## Abstract

Accurate position sensing is important for state estimation and control in robotics. Reliable and accurate position sensors are usually expensive and difficult to customize. Incorporating them into systems that have very tight volume constraints such as modular robots are particularly difficult. PaintPots are low-cost, reliable, and highly customizable position sensors, but their performance is highly dependent on the manufacturing and calibration process. This paper presents a Kalman filter with a simplified observation model developed to deal with the nonlinearity issues that result in the use of low-cost microcontrollers. In addition, a complete solution for the use of PaintPots in a variety of sensing modalities including manufacturing, characterization, and estimation is presented for an example modular robot, SMORES-EP. This solution can be easily adapted to a wide range of applications.

## 1 Introduction

It is necessary to have accurate position sensing for most robotics control processes. Commercial position sensors are available for many applications. However, the form factor often presents a challenge to use these commercial off-the-shelf sensors. This is especially true for compact robotic systems with tight space constraints such as modular robotic systems [1]. Customizable position sensors give more flexibility to fit these situations.

Servos have been used in many modular robotic systems, such as Molecule [2] and CKBot [3]. Servo motors have built-in position control circuits. Some modular robotic systems, such as 3D Fracta [4], M-TRAN III [5], and SuperBot [6], use rotary potentiometers for position feedback. Optical or hall-effector sensor encoders are used in PolyBot [7], Crystalline [8], and ATRON [9] to report position information. These devices are usually too large for highly space-constrained systems, such as modular robots like SMORES-EP [10].

PaintPots are highly customizable and low-cost position sensors that can be easily manufactured by widely accessible materials (spray paint and plastic sheets) and tools (laser cutters or scissors) [11]. The sensors can exist in different forms in terms of size, shape, and surface curvature. Thus, these sensors can be easily integrated with well-designed parts or systems. The sensing performance and cost of PaintPot sensors make them competitive with commercial potentiometers [11]; yet, the customizability enables the use in situations that are not possible with commercial potentiometers.

Two different designs of PaintPot sensors are used in SMORES-EP modular robots. In each SMORES-EP module, there are four degrees-of-freedom (DOFs) requiring position sensing shown in Fig. 1—three continuously rotating joints (LEFT DOF, RIGHT DOF, PAN DOF) and one bending joint with a 180-deg range of motion (TILT DOF) [10]. Conductive spray paint is used to generate a resistive track surface. The manufacturing process is easy enough for a person to make a sensor quickly, but often does not yield consistent measurement and performance. The terminal-to-terminal resistance can vary over a large range depending on the thickness of the paint with a nonlinear output. This fact complicates the position estimation for every DOF.

Fig. 1
Fig. 1
Close modal

For state estimation, stochastic techniques based on the probabilistic assumptions of the uncertainties in the system are widely applied. For linear systems, the Kalman filter [12] has been shown to be a reliable approach where uncertain parts in systems are assumed to have a particular probability distribution, usually Gaussian. Extensions including the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) [13] have been developed for nonlinear systems. For SMORES-EP DOF state estimation, we developed a new Kalman filter with a simpler observation model considering the nonlinearity of PaintPot sensors. A complete and convenient calibration process is developed to precisely characterize each position sensor quickly. Four estimators can run on a 72-MHz microcontroller at the same time to track the states of all DOFs in a SMORES-EP module.

There are many applications that could use potentiometers but are too size constrained. For example, goniometers in instrumented gloves for virtual reality currently use expensive strain gauges or nonlinear flex sensing technologies. A joint angle potentiometer would be a good low-cost solution except for size constraints. Any compact device with a hinge (a flip phone, smart eyeglasses, etc., or wearable devices) could have that joint angle measured to provide feedback. This paper presents an example solution of a class of potentiometers that is easily adapted to fit many low-profile applications. It is semi-custom in that it is easily scalable to short-run numbers. These sensors are easily characterized and combined with our state estimation method, PaintPot sensors can provide reliable position information with little computation cost. Our solutions in the SMORES-EP system show that PaintPot sensors are promising for a variety of robotic applications.

The paper is organized as follows. Section 2 reviews relevant and previous work. Section 3 introduces the manufacturing and customized design for the SMORES-EP system, as well as some necessary information. The characterization process is shown in Sec. 4, and the estimation method is presented in Sec. 5 for all DOFs. Some experiments are shown in Sec. 6. Finally, Sec. 7 talks about the conclusion.

## 2 Related Work

A potentiometer is a three-terminal resistor with a sliding or rotating contact (or wiper) that functions as a voltage divider [14]. There are three basic components in a potentiometer (Fig. 2): a resistive track, fixed electrical terminals on the track ends, and an electrical wiper. Different from most modern potentiometer tracks that are continuous semiconductive surfaces made of graphite, ceramic-metal composites (cermets), conductive plastics, or conductive polymer pastes, PaintPot sensors use an inexpensive carbon-embedded polymer spray paint [11].

Fig. 2
Fig. 2
Close modal

## 4 Sensor Characterization

Potentiometers used as voltage dividers typically model the input position as having a linear relationship with the output voltage. Close adherence to the linear model has to be achieved by ensuring that the resistance between two points along the track is constant, which requires uniform geometry, thickness, and material properties of the track. This is difficult for PaintPots which are manually spray-painted. In order to obtain accurate position control on all DOFs of a SMORES-EP module, a calibration process is needed to characterize the performance of the particular PaintPots installed.

One terminal of a wheel PaintPot is connected with 3.3 V, and the other terminal is connected with ground. Two wipers can contact the track and report current voltage (V0 and V1) in the form of two 10 bit analog-to-digital conversion values ranging from 0 to 1023, and wheel position θ = 0 rad is shown in Fig. 6 and the whole range of θ is from −π rad to π rad. When a wiper contacts on or around the V-shape gap, the voltage value is not usable. This is the reason for having two wipers, to enable sensing the full 360 deg range. So, V0 should be ignored when θ is in the range from $23πrad$ to $56πrad$ and V1 should be ignored when θ is in the range from $−56πrad$ to $−23πrad$.

Fig. 6
Fig. 6
Close modal

Similar to wheel PaintPots, the tilt PaintPot is also powered between ground and 3.3 V with one single wiper contacting the track all the times. The voltage V0 from the voltage divider goes through a 10 -bit analog-to-digital conversion (with a range from 0 to 1023). When the tilt position θ = 0 rad, the wiper is positioned in the middle of the track as shown in Fig. 7.

Fig. 7
Fig. 7
Close modal

An automatic sensor calibration setup is developed based on AprilTags tracking [22] shown in Fig. 8(a). Three tags are used to track the rigid bodies of a SMORES-EP module. Tag 2 is fixed to the base to be the reference frame, Tag 1 is fixed to the TOP Face of a SMORES-EP module for TILT DOF tracking, and Tag 0 can be fixed to LEFT Face, RIGHT Face, or TOP Face for wheel DOF tracking (Fig. 8(b)). During characterization, one DOF is moved at a time through its entire range of motion (2π rad for wheel DOF, π rad for TILT DOF) in both directions. The data, including θ and reported voltage, are recorded at 14 Hz (speed limited by the AprilTag ROS package).

Fig. 8
Fig. 8
Close modal

While the voltage data is not linear with DOF position, it is monotonic (piece-wise monotonic for wheel DOFs). A third-order polynomial provides a suitable model. For a wheel PaintPot, the data from both wipers are shown in Figs. 9(a) and 9(b), respectively. For wiper 0, the reported voltage V0 is not useful when DOF position θ is in the range from $23πrad$ to $56πrad$ (shown in red). Due to the gap of wheel PaintPots, θ = f0(V0) is a piece-wise function which can be converted into a continuous function $θ¯0=f¯0(V0)$ by shifting the segment ranging from $56πrad$ to π rad (shown in green) by 2π rad downward (shown in yellow). Similarly, for wiper 1, the segment when θ is in the range from $−56πrad$ to $−23πrad$ (shown in red) is meant to be trimmed, and the piece-wise function θ = f1(V1) is converted into a continuous function $θ¯1=f¯1(V1)$ by shifting the segment ranging from −π rad to $−56πrad$ (shown in green) by 2π rad upward (shown in yellow). After taking 50 s data in both directions (showing little hysteresis), $f¯0(V0)$ and $f¯1(V1)$ are shown in Figs. 10(a) and 10(b), respectively. An example run from a tilt PaintPot is shown in Fig. 11(a), and the characterization result θ = f(V0) is shown in Fig. 11(b).

Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal
Fig. 11
Fig. 11
Close modal

## 5 Position Estimation

### 5.1 Transition Model.

Four DC motors are used to drive four DOFs (LEFT DOF, RIGHT DOF, PAN DOF, and TILT DOF) with a geared drive train shown in Fig. 12. The drive has pinion gears driving four identical spur gears. Two outer spur gears are attached to the left wheel and right wheel, respectively, for the LEFT DOF and RIGHT DOF. The crown gear is coupled to the two inner spur gears. When these two inner gears spin in the opposite direction, the top wheel attached to the crown gear rotates which is the PAN DOF. TILT DOF rotates when these two inner spur gears spin in the same direction. The transmission ratio for each DOF is determined by the gear train, then a linear relationship between the DOF velocity and the motor angular velocity can be obtained
$θ˙=kω+n$
(1)
in which k is the transmission ratio of the DOF, ω is the angular velocity of the driving motor(s), nN(0, Q) is the additive Gaussian white noise, and $θ˙$ is the angular velocity of the DOF. The transition model of a DOF in discrete time can be derived by one-step Euler integration:
$θt=θt−1+θ˙t−1δt+nt−1δt=θt−1+kωt−1δt+nt−1δt=θt−1+Gωt−1+Unt−1$
(2)
in which δt is the finite time interval. Let θ be the state x and ω be the system input u, then the transition model is
$xt=xt−1+Gut−1+Unt−1$
(3)
Fig. 12
Fig. 12
Close modal

### 5.2 Observation Model.

In Sec. 4, a PaintPot can be characterized with a nonlinear model—a third-order polynomial θ = f(V). Here, V is the measurement, namely the reported voltage(s) from the wiper(s). Based on this, a nonlinear observation model z = h(x) where z is the measurement and x is the state can be derived. We can linearize z = h(x) about the current mean and variance of the DOF position x to apply an extended Kalman filter (EKF). However, this places some computational burden on the SMORES-EP microcontroller. Here, we present a simple approach that can generate a linear observation model directly without linearization of the original nonlinear observation model about the current mean and variance of the state to overcome this nonlinearity. With this new approach, we just need to check if features are available using a simple rule and compute the newly converted measurements by evaluating polynomials obtained from sensor characterization (Sec. 4).

#### 5.2.1 Wheel PaintPots.

For the wheel PaintPot sensor, the observation model is a piece-wise function due to the geometry of the sensor:

1. When $θ∈(−πrad,−56πrad)$$∪(−23πrad,23πrad)$$∪(56πrad,$$πrad)$, there are two valid measurements which are the reported voltages V0 and V1 from both wipers.

2. When $θ∈[−56πrad,−23πrad]$, only V1 is valid.

3. When $θ∈[23πrad,56πrad]$, only V0 is valid.

To accommodate the 2π rad shift to obtain a continuous function for sensor characterization (Sec. 4), the observation can be modeled as
$z={[V0V1]=[f¯0−1(x)f¯1−1(x+2π)],x<−56πradV1=f¯1−1(x),−56πrad≤x≤−23πrad[V0V1]=[f¯0−1(x)f¯1−1(x)],−23πrad56πrad$
(4)
in which x is the DOF position θ.
In order to avoid linearizing this piece-wise observation model, we change the measurement to be the reported states rather than the two reported voltages. During the motion, there is at least one feature available for tracking, namely at least one wiper is contacting the wheel PaintPot at any time. Let zi be the measurement from the ith feature, then the measurement model with additive Gaussian white noise is simplified as
$zi=xi+vi=hi(x,vi)i=0,1$
(5)
in which viN(0, Ri) and xi is the reported state from ith feature determined by state x in the following way:
$x0={xx≤56πx−2πx>56π$
(6a)
$x1={x+2πx<−56πxx≥−56π$
(6b)
Here, the measurement model is linear and the predicted measurement can be computed easily. The actual measurement for the ith feature can be obtained by evaluating $f¯i(Vi)$ derived from the sensor characterization process. Recall that when $x∈[23πrad,56πrad]$, V0 is not valid meaning this feature is not available. Otherwise, the actual measurement from this feature is simply $f¯0(V0)$ if $x∉[23πrad,t56πrad]$. And the valid range of V0 is from $Vmin0=f¯0−1(56π−2π)$ (because the segment from $56πrad$ to π rad is shifted downward by 2π rad) to $Vmax0=f¯0−1(23π)$. Similar procedures can be applied to V1, this feature is $f¯1(V1)$ if $x∉[−56πrad,−23πrad]$; otherwise, this feature is not available. The valid range of V1 is from $Vmin1=f¯1−1(−23π)$ to $Vmax1=f¯1−1(−56π+2π)$.

#### 5.2.2 Tilt PaintPots.

The observation model for tilt PaintPots is straightforward which is
$z=V0=f−1(x)$
(7)
and it is a nonlinear function. Similarly, in order to avoid linearizing this observation model, we change the measurement to be the reported state and there is only one feature for tracking. Then, the observation model with additive Gaussian white noise is simplified as
$z=x+v=h(x,v)$
(8)
in which vN(0, R) and the model is linear. The feature should always be available, and the actual measurement for this feature is obtained by evaluating f(V0).

### 5.3 Kalman Filter.

With the transition model and the new form of observation model, a Kalman filter framework can be applied for state estimation.

#### 5.3.1 Kalman Filter for Wheels.

The initial state of a wheel DOF can be derived by any available feature. First check V0, and if it is inside the valid range from $Vmin0$ to $Vmax0$, compute the initial state $x0=f¯0(V0)$, and shift x0 if necessary. That is, if x0 < −π, let x0 be x0 + 2π. If $V0∉(Vmin0,Vmax0)$, then $x0=f¯1(V1)$ because the wiper 1 must contact the valid range of the track at this time, and similarly shift x0 if necessary, namely if x0 > π, let x0 be x0 − 2π. The prior state can be represented as a Gaussian distribution p(x0) ∼ N(μ0, Σ0) where μ0 = x0 and Σ0 is initialized to an arbitrarily small value.

With Eq. (3), the prediction step is
$μ¯t=μt−1+Gut$
(9a)
$Σ¯t=Σt−1+U2Q$
(9b)
With Eqs. (5), (6a), and (6b), the predicted measurement $xti$ for the ith feature can be computed. The Analog-to-digital value Vi from the ith feature at time t is used to compute the actual measurement zi. If $Vi∈(Vmini,Vmaxi)$, $zti=f¯i(Vi)$. Otherwise, this feature is not available. If both features are available, then $zt=[zt0,zt1]⊤$, $Ct=[1,1]⊤$, and the Kalman gain is
$Kt=Σ¯tCt⊤(CtΣ¯tCt⊤+R)−1$
(10)
in which R = diag(R0, R1). If there is only one feature (e.g., the ith feature) available, then $zt=zti$, Ct = 1, and the Kalman gain is
$Kt=Σ¯tCt(Ct2Σ¯t+Ri)−1$
(11)
The state is then updated:
$μt=μ¯t+Kt(zt−Ctμ¯t)$
(12a)
$Σt=Σ¯t−KtCtΣ¯t$
(12b)
The estimated position for this wheel DOF at time t is μt.

#### 5.3.2 Kalman Filter for Tilt.

The initial state for a TILT DOF can be derived by evaluating f(V0). The prior state can be represented as a Gaussian distribution p(x0) ∼ N(μ0, Σ0) where μ0 = x0 and Σ0 is initialized with some small value. The prediction step is in the same form with wheel DOFs (Eqs. (9a) and (9b)). The predicted measurement can be computed from Eq. (8) which is simply $μ¯t$. The current actual measurement is computed by evaluating zt = f(V0) where V0 is the current reported voltage. Then, the Kalman gain is simply
$Kt=Σ¯t(Σ¯t+R)−1$
(13)
and the state is updated in the following:
$μt=μ¯t+Kt(zt−μ¯t)$
(14a)
$Σt=Σ¯t−KtΣ¯t$
(14b)
And, the estimated position for TILT DOF at time t is μt.

## 6 Experiment

The PaintPot sensors, including wheel PaintPots and tilt PaintPots, are installed in SMORES-EP modules for position sensing. Currently, 25 SMORES-EP modules have been assembled. In the experiments, both a wheel PaintPot and a tilt PaintPot are characterized first and then the newly developed Kalman filters are implemented to show the effectiveness of our low-cost position sensing solution.

The data from both wipers for a wheel PaintPot is shown in Figs. 13(a) and 13(b), respectively. The segments labeled by red color are useless, and the green segments are shifted to the yellow segments to generate continuous functions to describe the relationship between voltage and angular position. This sensor is installed for PAN DOF on a SMORES-EP module. All our sensors are painted manually, so the quality is not consistent with no guarantees on bounds. The sensor characterization results for both wipers are shown in Figs. 14(a) and 14(b), respectively.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

With our Kalman filter, we are still able to derive a good estimation of the angular position for the PAN DOF on this module. A simple controller is used to command the PAN DOF to a desired position from its current position along a fifth-order polynomial trajectory. The first experiment commands the PAN DOF to angular position 0 rad from π rad resulting in an average error of 0.0878 rad as shown in Fig. 15(a). In this experiment, V0 is not valid for a while because wiper 0 contacts the gap of the track. The second experiment commands the PAN DOF to angular position 0 rad from −π rad with average error being 0.0698 rad as shown in Fig. 15(b). In this experiment, V1 is not valid when wiper 1 contacts the gap of the track.

Fig. 15
Fig. 15
Close modal

The data from the wiper for a tilt PaintPot are shown in Fig. 16(a), and the sensor characterization result is shown in Fig. 16(b). In the experiment, this TILT DOF is commanded to traverse most of the range. The result from our estimator is shown in Fig. 17 with an average error being around 0.0325 rad.

Fig. 16
Fig. 16
Close modal
Fig. 17
Fig. 17
Close modal

## 7 Conclusion

In this paper, a complete low-cost and highly customizable position estimation solution is presented, especially suitable for highly space-constrained designs which are very common in modular robotic systems. PaintPots are low-cost, highly customizable, and can be manufactured easily by accessible materials and tools in low quantities. For the SMORES-EP system, two different types of PaintPot sensors are used, and a convenient automatic calibration approach is developed using AprilTags. A modified Kalman filter is developed to overcome the piece-wise nonlinearity of the sensors and some experiments show the accuracy. The successful application of PaintPots in the SMORES-EP system shows that they can provide reliable position information with a simple hardware setup. PaintPots can be easily adapted to and installed on a variety of systems, the not consistent performance due to the manufacturing process can be resolved by our characterization process which can be easily set up, and reliable state estimation can be derived by our modified Kalman filter that can be running on a low-cost microcontroller. The overall solution provides a new position sensing technique for a wide range of applications.

## Acknowledgment

We would like to thank Hyun Kim for experiments and hardware maintenance. This work was funded in part by NSF grant No. CNS-1329620.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data and information that support the findings of this article are freely available online.4 Data provided by a third party are listed in Acknowledgment.

## Nomenclature

k =

transmission ratio

n =

Gaussian white noise in transition model

z =

measurement in Kalman filter

xi =

reported state from the ith feature

zi =

measurement from the ith feature

vi =

observation noise of the ith feature

x or θ =

angular position

u or ω =

angular velocity of the driving motor(s)

Vi =

voltage measurement from wiper i

$θ¯i$ =

shifted position measurement from wiper i

$N(μ¯,Σ¯)$ =

predicted Gaussian distribution of the angular position

N(μ, Σ) =

updated Gaussian distribution of the angular position

Kt =

Kalman gain at time t

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