into
configuration space, even for multiple particles. (Montina’s recent work makes
a related argument Montina .) The best example of this is the classical
Maxwell field, which manages to reproduce infinite-photon-number quantum
theory using fields that still exist in physical spacetime (as opposed to
infinite-dimensional configuration space).

The most important lesson from the above analysis is probably the following
statement: Violations of Bell inequalities do not necessarily imply the
failure of locality (in the sense used in general relativity). Although
locality is one assumption that underlies Bell’s theorem, it is not the only
one; the theorem also assumes that any hidden variables will not be dependent
on the future measurement settings Bell . It is natural to violate such an
assumption in this framework – indeed, one expects it to be violated, given a
true FBC. Experimentally-observed violations of Bell-type inequalities are
therefore further motivation for considering FBCs, especially if one wants to
retain basic notions of locality.

It is notable that this approach maps nicely to quantum field theory, which
also uses an invariant expression for the joint probability of two consecutive
measurements. For a quantum scalar field $\phi$ (promoted to an operator, with
imposed commutation constraints), this joint probability is

$P[\phi(t_{0}),\phi(t_{f})]=\left|\int{\cal{D}}\phi\,e^{iS[\phi]/\hbar}\right|^{2}$
(7)

Here the functional integral is over all field configurations consistent with
the IBC $\phi(t_{0})$ and the final measurement result $\phi(t_{f})$; $S$ is
the classical action. It is notable that this mathematics naturally
incorporates FBCs. This expression can be made even more general by imposing
the boundaries on a closed hypersurface rather than two parallel “instants”.
Note that there is nothing, in principle, that would prevent the evaluation of
(7) for a purely classical field constrained by IBCs and FBCs. In this
perspective, the dominant contribution from quantum theory is on the boundary
itself – the limitation of the allowed measurement values on a given field.

But the classical framework presented here provides another way to constrain
those measurements, without the use of quantum measurement theory (or
operators of any sort). If there is some pair of measurement results
$\phi(t_{0})$ and $\phi(t_{f})$ with no allowable solutions at all (such as
certain values of $\Delta t$ and $L$ in (6)), then the integral in (7) will
simply be zero, and those pairs of measurements would therefore have zero
probability of occurring together. Furthermore, if there were pairs of
possible measurements for which there was no way to extremize the classical
action $S$ (such as angular momentum measurements that were not near a
multiple of $\hbar/2$ Wharton3 ), then the value of the integral in (7) would
be quite small due to the lack of stationary phase terms. The net effect would
be an apparent quantization of an underlying continuous system – it would only
be quantized when one “looks”, because it is the externally imposed
observation/measurement that is the source of the quantization constraints.
Indeed, seeming paradoxes like “delayed-choice” experiments become trivial to
interpret in this framework, as it is the future measurements themselves which
are defining the evolution of the system.

The preliminary results presented here (and in related research Wharton ;
Wharton3 ) are certainly nowhere close to being a replacement for all of
quantum theory. But there is every indication that this is a promising
research avenue to pursue. The only obvious conceptual disadvantage is the
retrocausal aspect of FBCs, and even that might be considered an advantage if
one is concerned about maintaining a temporal symmetry in theories that
purport to describe time-symmetric events. Indeed, taking Eqn. (7) seriously
can even be used as a strong case for retrocausation EPW ; WMP . Furthermore,
the framework advocated here is probably more compatible with general
relativity than is quantum field theory, as classical fields in curved
spacetime do not come with the conceptual problems of quantum fields. It is
even possible that, instead of quantizing spacetime, an approach to quantum
gravity might be found in the continuous ontology of classical field theory.

For now, though, more modest efforts are called for: the construction of a
time-neutral theory of field-field measurement, explicitly incorporating
potentials/interactions, and searching for experimental tests of this overall
framework. Two-time boundaries imposed on classical fields is a subject that
has been curiously neglected for the past hundred years. Based on these
preliminary results, it seems reasonable to venture further down this
particular path.

##  VI Acknowledgements

The author is grateful to Rob Spekkens and the Perimeter Institute for the
invitation to give the seminar on which this paper was based PItalk . Further
thanks goes to Huw Price for conceptual inspiration and the Eddington quote.

## References

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#  Vision-based Control of a Soft Robot for Maskless Head and Neck Cancer
Radiotherapy

Olalekan P. Ogunmolu1, Xuejun Gu2, Steve Jiang2, and Nicholas R. Gans1
1Olalekan P. Ogunmolu and Nicholas R. Gans are with the Department of
Electrical Engineering, University of Texas at Dallas, Richardson, TX 75080,
USA {olalekan.ogunmolu<EMAIL_ADDRESS>Gu and Steve Jiang are with
the Department of Radiation Oncology, University of Texas Southwestern Medical
Center, Dallas TX 75390, USA {Xuejun.Gu<EMAIL_ADDRESS>
###### Abstract

This work presents an on-going investigation of the control of a pneumatic
soft-robot actuator addressing accurate patient positioning systems in
maskless head and neck cancer radiotherapy. We employ two RGB-D sensors in a
sensor fusion scheme to better estimate a patient’s head pitch motion. A
system identification prediction error model is used to obtain a linear time
invariant state space model. We then use the model to design a linear
quadratic Gaussian feedback controller to manipulate the patient head position
based on sensed head pitch motion. Experiments demonstrate the success of our
approach.

##  I Introduction

This paper presents a continuation of our investigation of an image-guided
soft robot patient positioning system for use in head and neck (H&N) cancer
radiotherapy (RT). In 2014, over 1.6 million patients developed pharynx and
oral cavity cancers in the United States, which led to over 580,000 deaths
[1]. Typical H&N cancer treatment involves intensity-modulated radiotherapy
(IMRT), which delivers high potent dose to tumors while simultaneously
minimizing dose to adjacent critical organs such as spinal cord, parotids
glands, and optical nerves. Typically, a patient lies on a 6-DOF movable
treatment couch, and laser or image-guidance systems are used to ensure the
patient is in the proper position.

IMRT requires accurate patient positioning. An examination of patient
displacement and beam angle misalignment during IMRT showed errors as small as
3-mm caused 38% decrease in minimum target dose or 41% increase in the maximum
spinal cord dose [2]. Image-guided radiotherapy (IGRT) has improved IMRT
accuracy while reducing set-up times [3, 4, 5]. However, current IGRT
practices focus on using images acquired before treatment to confirm beam
placement [6]. The discomfort caused by head masks in prolonged IMRT treatment
can increase patients’ voluntary and involuntary motion. Studies show that
translational errors caused by patient motion can be larger than 6mm, and
rotational errors can be as high as 2∘ [7]. Current motion-tracking systems,
such as Cyberknife and Novalis are not compatible with conventional linear
particle accelerators used at the majority of cancer centers. Moreover, these
two systems are limited to assuming the patient’s body is rigid during motion
tracking and compensation. Recently, a robotic real-time surface image-guided
positioning system was studied for feasibility in frameless and maskless
cranial stereotactic radiosurgery [8]. While it achieved similar accuracy as
the existing clinical methods, the system may not be suitable to IMRT due to
the presence of mechanical and electrical parts in the path of the radiation
beam.

Soft robot systems are deformable polymer enclosures with fluid-filled
chambers that enable manipulation and locomotion tasks by a proportional
control of the amount of fluid in the chamber [9, 10]. Their customizable,
deformable nature and compliance make them suitable to biomedical applications
as opposed to rigid and stiff mechanical robot components. They can also be
made radiotransparent, which is necessary in IMRT.

The long term goal of our work is to address the non-rigid motion compensation
during H&N RT. As we continue our initial investigation, we control one degree
of freedom, raising or lowering of a generic patient’s head, lying in a supine
position, to a desired height above a table. The current system consists of a
single inflatable air bladder (IAB), a mannequin head and a neck/torso motion
simulator, two different Kinect RGB-D cameras to measure patient position, two
current-controlled pneumatic valve actuators, and a National Instruments myRIO
microcontroller. In this work, we extended and improve our previous work [11].
This paper contributes better vision tracking and localization methods via
filtering and fusion of the two RGB-D estimates. We improve on the system
identification of the soft-robot system and now incorporate an optimal control
network. The result is a much improved motion control.

Section II of this paper briefly presents the design of the soft robot system.
Section III discusses the computer vision algorithms to detect the patient’s
face and fusion of measurements from the RB-D images. Section IV presents
results of system identification for the soft-robot system. Section V presents
design of the linear quadratic Gaussian (LQG) controller, and Section VI
presents several experiments to demonstrate the system.

##  II Soft Robot Design Overview

The soft robot actuation mechanism combines an IAB (19” x 12”) made of
lightweight, durable and deformable polyester and PVC, two current-controlled
proportional solenoid valves, and a pair of silicone rubber tubes (attached to
a T-port connector at the orifice of the IAB) in order to convey air in/out of
the IAB. A 1HP air compressor supplied regulated air at 30 psi to the inlet
actuating valve, while an interconnection of a 60W micro-diaphragm pump and a
valve removed air from the outlet terminal of the IAB. The RGB-D sensors are
mounted directly above the head for raw head position and velocity
measurements, while local Kalman filters (KFs) provide two estimates of the
head position and velocity. The sensor estimates are aggregated using a track-
to-track KF-based sensor fusion algorithm. We apply the fusion result in a new
robust control law for the pneumatic actuator valves, thereby regulating air
pressure within the IAB and moving the patient’s head as desired. The real-
time controller was deployed on a National Instruments myRIO embedded system
running LabVIEW 2015. The LabVIEW algorithms were processed within a Windows 7
virtualbox running on the Ubuntu host workstation.

We use a Kinect Xbox 360, and a Kinect for Windows v2 sensor to estimate head
position and velocity. The two sensors use different electronic perception
technologies to determine distance of an object from the camera origin. They
therefore have different lateral and range resolutions as well as different
noise characteristics. Image processing for both cameras is executed on a 22GB
RAM mobile workstation with Intel Core i7-4800MQ processor running 64-bit
Ubuntu Trusty on a Linux 4.04 kernel.

##  III Image-Based Patient Position Estimation

We perform recursive filter estimations of the RGB-D measurements and improve
position estimates by using an additional sensor to better localize tracked
features.

Figure 1: Noise floor of Kinect Xbox Sensor vs. Kinect v2 Sensor

We add the Kinect v2 sensor (henceforth called the v2 sensor), based on the
time-of-flight (ToF) electronic perception principle. In ToF, light pulses
illuminate a scene, and depth is calculated by determining the phase shift of
the returned light signals. The active infra-red reduces the dependence on
ambient lighting [12], and this sensor has a higher spatial depth resolution
of 512$\times$424 pixels at 30Hz interactive rate, compared to the Xbox’s 320
$\times$ 240 pixels [13]. To minimize the noise due to the limited sensor
resolution, the v2 has in-built noise improvement capabilities [14].

The v2 provides a higher depth-map accuracy and lower noise floor compared to
Kinect for Xbox, as can be seen from Fig. 1, where the v2 exhibits a noise
auto-covariance of 11.4707$mm^{2}$ compared with 22.7057$mm^{2}$ for the Xbox.
Despite the improved performance of the v2, noise remains an issue, as is the
case for every electronic perception system. To alleviate this, we employ a
multisensor data fusion of both Kinect sensors’ observations. We achieved this
by local Kalman Filter estimates of each sensor’s observations, and we fuse
the estimates via a variance-weighted multisensor Kalman filter fusion scheme
described later in this section.

###  III-A Face Detection and eye-feature tracking

We approached face detection using Haar Cascade Classifiers (HCC) [15]. HCC’s
are based on integral image representation, which allow for features
evaluation while maintaining high detection rates. The features resemble Haar
basis functions. A classifier is formed by choosing a small number of crucial
features with AdaBoost, and a weighted sum of individual classifiers is used
to construct a strong object detection classifier in a cascade manner. This
increases the detector’s speed by concentrating on areas within an image with
high probability of features of interest.

Figure 2: Original colored image retrieved from the Kinect v2 Sensor.

A drawback of HCC’s is the memory consumed on computing devices when searching
through image pixels for specific regions of interest. Searching through a 640
$\times$ 480 pixels gray-scale image for specific features caused a 90%
reduction in the frame rates of either sensor, when the algorithm is run on a
CPU. To overcome this, both sensor’s images were spatially down-sampled via
linear interpolation before HCCs were applied. Face detection was performed on
a single NVIDIA Quadro K1100M GPU. We retrieve each detected face from the
GPU, and then detect eyes within detected faces using the same procedure.

To achieve robust detection, the minimum number of neighbors in each candidate
rectangle feature was determined based on our experience. The search area
within an image was chosen to be within the range of (5 $\times$ 5) pixels and
(20 $\times$ 20) pixels. This gave us more than 90% face detection rate for
both sensors. A similar approach was used for the eye classifier. The final
implementation achieved a frame rate of 15Hz for each sensor running
independently on the Linux host computer. Further improvement in frame rates
is an avenue for future work.

###  III-B Local Kalman Filters

From Fig. 1, we see that both RGB-D sensors suffer from notable associated
noise, which is not suitable for our control requirement. To refine the
observation, local Kalman Filter (KF) estimates for each sensor were computed
to determine state estimates $\hat{\textbf{x}}(i)$ that minimizes the mean-
squared error to the true state $\textbf{x}(i)$, given a measurement sequence
$z(1),\cdots,z(j)$, that is

$\displaystyle\hat{\textbf{x}}(i|j)$ $\displaystyle=\text{arg
}\min_{\hat{\textbf{x}}(i|j)\in\mathbb{R}^{n}}\mathbb{E}\\{(\textbf{x}(i)-\hat{\textbf{x}})(\textbf{x}(i)-\hat{\textbf{x}})|z(1),\cdots,z(j)\\}$
$\displaystyle\triangleq\mathbb{E}\\{\textbf{x}(i)|{z}(1),\cdots,{z}(j)\\}\triangleq\mathbb{E}\\{\textbf{x}(i)|{Z}^{j}\\}$
(1)

where the obtained estimate is the expected value of the state at time $i$
given observations up to time $j$. The covariance of the estimation error is
given by

$\displaystyle\textbf{P}(i|j)\triangleq\mathbb{E}\\{(\textbf{x}(i)-\hat{\textbf{x}}(i|j)(\textbf{x}(i)-\hat{\textbf{x}}(i|j)^{T}|Z^{j}\\}.$
(2)

Assuming the model of the state is common to both sensors, and denoting the
distance from the v2 to the head as $d(k)$, we define
$\textbf{x}(k)=[d(k),\,\dot{d}(k)]^{T}\in\mathbb{R}^{2}$ as the state vector
of interest, and let $\Delta T$ be the time between steps $k-1$ and $k$. The
model state update equations are given by

$\textbf{x}_{k}=\textbf{F}_{k}\textbf{x}_{k-1}+\textbf{B}_{k}\textbf{u}_{k}+\textbf{G}_{k}\textbf{w}_{k}$
(3)

where $\textbf{F}(k)\in\mathbb{R}^{2\times 2}$ is the state transition matrix
given by

$\textbf{F}=\begin{bmatrix}1&\Delta T\\\ 0&1\end{bmatrix}$ (4)

$\textbf{u}(k)\in\mathbb{R}^{2}$ is the control input, $\textbf{B}(k)$ is the
control input matrix that maps inputs to system states,
$\textbf{G}(k)\in\mathbb{R}^{2\times 2}$ process noise matrix, and
$\textbf{w}(k)\in\mathbb{R}^{2}$ is a random variable that models the state
uncertainty. In the absence of inputs $\textbf{B}_{k}\textbf{u}_{k}=0$, and
the model becomes

$\displaystyle\textbf{x}_{k}=\textbf{F}_{k}\textbf{x}_{k-1}+\textbf{G}_{k}\textbf{w}_{k}$
(5)

where $\textbf{w}_{k}$ is the effect of an unknown input and $\textbf{G}_{k}$
applies that effect to the state vector, $\textbf{x}_{k}$. The process noise
is assumed unknown and is modeled as uncontrolled forces causing an
acceleration $a_{k}$ in the head position ($a_{k}$ is thus a scalar random
variable with normal distribution, zero mean and standard deviation
$\sigma_{a}$). We model this into (3) by setting $\textbf{G}_{k}$ to identity
and set $\textbf{w}(k)\sim\mathcal{N}(0,\textbf{Q}(k))$ where the covariance
matrix $\textbf{Q}(k)$ is set to a random walk sequence defined by
$\textbf{W}_{k}={[\frac{{\Delta T}^{2}}{2},\Delta T]}^{T}$ . Therefore, we
find that

Q
$\displaystyle=\textbf{W}\textbf{W}^{T}{\sigma_{a}}^{2}=\begin{bmatrix}\dfrac{{\Delta
T}^{4}}{4}&\dfrac{{\Delta T}^{3}}{2}\\\ \dfrac{{\Delta T}^{3}}{2}&{\Delta
T}^{2}\end{bmatrix}{\sigma_{a}}^{2}.$ (6)

Denoting the head displacement at time $k$ as measured by the Xbox and v2 as
${z}_{1}(k)$ and ${z}_{2}(k)$ respectively, the sensors’ measurements were
mapped to the v2 reference frame and modeled as

${z}_{s}=\textbf{H}_{s}(k)\textbf{x}(k)+{v}_{s}(k)\qquad\qquad s=1,2$ (7)

where $\textbf{H}_{s}(k)={\begin{bmatrix}1&0\end{bmatrix}}^{T}$ maps the
system’s state space into the observed space, and ${v}_{s}(k)\in\mathbb{R}$ is
a random variable that models the sensor error. We define ${v}_{s}(k)$ as a
normally distributed random variable with zero mean and variance
$\sigma_{rs}^{2}$. We assume the random sequences ${v}_{1}(k)$,${v}_{2}(k)$,
$\textbf{w}(k$) are independent and uncorrelated in time.

At each time step, $k$, each local KF’s priori and posteriori estimates are
computed through the typical prediction and update phases

Prediction Phase:

$\displaystyle\hat{\textbf{x}}_{k|k-1}$
$\displaystyle=\textbf{F}\hat{\textbf{x}}_{k-1|k-1}+\textbf{B}_{k}\textbf{u}_{k}$
$\displaystyle\textbf{P}_{k|k-1}$
$\displaystyle=\textbf{F}_{k}\textbf{P}_{k-1|k-1}{\textbf{F}_{k}}^{T}+\textbf{Q}_{k}$
(8)

where $\hat{\textbf{x}}_{k|k-1}$ and $\textbf{P}_{k|k-1}$ are the state
prediction vector and the prediction covariance matrix respectively.

Update Phase:

$\displaystyle\textbf{K}_{k}$
$\displaystyle=\textbf{P}_{k|k-1}{\textbf{H}_{k}}^{T}{[\textbf{H}_{k}\textbf{P}_{k|k-1}{\textbf{H}_{k}}^{T}+\textbf{R}_{k}]}^{-1}$
$\displaystyle\hat{\textbf{x}}_{k|k}$
$\displaystyle=\hat{\textbf{x}}_{k|k-1}+\textbf{K}_{k}(\textbf{z}_{k}-\textbf{H}_{k}\hat{\textbf{x}}_{k|k-1})$
$\displaystyle\textbf{P}_{k|k}$
$\displaystyle=(\textbf{I}-\textbf{K}_{k}\textbf{H}_{k})\textbf{P}_{k|k-1}$
(9)

where $\textbf{K}_{k}$, $\hat{\textbf{x}}_{k|k}$, and $\textbf{P}_{k|k}$ are
respectively the KF gain, posteriori state estimate and its state covariance
matrix. In implementing the KF of (III-B) and (III-B), the variance of the
process noise/signal noise of each local KF was informed by our knowledge of
the physics of both sensors (electronic perception methods, range resolutions
and examining each sensor’s depth map to understand the data available to the
filter), engineering judgment, and kinematics of the process model. We found
these values sufficiently modeled the underlying process dynamics

$\displaystyle\sigma_{a}$
$\displaystyle=2000mm^{2};\qquad{\sigma_{r1}}^{2}=70mm^{2}\text{for the Xbox,
and}$ $\displaystyle{\sigma_{r2}}^{2}$ $\displaystyle=60mm^{2}\qquad\text{for
the Kinect v2 sensor}.$

Figure 3:  KF results for the Xbox observation Figure 4:  KF results of Kinect
v2’s observation

Figs. 3 and 4 show the local filter estimate results of the observation from
both the Kinect Xbox and v2 sensors post-filtering. The noise floor becomes
noticeably reduced by each sensor after the KF filtering. The steady-state
performance of both sensors include a reduction in the variance of the
observation sequence by 80.81%, while the Kinect v2 shows an improvement in
noise rejection by almost 60% .

###  III-C Data Fusion

Each local KF estimate was combined at a central fusion site to obtain a
track-to-track fused global estimate. To communicate each estimate and
associated covariance matrix, we create Unix FIFO special files (i.e. named
pipes) on the kernel file system, write the estimates and covariance matrices
to the pipes at each local site and retrieve the values at the central site.

Named pipes are low-level file I/O systems that can be shared by processes
with different ancestry. During data exchange through a FIFO, the kernel
forwards all data internally without having to write it to the file system.
Since they exist within the kernel and the file system is just an entry
serving as a reference point for the processes to access the pipe with a file
system name, there is practically no delay in data communication.

Local tracks are generated at each sensor site according to (III-B), resulting
in two local state predictions from the Kalman filters (3). At the central
fusion site, we assume a state model common to both sensors given by (III-B)
and adopt a variance-weighted average of each local track in the global track
fusion algorithm [16]

$\displaystyle\hat{\textbf{x}}_{F}(k|k)$
$\displaystyle=\textbf{P}_{F}(k|k)\sum\limits_{i=1}^{N}\left[{\textbf{P}_{s}}^{-1}(k|k)\hat{\textbf{x}}_{s}(k|k)\right]$
$\displaystyle\text{where }\textbf{P}_{F}(k|k)$
$\displaystyle=\left[\sum\limits_{i=1}^{N}{\textbf{P}_{s}}^{-1}(k|k)\right]^{-1}.$
(10)

Fig. 5 shows the output of the fusion scheme compared against the single
Kalman filters during a head-raising motion. The fusion of the local tracks
produces better estimates, with improved signal to noise ratio. The fused
estimate assigns more weight to the less noisy signal from Kinect v2. Through
the implementation of the local tracks and a global track KF estimator, we
improved the accuracy of the effective signal to be used in our control
algorithm to no more than a standard deviation of 0.75mm from the true
position of an object. The noise spikes in the fused tracks when the process
state estimates are yet to converge as noticeable in Fig. 5 can be attributed
to the noisy initialization of pixels in the sensors before they attain their
steady state values. On average, it takes approximately 30 seconds for the
pixel values in the Kinect sensor to reach their final steady state values
[17]. This can be avoided by running the fusion algorithm for at least 2
minutes before the fused signal is used for any control purposes.

Figure 5: Kalman filter Track-to-Track fusion of Kinect Xbox and v2’s local
tracks

The code for the multisensor fusion experiment is available on the git repos
[18] and [19].

##  IV Soft Robot System Identification

We approach the modeling procedure with an identification prediction error
(PEM) approach, where we estimate a mathematical model, $G(t)$, based on the
minimization of the sum of squared errors between estimates of the head
height, $\hat{y(t)}$, and true head height, $y(t)$, from the fusion _i.e._

$G(t)=\text{arg }\min_{\theta}\,{V_{N}(\theta,Z^{N})}\\\ $
$\displaystyle\text{ where
}V_{N}(\theta,Z^{N})=\sum_{k=1}^{\mathcal{K}}\sum_{i=1}^{n}\dfrac{1}{2}(\hat{y_{i}}(k)-y_{i}(k))^{2}.$
(11)

$Z^{N}=\\{u(1)\cdots u(N)\quad y(1)\cdots y(N)\\}$ is the vector of past input
and output (fused estimates) measurements over a bounded interval $[1,N]$ and
$\theta$ is the greedy vector of parameters that approximate the model we seek
to build. (11) is a special case of the least squares criterion.

###  IV-A Model Structure

Following Ljung’s formulation in [20, §4.5], we pose the identification
problem as determining the “best model” from a set of candidate model sets via
an iterative approach that parametrizes the noncountable model sets smoothly
over an area with the assumption that the underlying system is linear time-
invariant. Here, our model structure is a differentiable mapping from a
connected, compact subset $\mathcal{D}_{\mathfrak{M}}$ of $\mathcal{R}^{d}$ to
a model set $\mathfrak{M}^{*}$, such that the gradients of the predictor
functions are stable. This procedure is included in the MATLAB system
identification toolbox, and since the method is well-documented in [21] we
omit details.

External disturbances and stochastic variables are modeled as additive white
noise sequence, $e(k)$, based on lagged inputs and outputs, and our objective
is to estimate a stochastic state space model structure of the form

$\displaystyle\mathbf{x}(k+1)=\mathbf{Ax}(k)+\mathbf{Bu}(k)+\mathbf{w}(k)$
$\displaystyle\mathbf{y}(k)=\mathbf{Cx}(k)+\mathbf{Du}(k)+\mathbf{v}(k)$ (12)

where the noise terms $\mathbf{w}(k)\text{ and }\mathbf{v}(k)$ compensate for
the effect of disturbances beyond frequencies of interest to system dynamics
and make the model robust to model uncertainties. Since $u$ and $y$ alone are
measurable in our setup, the states $\mathbf{x}(k)$ are estimated and (IV-A)
becomes a linear regression problem, where all the unknown matrix entries are
linear combinations of the measured inputs and output variables. This can be
written as

$\displaystyle Y(k)=\Theta\Phi(k)+E(k)$ (13)

where

$Y(k)=\left[\begin{array}[]{c}\mathbf{x}(k+1)\\\
\mathbf{y}(k)\end{array}\right],\hskip
3.00003pt\Theta=\left[\begin{array}[]{cc}\mathbf{A}&\mathbf{B}\\\
\mathbf{C}&\mathbf{D}\end{array}\right]$

$\Phi(k)=\left[\begin{array}[]{c}\mathbf{x}(k)\\\
\mathbf{u}(k)\end{array}\right]\hskip 1.99997pt\text{ and
}{E}(k)=\left[\begin{array}[]{c}\mathbb{E}(w(k))\\\
\mathbb{E}(v(k))\end{array}\right].$

We assume the noise term is white in order to assure an unbiased model. The
parameter estimation problem is then to estimate the
$\mathbf{A},\mathbf{B},\mathbf{C},\text{ and }\mathbf{D}$ matrices by the
linear least squares regression of (13) assuming no physical insight into the
system (i.e. a black box model). $\mathbb{E}(\mathbf{w}(k))$ and
$\mathbb{E}(\mathbf{v}(k))$ are estimated as a sampled sum of squared errors
of the residuals.

###  IV-B Parameter Estimation

The input, $u(k)$, and output signals, $y(k)$, can be characterized by a
linear difference equation of the form

$\displaystyle y(k)$ $\displaystyle=-a_{1}y(k-1)-\cdots-a_{n_{a}}y(k-n_{a})$
(14) $\displaystyle-b_{1}u(k-1)-\cdots-b_{n_{b}}u(k-n_{b})-e(k)$
$\displaystyle-c_{1}e(k-1)-c_{n_{c}}e(k-n_{c})$

where $e(k)$ describes the equation error as a moving average of white noise,
and we assume $e(k)$ has a bias-variance term $\lambda$. We can rearrange (14)
using the vectors

$\displaystyle\begin{split}\psi(k,\theta)&=[-y(k-1)\cdots-y(k-n_{a})\,\,u(k-1)\cdots\\\
&\qquad u(k-n_{b}),\,\,e(k-1,\theta),\cdots,e(k-n_{c},\theta)]^{T}\end{split}$
(15)
$\displaystyle\theta=[-a_{1},\cdots,-a_{n_{a}},-b_{1},\cdots,-b_{n_{b}},-c_{1},\cdots,-c_{n_{c}}].$
(16)

The adjustable parameters of (15) are elements of $\theta$. In our prediction
model, it is convenient to write (14) as a one-step-ahead predictor of the
form

$\displaystyle\hat{y}(k)$
$\displaystyle=G(q,\theta)u(k)+H(q,\theta)\hat{e}(k)$ (17)
$\displaystyle\text{with}\quad G(q,\theta)$
$\displaystyle=\dfrac{B(q)}{A(q)},\text{ }H(q,\theta)=\dfrac{C(q)}{A(q)}$

which is a complete autoregressive moving average with exogenous input (ARMAX)
model. $G(q,\theta)$ represents the transfer function from input to output
predictions, and $H(q,\theta)$ denotes the transfer function of prediction
errors to the output model, $\hat{y}(k)$; $q$ is the z-transform, $z^{-1}$,
while $A(q)$, $B(q)$, and $C(q)$ are polynomials defined as

$\displaystyle A(q)$ $\displaystyle=1+a_{1}q^{-1}+\cdots+a_{n_{a}}q^{-n_{a}},$
$\displaystyle B(q)$ $\displaystyle=b_{1}q^{-1}+\cdots+b_{n_{b}}q^{n_{b}},$
$\displaystyle C(q)$ $\displaystyle=1+c_{1}q^{-1}+\cdots+c_{n_{c}}q^{-n_{c}}$
(18)

[22]. The predictor turns out to be a linear filter of the form

$\displaystyle\hat{y}(k|\theta)=W_{y}(q,\theta)y(k)+W_{u}(q,\theta)u(k)$ (19)
$\displaystyle\text{and }y(k)=G(q,\theta)u(k)+H(q,\theta)[y(k)-\hat{y}(k)]$
(20)

where $H(q,\theta)$ is the noise model and $\hat{y}(k)$ above can be regarded
as the one-step ahead predictor. After rearranging (19),we find that

$W_{y}=1-H^{-1}(q,\theta)\\\ \text{ and
}W_{u}(q,\theta)=G(q,\theta)H^{-1}(q,\theta)$

such that the residual errors from (19) become

$\displaystyle e(k)=[y(k)-G(q,\theta)u(k)]H^{-1}(q,\theta).$ (21)

We can consider (21) as passing the prediction errors through a linear filter
that allows extra freedom in dealing with non-momentary properties of the
prediction errors. Since the model is that of a linear system, (21) satisfies
our objective by approximating the prefilter with the choice of the noise
model in (13).

The estimation problem is to predict the estimates, $\hat{y}(k|\theta)$ so
that the errors, $\varepsilon(t,\theta)=\parallel
y(t)-\hat{y}(t|\theta)\parallel_{p}$ are minimized by the choice of an
appropriate p-norm criterion function, such as the mean squared error proposed
in (11).

####  IV-B1 Input Signal Design

The input signal choice for a system identification experiment will determine
a system’s operating point and model accuracy. Therefore, the input should be
rich enough to excite a system and force it to show properties needed for the
model’s purpose. For the model to be informative across all the desired
frequency range, a periodic, persistently exciting uniform Gaussian White
noise (UGWN) signal with clipped amplitudes corresponding to the bandwidth of
the valves was designed offline, and its frequency spectrum analyzed to ensure
it had as small a crest factor as possible (since the asymptotic properties of
the model will be mostly influenced by the spectrum rather than the waveform’s
time-series shape). Gaussian White Noise signals (GWN) and Pseudo-Random
Binary Signals (PRBS) are well-known to achieve virtually any signal spectrum
without very narrow pass bands. Therefore, pseudo-random uniform white noise
sequences were generated using the very-long cycle random number generator
algorithm. Given that the probability density function, $f(x)$, of the
uniformly distributed uniform white noise is

$\displaystyle f(x)$ $\displaystyle=\frac{1}{2}A\qquad\text{if $x<|A|$ and}$
$\displaystyle u(x)$ $\displaystyle=0\qquad\text{ if $x>|A|$ }$ (22)

where $A$ is the amplitude. The expected mean, $\mu$, and the expected
standard deviation, $\sigma$ of the sequence are [23]

$\displaystyle\mu=\mathbb{E}(x)=0\text{,}\qquad\sigma=\left[\mathbb{E}\\{(x-\mu)\\}\right]^{1\over
2}=\frac{A}{\sqrt{3}}.$ (23)

Figure 6: Time/Frequency-Domain Properties of the Input Signal

The spectrum of the resulting signal in Fig. 6 gives good signal power, which
nicely relates to the bandwidth of the pneumatic valves and achieves virtually
all signal spectrum with little narrow pass bands.

We therefore use the signal of (IV-B1) to model the desired asymptotic
estimates, $\hat{y}(t)$, of (17). We sampled $y(t)$, the fused measurement
described in (III-C) well-above the system’s Nyquist frequencyand acquired
enough samples to make $Z^{N}$ asymptotically approach $\hat{\theta}_{N}$ as
$N\rightarrow\infty$. The data collection procedure closely follows that
described in our previous paper and we refer readers to [11, §IV.A] for a more
detailed treatment.

The collected data was separated in a 60:40% ratio for training and testing
purposes, respectively, to assure a training model that generalizes well.

####  IV-B2 State Space Realization

If we define

$\hat{Y}_{r}(k)=[\hat{y}(k|k-1),\cdots,\hat{y}(k+r-1)|k-1]^{T}$
$\hat{Y}=[\hat{Y}_{r}(1)\cdots\hat{Y}(N)],$

it follows that 1) as $N\rightarrow\infty$, there are $n$-th order minimal
state space descriptions of the system if and only if the rank of the matrix
of prediction vectors, $\hat{Y}$, is equal to $n$ for all $r\geq n$; and 2)
the state vector of any minimal realization in innovations can be chosen as
linear combinations of $\hat{Y}_{r}$ that form a row basis for $\hat{Y}$,
i.e.,

$x(t)=L\hat{Y}_{r}(k)$

with $L$ being an $n\times pr$ matrix ($p$ is the dimension of $y(k)$) [20,
§7.3]. The true prediction is given by (17) with innovations $e(j)$ written as
a linear combination of past input-output data. The predictor can thus be
expressed as a linear function of $u(i),\,y(i),\,i\leq k-1$. In practice, the
predictor is approximated so that it depends on a finite amount past data such
as $s_{1}$ past outputs and $s_{2}$ past inputs of the form

$\displaystyle\hat{y}(k|k-1)$
$\displaystyle=\alpha_{1}y(k-1)+\cdots+\alpha_{s_{1}}y(k-s_{1})$ (24)
$\displaystyle+\beta_{1}u(k-1)+\cdots+\beta_{s_{2}}u(k-s_{2}).$

Piping the identification data through the MATLAB function ‘ssest’ and testing
various model orders based on the ranking of singular values of the Hankel
matrix of input-output measurements [20], we obtained the results listed in
Table I on training and testing dataset. The MATLAB system identification
script is provided on a github repo [24] and contains the dataset used for the
experiment.

TABLE I: Model estimates Data Type | Expts | MO111Model Order | MSE222Mean Squared Error ($mm^{2}$). | Fit (% ) |  FPE 333Akaike Final Prediction Error ([20, Secs 7.4 and 16.4]).
---|---|---|---|---|---
Training | i | 2 | 0.001437 | 97.64 | 0.001438
| ii | 4 | 0.001454 | 97.62 | 0.00145584
| iii | 6 | 0.001333 | 97.72 | 0.001336
| iv | 8 | 0.001298 | 97.76 | 0.001298
Testing | i | 2 | 0.000963 | 98.47 | 0.000964
| ii | 4 | 0.0008574 | 98.56 | 0.008594
| iii | 6 | 0.000846 | 98.57 | 0.000849
| iv | 8 | 0.000843 | 98.57 | 0.000848

The model set above exhibit a high-fit of estimate to fed data with generally
good mean-square errors and final prediction errors for a control experiment.
With increasing model order starting from 4, we see that the fits start
reaching convergence, as the mean-square errors and final prediction errors
become constant. In the frequency-domain, this is the equivalent to having
pole-zero cancellations for higher-order models. We therefore conclude there
is no useful properties a higher-order model could predict beyond an order of
8. The second-order model sufficiently approximates the system and is not
significantly outperformed by the higher order models –which would contribute
higher complexity to the control design. We therefore pick the 2nd order state
space model (IV-A) as

$\displaystyle\textbf{x}(k+Ts)=\textbf{A}\textbf{x}(k)+\textbf{B}\textbf{u}(k)+\textbf{K}\textbf{e}(k)$
$\displaystyle\textbf{y}(k)=\textbf{C}\textbf{x}(k)+\textbf{D}\textbf{u}(k)+\textbf{e}(k)$
(25)

where $Ts$ is the sampling period, $\textbf{e}(k)$ is the modeled zero-mean
Gaussian white noise with non-zero variance,

$\displaystyle\textbf{A}=\begin{bmatrix}0&1\\\
-0.9883&1.988\end{bmatrix},\hskip
3.99994pt\textbf{B}=\begin{bmatrix}-3.03e-07\\\ -4.254e-07\end{bmatrix}$
$\displaystyle\textbf{C}=\begin{bmatrix}1&0\end{bmatrix},\hskip
3.99994ptD=0,\text{and}\hskip
3.99994pt\textbf{K}=\begin{bmatrix}0.9253&0.9604\end{bmatrix}^{T}.$ (26)

The pair $(A,B)$ is stabilizable and and the pair $(A,C)$ is detectable.

##  V LQG Control

We employ a LQG controller and estimator to minimize the following cost
function subject to the state equation (IV-B2)

$J=\sum\limits_{k=0}^{\mathcal{K}}x^{T}(k)\,Q\,x(k)+R\,u(k)^{T}\,u(k)+2x(k)^{T}\,N\,u(k)$
(27)

where $\mathcal{K}$ is the terminal sampling instant, $Q$ is a symmetric,
positive semi-definite matrix that weights the $n$-states of the $A$ matrix,
$N$ specifies a matrix of appropriate dimensions that penalizes the cross-
product between the input and state vectors, while $R$ is a symmetric,
positive definite weighting matrix on the control vector $u$. The quadratic
cost function in (27) allows us to find an analytical solution (controller
sequence) to the minimization of $J$ over the prediction horizon, $n_{y}$

$\Delta u=\text{arg }\underset{\Delta u}{\text{min }}J$ (28)

where $\Delta{u}$ is the future control sequence and the first element in the
sequence is used in the control law at every time instant. We model additive
white noise disturbances into the discrete estimator’s states; therefore the
optimization problem becomes a stochastic optimization problem that must be
solved.

The separation theorem ensures that we can construct a state estimator which
asymptotically tracks the internal states from observed outputs, $y(k)$, using
the algebraic Riccati equation given as

$\begin{split}A^{T}PA\mbox{-}(A^{T}PB\mbox{+}N)(R\mbox{+}B^{T}PB)^{-1}(B^{T}PA\mbox{+}N)\mbox{+}&Q.\end{split}$
(29)

where $P$ is an unknown $n\times n$ symmetric matrix and $A$, $B$, $Q$, and
$R$ are known coefficient matrices as in (IV-B2) and (27). We find an optimal
control law by solving the minimization of the LQ problem, (27) which we then
feed into the states.

In practice, it is a good idea to start with an identity matrix, $Q$, a zero
penalty matrix, $N$, and tune $R$ till one obtains convergence by the state
estimator. The following optimal values were used after a heuristic search

$\displaystyle Q=\begin{bmatrix}1.0566&0\\\ 0&1.0566\end{bmatrix},\hskip
3.99994ptR=\begin{bmatrix}0.058006\end{bmatrix}.$ (30)

$\textbf{B}(k)$$\textbf{u}(k)$$\textbf{K}_{opt}$$\hat{\textbf{y}}(k)$$\int$$C(k)$$x(k)$$A(k)$$w(k)$$y(k)$$v(k)$$B(k)$$\int$$C(k)$$\hat{y}(k)$$A(k)$$K_{obs}$$\epsilon=\hat{y}(k)-y(k)$$\hat{x}(k)$$-1$$y(k)$
Figure 7: Full Linear Quadratic Gaussian Plant Estimator

We construct a full online estimator for the identified plant as in Fig. 7,
whereby the noise processes are assumed to be independent, white, Gaussian, of
zero mean and known covariances. The optimal controller gains, $K_{opt}$, are
determined from the equation

$K_{opt}={R}^{-1}(B^{T}\,P+N^{T})$ (31)

[25] where $P$ is the solution to the algebraic Riccati equation (29) and
$\mathbb{E}[w(k)w^{\prime}(\tau)]=R(k)\delta(k-\tau).$ Therefore, the online
optimal estimate, $\hat{x}(k+1)$ of $x(k)$ is

$\displaystyle\hat{x}(k+1)=A(k)\hat{x}(k)+K_{lqg}\left[C(k)\hat{x}(k)-y(k)\right]$
(32)

where $\hat{x}(k_{0})=\mathbb{E}\left[x(k_{0})\right]$ The observer is
equivalent to a discrete stochastic Kalman filter that estimates the optimal
state $\hat{x}(k|k)$ as shown in Fig. 7. The estimator equations are similar
to equations (III-B) and (III-B) and the online, unbiased estimate is

$\displaystyle\hat{x}(k+1)$
$\displaystyle=A(k)\hat{x}(k)-K_{obs}[\hat{y}(k)-y(k)]+B(k)u(k)$
$\displaystyle\hat{y}(k)$ $\displaystyle=C(k)\hat{x}(k)$ (33)

$\implies$

$\displaystyle\begin{split}\hat{x}(k+1)&=A(k)\hat{x}(k)-K_{obs}[C(k)\hat{x}(k)-y(k)]\\\
&\qquad\qquad\qquad+B(k)u(k).\end{split}$ (34)

Through heuristics, we found the following variances of the online estimator
to be useful:

$\displaystyle Q_{e}=\begin{bmatrix}0.4511&0\\\ 0&0.4511\end{bmatrix},\hskip
3.99994ptR_{e}=\begin{bmatrix}0.01\end{bmatrix}$

##  VI Experimental Results and Discussion

The control network was implemented on an NI-myRIO running LabVIEW 2015. We
initialized the Kinect sensors to allow for all pixels within the depth
cameras to reach steady state under ambient light. We performed multiple
experiments to evaluate the developed state space model of IV-B and LQG
controller of V. 111The LabVIEW identification and control codes are available
on the git repo [26].. The input variable is the current that excites the
valve, which in turn actuates the bladder; the head moves in response to
bladder actuation. The fused estimate of the Kinect sensors are used to
estimate the real-time head pitch motion as described in III-C; this is in
turn used in a feedback to the LQG controller.

Fig. 8 shows the results from a constant reference trajectory, which the head
is meant to track. We notice a settling time of approximately 24 seconds
before we reach steady state. The delay arises from our design requirements
and is not a drawback in clinical trajectory tracking where we must ensure
smooth head motion to desired target. It is also seen that the controller
exhibits relatively smooth tracking within a 1.5 mm standard deviation over
time after a relative overshoot of 5mm in bottom graph of Fig. 8. The
overshoot can be explained by the estimator’s search for a steady state region
based on the time it takes for the pixel values of the sensors to reach steady
state. The controller tracks the reference to within $\pm 2mm$.

Figure 8:  LQG Controller on Manikin Head

However, we noticed an inconsistency at certain operating ranges in the
current LTI model. The applied current based on fusion feedback occasionally
reaches a steady state error, as can be seen from Fig. 9. We conjecture this
is due to an unmodeled nonlinearity at the inlet valve that maps input
currents to system states. To better approximate the nonlinearity from input
to output, we will investigate using a Hammerstein block-structured model that
better approximates the nonlinearity from inputs to states and states to
output of the system.

The fusion algorithm proved useful to cancel jitter in the depth measurement
of the sensor over our previous results, but it falls short of the 1mm
accuracy in head and neck cancer RT. Having established proof of concept in
this investigation, we will begin investigation of better head localization by
using sophisticated motion capture systems or laser scanners, such as those
actively employed in IGRT.

Figure 9:  LQG Controller on Manikin Head

##  VII Summary

We have presented a continuation of or initial investigation into a
pneumatically-driven soft robot system for head and neck radiotherapy.
Measurements from two Kinect RGB-D cameras are fused to refine the accuracy of
observations. System identification on the soft robot was combined with LQG
controller design to provide an optimal controller. Experiments showed we
could actuate the patient head within 2.5mm accuracy. Future work will
investigate accurate multi-axis positioning using better 3D sensor, control of
multiple IABs, and nonlinear modeling or model free-methods to overcome
limitations of the LTI model.

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