=Paper=
{{Paper
|id=Vol-2525/paper16
|storemode=property
|title=Compliance errors estimation for robotic manipulator using the Quantum Monte Carlo method
|pdfUrl=https://ceur-ws.org/Vol-2525/ITTCS-19_paper_31.pdf
|volume=Vol-2525
|authors=Ramil Dautov,Alexandr Klimchik
|dblpUrl=https://dblp.org/rec/conf/ittcs/DautovK19
}}
==Compliance errors estimation for robotic manipulator using the Quantum Monte Carlo method==
https://ceur-ws.org/Vol-2525/ITTCS-19_paper_31.pdf
Compliance errors estimation for robotic manipulator using the
Quantum Monte Carlo method *
Ramil Dautov Alexandr Klimchik
Innopolis University Innopolis University
Innopolis, Russia Innopolis, Russia
r.dautov@innopolis.ru a.klimchik@innopolis.ru
Abstract
The paper presents new approach to estimating the compliance errors of
robotic manipulator. The existing methods are complicated with the non-
linear character of equations coming from the theory. Quantum Monte Carlo
computational technique is shown to overcome this problem. The algorithm
of evaluating and compensating compliance errors using this technique is
presented.
1 Introduction
The precision of industrial robots plays an essential role in manufacturing and high-quality machining. There are many
factors that influence on the accuracy of robotβs performance. The essential part of deviations originates from the
interactions between the robot and a workpiece (loading). These interactions cause the deformations of manipulator
components that lead to errors in performance (compliance errors).
Compliance errors are in high dependence on the configuration of the robot and this fact complicates their analysis.
Different models have been developed to estimate and compensate the compliance errors [1, 2, 3]. The theory appears to
be non-linear and the problem is usually resolved by using the approximations and implementing the computational
schemes to solve equations.
However, there exists one computational method that provides the possibility to avoid such complexities and consider
the theory in its original form β Quantum Monte Carlo scheme [4, 5].
Quantum Monte Carlo method is broadly used in condensed matter physics and physics of elementary particles. This
technique allows us to reduce the problem of solving the highly non-linear equations to the problem that is common in
statistical physics.
The main role here plays the action functional of the system. The equations of motion are solved indirectly by the help
of statistical computational algorithms that find the solution by minimizing the action.
In this article we propose the possible implementation of Quantum Monte Carlo technique to the problem of
compensation the compliance errors. The article is structured as follows. We begin with the brief review of the Quantum
Monte Carlo method in section 2. Our review is based on the two well-developed introductory courses on implementations
of the Quantum Monte Carlo technique [4, 5]. In Section 3 we show how this technique can be applied to evaluation and
compensation the compliance errors. The corresponding algorithm is also presented.
*
Copyright Β© 2019 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0
International (CC BY 4.0).
2 Quantum Monte Carlo method
In quantum physics we have a well-developed formalism that generalizes the action principle of classical mechanics. This
formalism is called Path Integral formulation of quantum mechanics and was proposed by Richard Feynman in 1948 [6].
Path integrals replace the notion of classical trajectory with the functional integral over an infinite number of mechanically
possible trajectories. These integrals are used to obtain the value of particular parameters of the system.
The path integrals reveal the correspondence between quantum and stochastic processes and also can be considered as
an analytical continuation of a method of random walks. This property allows to transform path integral theory to the form
analogous to the theory of statistical physics and molecular dynamics.
2.1 Path Integrals
Path integrals provide us possibility to calculate the value of a particular parameter of the system. Let us denote the
parameter that we want to evaluate as π΄. In general, π΄(π₯) is a function of coordinates. Then we can find the average value
of π΄(π₯) by following formula:
π
π(π₯)
β« π·π₯(π‘)π΄(π₯)π β
< π΄(π₯) >= π
(1)
π(π₯)
β« π·π₯(π‘) π β
Here π β is an imaginary unit, β - is a Planck constant (usually quantum physicists work in a system of units with β = 1),
π‘
π(π₯) = β«π‘ π ππ‘ πΏ(π₯, π₯Μ ) β is an action functional. β« π·π₯(π‘) designates the sum over all possible particle paths from the initial
π
time π‘π to the final π‘π .
Although there are some theoretical advantages of formula (1), it is difficult to calculate the value of π΄(π₯) analytically.
To overcome this obstacle there has been developed a numerical method based on the Monte Carlo scheme. It has been
already pointed out that path integral formulation reveals the correspondence between the quantum physics and statistical
physics. To show this correspondence explicitly let us replace the time variable: π‘ β ππ‘ β so-called Wickβs rotation (it is
actually rotation in an imaginary plane). After Wickβs rotation, considering β = 1 we receive βEuclidianβ path integrals:
β« π·π₯(π‘)π΄(π₯)π βπ(π₯)
< π΄(π₯) >= (2)
β« π·π₯(π‘) π βπ(π₯)
Euclidian path integrals look very similar to the integrals of thermal averages that we have in Thermodynamics and
Statistical physics. This representation is much better for numerical work as it has no oscillations in sign within the powers
of exponents.
It is important to take into account the changes in the action π(π₯) caused by the Wickβs rotation. For instance, let us
consider the action of point-like particle with mass π. Before the Wickβs rotation the action has the form:
π‘π
ππ₯Μ 2
π(π₯) = β« ππ‘ ( β π(π₯)) (3)
π‘π 2
After the Wickβs rotation π‘ β ππ‘ we will have:
π‘π
ππ₯Μ 2
π(π₯) = β« ππ‘ ( + π(π₯)) (4)
π‘π 2
We see that Lagrangian πΏ(π₯, π₯Μ ) has changed its form and became actually the Hamiltonian of the system. And
continuing the analogy between Euclidian path integrals and thermodynamics one can easily show that the time π‘ now has
relation with the βtemperatureβ π [4, 5]:
1
π‘βπ½β‘
ππ΅ π (5)
Thus any code developed to conduct calculations in thermal physics can be applied for calculation path integrals (2).
2.2 Quantum Monte Carlo algorithm
Quantum Monte Carlo technique provides a numerical procedure for calculating the path integrals using the stochastic
methods developed in statistical physics. To reduce our continuous formula (2) into the numerical discretized form we
must address two issues.
Firstly, we need to find a convenient representation of an arbitrary path {π₯(π‘), π‘π β€ π‘ β€ π‘π } in the computer. A path is
given by the π₯(π‘) function which, in principle, can be complicated enough for computers. We approximate the path by
specifying π₯(π‘) only at the nodes on a discretized time axis:
π‘π = π‘π + ππ, for π = 0,1, β¦ , π (6)
Here π is a grid spacing:
π‘π β π‘π
π=
π (7)
Then the path can be described by a vector:
π₯ = {π₯(π‘0 ), π₯(π‘1 ), β¦ , π₯(π‘π ) } (8)
In lattice theories this vector is usually called βconfigurationβ. The integral over all paths becomes an ordinary
multidimensional integral over all possible values for each of the π₯(π‘π )βs:
+β
β« π·π₯(π‘) β β« ππ₯1 ππ₯2 β¦ ππ₯πβ1 (9)
ββ
Here we denoted π₯(π‘π ) = π₯π . We donβt integrate over endpoints π₯0 and π₯π since they are held fixed.
The second issue is related to the evaluation of the action on the discretized path (8). In fact, here we just need to
substitute the derivatives in the action with their finite differences approximations. Integrals are replaced with the
summation and the potential π(π₯) should be calculated in each node π(π₯π ). For instance, the discretized version of the
action (4):
πβ1
π
ππππ‘ (π₯) = β [ (π₯π+1 β π₯π )2 + ππ(π₯π )] (10)
2π
π=0
We have reduced mechanics to the problem of numerical integration. Now we can evaluate the path integrals using the
standard methods of numerical multidimensional integration. The Monte Carlo procedure appears to be the more general
and convenient technique for this purpose.
One can note that path integral of the form (2) is actually a weighted average over paths with weight exp(βπ(π₯)).
Taking into account this fact we generate a large number πππ of random paths (or configurations)
(πΌ) (πΌ) (πΌ) (11)
π₯ (πΌ) β‘ {π₯0 , π₯1 , β¦ , π₯πβ1 } πΌ = 1, β¦ , πππ ,
on our grid in such a way that the probability π[π₯ (πΌ) ] for obtaining the particular path π₯ (πΌ) is:
(πΌ) ]
π[π₯ (πΌ) ]~π βπ[π₯ (12)
Then unweighted average of π΄(π₯) over this set of paths approximates the weighted average over uniformly distributed
paths:
πππ
1
< π΄(π₯) >β π΄Μ
= β π΄[π₯ (πΌ) ] (13)
πππ
πΌ=1
π΄Μ
is the Monte Carlo estimation for < π΄(π₯) > on our lattice. The estimation will never be the exact since we use the
1
discretized paths and πππ will never be infinite. One can show that statistical errors will be of the order [4, 5].
βπππ
There are many ways of how to generate the set of random paths π₯ (πΌ) [4, 5]. We review only the main idea of these
methods. For the detailed review the reader can refer to the articles [4, 5].
So, the general idea of the methods is the following:
(0) (0) (0)
ο· first, generate an arbitrary path π₯ (0) = {π₯0 , β¦ , π₯π , β¦ , π₯πβ1 };
ο· at each ππ‘β site we generate the random number β¦, with probability uniformly distributed between β π and π,
π β is a some constant;
(0) (0)
ο· replace π₯π β π₯π + β¦ and calculate the change βπ in the action caused by this replacement;
(0)
ο· if βπ < 0 then retain the new value for π₯π ;
(0)
ο· if βπ > 0 generate a random number π uniformly distributed between 0 and 1; retain the new value for π₯π if
exp(ββπ) > π, otherwise restore the old value;
This algorithm is repeated some number of times and at the end we obtain the path that minimizes the action and can be
considered as the solution to the equations of motion. The constant π is tuned experimentally by running the algorithm so
that 40-60% of new π₯π βs are retained.
3 Compliance errors estimation
The main problem of estimating the compliance errors is the necessity to deal with complicated non-linear equations of
motion. In Quantum Monte Carlo method we have no need to work with the exact equations of motion. All the dynamical
and kinematical information of the system is already included into the Lagrangian function. Using the Quantum Monte
Carlo technique we can minimize the action and obtain the configuration of the system that represents the solution of the
equations of motion.
To estimate the compliance errors, we must take into account two components. The first component represents the
stiffness properties of the robot. These properties are defined by the stiffness matrix. In this work we assume that the
stiffness matrix of the system is known. The stiffness matrix can be found by implementing the CAD-based approach [7].
The second component represents the forces/torques acting on the end-effector. Further, we also assume that this
component is given. For instance, it can be obtained by direct measurements using the force/torque sensor adjusted on the
end-effector.
Compliance errors occur due to the interaction between a workpiece and the end-effector. The control system of the
robot knows the configuration of the manipulator at the given time and manages its movements to make the end-effector
follow the desired trajectory. However, because of the compliance the deflections appear and the end-effector follows
another trajectory (real trajectory) that differs from the desired one.
The motion of the end-effector can be described by the Lagrange equations of the second kind:
π ππΏ ππΏ (14)
( )β = ππ , π = 1, β¦ , π
ππ‘ ππΜ π πππ
π β the number of degrees of freedom, ππ β the generalized force acting on the end-effector along the direction of the
ππ coordinate, πΏ = π(π, πΜ ) β π(π) β Lagrangian of the system. π(π, πΜ ) β kinetic energy and π(π) β potential energy of
the system that contains contributions of a gravitational field and potential energy of elastic deformations caused by
compliance.
In principle, we can decompose the movement of the end-effector into the sequence of quasistatic translations: within
a sufficiently small period of time ππ can be considered as constants and during that time the motion of the end-effector
effectively is ruled by the Lagrangian of the form:
π
πΏ = π(π, πΜ ) β π(π) + β ππ ππ (15)
π=1
We see that the equation (14) can be derived from the (15) assuming the absence of non-conservative forces. Actually,
to use the Quantum Monte Carlo technique we have to rewrite the Lagrangian (15) in a Euclidian way (compare with (4)):
π
πΏ = π(π, πΜ ) + π(π) + β ππ ππ (16)
π=1
Then within that small period of time we can use the Lagrangian (16) to write down the Euclidian action functional and
using the Quantum Monte Carlo technique minimize it to obtain the real configuration π = {ππ } of the robot. Then the
obtained real configuration is compared with trajectory stored in the control system. The compliance errors are estimated
as the differences between the real and desired (stored in the control system) trajectories. After evaluating the compliance
errors, the control system is able to adjust the trajectory of the end-effector to eliminate the deflections.
The main difficulty that can arise here is the time consumption of the algorithm. The external forces acting on the end-
effector can vary with high rate. The Monte Carlo algorithm takes some time to find the configuration minimizing the
action. By the time the configuration is found the end effector can experience another external force πβ²π , and our obtained
configuration π = {ππ } becomes outdated.
In fact, we can shorten the time that is needed for the Monte Carlo algorithm to minimize the action by taking as a
starting configuration π (0) in the Monte Carlo algorithm the desired configuration (just read it from the control system).
On practice, the deviations of the desired trajectory from the real trajectory are not so large and taking the desired
configuration as an initial will put the action into position near the minimum. So, a few iterations will be needed to minimize
the action and find the real trajectory. This feature may help us to adjust the trajectory of end-effector in the on-line regime.
The whole algorithm is represented on the Figure 1.
Figure 1: The algorithm for estimation and compensation the compliance errors
Summary
Compliance errors appear due to the interactions between the manipulator and the workpiece and they are in high
dependence on the configuration of the robot. This fact makes the analysis and estimation of the errors problematic. We
need to take into account the non-linear character of the compliance mechanism that leads us to the complicated math and
implementation of different approximations.
Quantum Monte Carlo method has been proposed to overcome these issues. The developed algorithm provides an
elegant way to find the real trajectory of the manipulator under the applied loadings. The obtained data then can be used
by the control system to adjust the trajectory and compensate the compliance errors. However, Quantum Monte Carlo
iterations can take considerable amount of time and the deeper analysis is required.
In future, the developed technique will be tested on the simulation of the compliance errors compensation of the
Stewartβs platform. Testing the Quantum Monte Carlo approach on that toy model will show the prospects of this method.
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