Data-Driven Modeling of a Track-basedStair-Climbing Wheelchair

Yogita Choudhary1, Nidhi Malhotra1, Pratyush Kumar Sahoo2, and Shyam Kamal1

Abstract—A stair-climbing wheelchair can notably enhance theautonomy in terms of mobility for the aged and disabled. Thispaper presents a data-driven system identification approach anda vision-based heading control algorithm for reliable operation ofa stair-climbing wheelchair. We develop a track-based wheelchairmodel and propose a methodology for autonomous stair traversal.Modeling the dynamics of track-based systems is a challengingtask. Hence, a data-driven approach based on the ObserverKalman filter Identification/Eigensystem Realization Algorithmis employed for modeling the complex dynamics of the system.For developing an affordable system, low-cost sensors such asMicrosoft Kinect and IMU are utilized. We employ a compu-tationally efficient image processing algorithm, and the headingangle is controlled using Linear Quadratic Regulator (LQR).The effectiveness of the proposed methodology for a safe stairtraversal is verified in the ROS-Gazebo environment.

I. INTRODUCTION

Traversing stairs pose a significant challenge to people withlower-limb impairments and reduces their independence incommuting from one place to another. The lack of properinfrastructures, such as ramps or elevators, for wheelchairusers leads to the need for stair-climbing wheelchairs. Severalresearchers have taken an interest in solving this problem in re-cent years [1], [2], [3]. The existing stair-climbing wheelchairsare either too expensive or require human assistance leadingto a trade-off between affordability and functionality. Thus,in this work, we employ low-cost sensors and perform au-tonomous stair traversal to enhance the functionality of suchsystems.

For stair-climbing wheelchairs, various mechanisms havebeen explored [4], [5], [6]. In a track-based mechanism, tracksmaintain contact with the ground and are driven by a setof wheels. As compared to wheel-based systems, there is alarger ground contact that increases the stability and traction[6]. However, in the case of tracked vehicles, slippage occurswhile turning. Thus, for estimating the system’s positionand orientation, the conventional methods applied to wheeledmobile robots, when applied to tracked-based systems, do notproduce reliable results. In [7], gyro sensor and empiricalrelations between slip ratios are used for slip estimation. In

This work was supported by the project titled “Construction ofNon-Monotonic Lyapunov Function for the Dynamical Systems Gov-erned by Differential Inclusions:” Mathematical Research Impact Cen-tric Support (MATRICS) to the Science and Engineering Research Board(SERB), India, (2019–2021) Project’s under Grant MTR/2018/000799.1Department of Electrical Engineering, Indian Institute of Technol-ogy (BHU) Varanasi, India (email: yogita.choudhary.eee16@iitbhu.ac.in,nidhi.malhotra.eee16@iitbhu.ac.in, shyamkamal.eee@iitbhu.ac.in)

2 Department of Mining Engineering, Indian Institute of Technology (BHU)Varanasi, India (email: pratyushk.sahoo.min16@iitbhu.ac.in)

[8], a sliding mode observer is employed to estimate theslip. In [22], terramechanics models that model the interactionbetween the track and the terrain are used to represent slippagebehavior. In the aforementioned approaches, slip estimationmethods are required to consider the slippage effect, makingthe model-based approaches for exact system identificationintractable and time-consuming. Thus, the main motivation ofthis work was to simplify the model identification process fortrack-based systems.

Recently, data-driven based approaches are increasinglybeing used for identifying complex systems since they donot require any assumptions regarding the system behavioras opposed to modeling by first principles [9], [10], [11]. Inthis paper, we employ a data-driven approach to identify thesystem as it eliminates the need to model the external motionprofile (slip, turning resistance, etc.). Thus, by generatingsufficient data, the dynamics of the system are captured inthe input-output data. To the author’s best knowledge, a data-driven approach has not yet been studied to model track-basedsystems.

Figure 1: Hardware model of stair-climbing wheelchair

The core contributions of this work include: (i) the designof a track-based stair-climbing wheelchair by modeling thesystem in SolidWorks followed by a hardware implementa-tion as shown in Fig. 1, (ii) an implementation of a data-driven system identification approach based on the ObserverKalman filter Identification (OKID)/Eigensystem RealizationAlgorithm (ERA), and (iii) a heading correction algorithm toensure a safe traversal of the wheelchair.

The heading correction algorithm consists of: (i) a vision-based heading angle estimator, and (ii) a Linear QuadraticRegulator based optimal feedback controller. Fig. 2 shows

2021 IEEE/ASME International Conference onAdvanced Intelligent Mechatronics (AIM)

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Figure 2: Stair-climbing wheelchair control scheme

the block diagram of the control scheme. Also, to reducethe system’s cost, low-cost sensors, i.e., the Microsoft KinectXbox 360 (RGB-D camera) and an Inertial Measurement Unit(IMU), are used to collect information about the environment.Sensors such as LiDAR could increase the functionality andreliability but would have dramatically increased the cost andthus, were not chosen for this system. The control scheme isimplemented and verified for safe stair traversal in the ROS-Gazebo environment.

II. MECHANICAL DESIGN

The mechanical model of the system is designed in Solid-Works. It consists of a Track-based Mechanism and a SeatInclination Mechanism.

A. Track-based Mechanism

In the track-based mechanism, to ensure stability whileclimbing the staircase, the wheelchair tracks must be in contactwith atleast two stair edges throughout the ascend [6]. Thus,for a stair with height h and width w (Fig. 3(a)), the minimumlength L of the wheelchair’s track in contact with ground, asshown in Fig. 3(b) must satisfy the following relation:

L > 2√h2 + w2.

For increasing the traction with stairs while climbing, a three-pulley mechanism with variable angle of attack is designed.

B. Seat Inclination Mechanism

During the stair traversal, the seat of the wheelchair getsinclined at the angle of the staircase. Thus, for ensuring asafe and comfortable traversal, a seat inclination mechanismis designed so that the seat always remains horizontal withrespect to the ground. The mechanism consists of a linearactuator, IMU, and a slider-bar system. The two ends of thebars are connected to the sliders and the seat, respectively. Thecurrent seat angle is sensed using IMU, and the linear actuatormoves the slider resulting in a change in the seat’s inclination.The horizontal level is maintained through feedback controlfrom IMU. Fig. 4 displays the seat at two different inclinations;

(a) Stair Model (b) SolidWorks Model

Figure 3: Track-based Mechanism

one when it is horizontal and the other when it is at some anglewith respect to ground.

(a) Seat horizontal (b) Seat at some inclination

Figure 4: Seat Inclination Mechanism

III. VISION ALGORITHM

Exteroceptive sensors such as RGB-D cameras provide vi-sual information to estimate the relative pose of the wheelchairwith respect to the stair. In our approach, we use the Cannyedge detection algorithm [12] on both the RGB and Depthchannels. An adaptive parameter estimation technique de-scribed in [13] is used to calculate the thresholds for edgedetection. Further, we employ the Progressive Probabilistic

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Hough Transform (PPHT) [14] on the output of the edgedetection algorithm to identify lines present in both the RGBand Depth channels. PPHT is a compute-efficient algorithmthat detects line segments based on a voting strategy ondetected edge points. PPHT provides the initial line segmentestimates, and only line segments with zero or near zeroslopes are considered for processing. We further propose a linemerging algorithm that defines a merging strategy based oncertain constraints on the line parameters. The vision algorithmhas been described below.

Algorithm 1 Stair Edge Detection Algorithm

• Input: RGB (R) and Depth (D) image from the Kinectsensor as in Fig. 5(a).

• Calculate the lower and upper threshold dynamically forboth R and D. Apply the Canny edge detection algorithmwith the thresholds on R and D as in Fig. 5(b).

• Remove the edge points having an abrupt change in thedepth values to generate RGB edge image (ER) andDepth edge image (ED) as in Fig. 5(c).

• Apply Probabilistic Hough Transform on ER and ED togenerate a set of lines L. Conserve line segments havingnear zero slopes to get L

′as in Fig. 5(d).

• Merge collinear line segments, merge a pair of lineswhenever the difference between their slope is less than2 pixels and the difference between the offset is less than10 pixels to get L

′′. Calculate the frequency of slopes and

conserve line segments having the most frequent slope asin Fig. 5(e).

• For the set of end points X ′ and Y ′ in the image frameof line segments in L

′′, find (xmax, ymax) = (max(X ′),

max(Y ′)) and (xmin, ymin) = (min(X ′), min(Y ′)) todefine the bounding box on the stair as in Fig. 5(f).

(a) Depth Image (b) Edge Detection (c) Line Detection

(d) Noise removal (e) Line Merging (f) Localization

Figure 5: Overview of the algorithm

A. Heading Angle Estimation - RGB-D Camera

The final set consists of lines L′′

representing particular stairedges. Points present on a stair step would have similar depthvalue readings when a depth sensor is wholly aligned with the

stair. However, the same points would provide distinct depthvalues when the depth sensor is not aligned with the stair. Weutilize this depth difference to calculate the heading angle ofthe wheelchair with respect to the staircase. The x-coordinatein the world frame in centimeters can be found as follows

X = aD(X ′ − 320) (1)

where X denotes the x-coordinate of any point P in the worldframe, and X ′ is the x-coordinate of P in the image frame. Dis the depth value of point P obtained from the depth channelof Microsoft Kinect, and a > 0 is the adjustment parameter.The heading angle θ can be calculated as

θ = cos−1∆D

∆X(2)

where ∆X denotes the difference between the x-coordinatesfor the smallest line segment in the set L

′′having endpoints

P2 and P1, and ∆D represents the average depth differencevalue for the corresponding points on all the stair edges (Fig.6). The following equations are used to obtain ∆X and ∆D.

∆X = a(X2′D2 −X1

′D1 + 320(D1 −D2)) (3)

∆D =

∑Ni=1(D2(i) −D1(i))

N(4)

Here N denotes the total number of stair edges, and D2(i)

and D1(i) represent the depth values of points P2 and P1

respectively of the ith stair edge.

Figure 6: Top view of the wheelchair

B. Heading Angle Estimation - IMU

The IMU provides the orientation and angular velocity ofthe wheelchair with respect to a global reference frame. Inorder to ensure robust heading angle estimation, we outline astrategy to use the yaw measurements from the IMU sensoralong with the heading angle calculated from the visionalgorithm. We calculate the offset φ, which is defined as

φ = θ − γ (5)

where θ is the heading angle estimated by the vision algorithmand γ is the yaw measured by the IMU sensor. The offset isupdated when the θ and γ are synchronized. In run time, weuse γ and φ in case of an aberration in θ estimated by thevision algorithm.

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IV. DATA-DRIVEN SYSTEM IDENTIFICATION

In this work, we use a data-driven system identificationapproach instead of employing model-based approaches tosimplify the tedious process of modeling the complex dynam-ics of the track-based wheelchair. The OKID/ERA approachcan be used for identifying linear systems with random input-output data [9], [10]. We assume that the system can beapproximated as a linear system for small heading angles onstairs and is controllable and observable. Thus, we employthe OKID/ERA approach. The ERA is based on the minimalrealization theory proposed by Ho and Kalman, where lowdimensional system identification can be done based on themeasured outputs of the impulse response [15]. The discrete-time system can be expressed in state-space form as

xk+1 = Axk +Buk (6a)

yk = Cxk +Duk (6b)

where x denotes the state vector, y denotes the measuredoutput vector, u is the control input, k is the time step, and A,B, C, D are the discrete-time system matrices. Consideringan unit impulse input and zero initial states, the output Yk(Markov parameter) can be represented as

Y0 = D (7a)

Yk = CAk−1B. (7b)

The time-shifted response for n measurements are stored inthe generalized Hankel Matrix H and in another shifted matrixH ′ as in (8), where m = bn−12 c.

H =

Y1 Y2 .. YmY2 Y3 .. Ym+1

.. .. .. ..Ym Ym+1 .. Yn−1

(8a)

H ′ =

Y2 Y3 .. Ym+1

Y3 Y4 .. Ym+2

.. .. .. ..Ym+1 Ym+2 .. Yn

(8b)

Since the data consists of noisy measurements, the SingularValue Decomposition (SVD) of the H matrix is taken totruncate the matrix and obtain the rank of the model [16]. TheA,B, and C matrices are obtained based on the truncated Hand the H ′ matrices. A detailed explanation of the method canbe found in [17].

The ERA requires impulse response experiments which arechallenging to perform in practice. Thus, OKID can be usedprior to ERA to observe the system’s impulse response withrandomized inputs [18]. For a random input with zero initialstates, the output vector can be given as in (9a) and (9b),and the output response matrix as in (9c) can be denoted asa product of the matrix with impulse response measurementsand the matrix containing control inputs. Solving (9c) givesthe original Markov parameters which can be used to constructthe Hankel matrix H in the ERA [17].

y0 = Du0 (9a)

yk =

k∑i=1

CAi−1Buk−i +Duk (9b)

[y0 y1 .. ym

]=

[Y0 Y1 .. Ym

] u0 u1 .. um0 u0 .. um−10 .. .. ..0 0 .. u0

(9c)

V. HEADING CONTROL

The main objective of the heading controller is to reduce theheading error of the system approximately to zero to ensurea safe traversal on stairs. Following system identification, aclosed-loop feedback controller is designed to control theheading angle of the system. For a linear system, severalmethods can be used for control design [23]. For tuning thegains of the feedback controller, we use the Linear QuadraticRegulator (LQR), which is a widely used design technique thatprovides optimal feedback gains for the system. In the LQR,the quadratic cost function J(x, u) is to be minimized.

J(x, u) =

∞∑k=0

(xkTQxk + uk

TRuk) (10)

The positive semi-definite matrix Q is the cost of the transientstate, and the positive definite matrix R is the cost for thecontrol input of proper dimension. The control input vectoruk is set as in (11) where K is the proper gain matrix.

uk = −Kxk (11a)

xk = [C −DK]−1yk (11b)

VI. SIMULATION AND RESULTS

A. ROS-Gazebo SimulationThe Robot Operating System (ROS) is a set of open-

source software packages used to program robots [24] andsimulate them in Gazebo, a simulator that enables the creationof realistic scenarios. The Computer-Aided Design (CAD)model designed using SolidWorks is imported in the UniversalRobotic Description Format (URDF) using the SW2URDFplugin. For modeling a track-based geometry in Gazebo, [19]proposes an approach where the tracks are modeled as a set offake wheels moving at the same speed. However, this methodfails on the stair nose while ascending due to unrealisticfriction that arises between the gaps of fake wheels which arenot present in real tracks. Ref. [20] presents a computationallyefficient method for simulating non-deformable tracks and isavailable as an open-source gazebo model plugin. We create aROS wrapper for this gazebo plugin to interact with otherROS nodes [21]. Fig. 7 shows the ROS node architecturewherein the /imu topic publishes the pitch of the wheelchairused for closed-loop feedback control of the seat inclinationmechanism. Further, for heading control, the /imu topicpublishes the yaw and angular velocity of the system, andthe /camera topic publishes the image which is processed bythe /depth node for providing the heading angle.

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Figure 7: ROS Node architecture

B. System Identification and Control

For identifying the A, B, C, and D matrices for oursystem, we consider a two-input two-output system where thecontrol inputs u1 and u2 are the left and right track velocities,respectively, and the measured outputs are the heading angle(θ) and the angular velocity of the system (ω). The θ isestimated as given in Section III and ω is measured by theIMU. We provide a set of random velocities to the system onstairs in the ROS-Gazebo environment. The measured outputsfor 1000 random inputs are stored and using the OKID/ERA,the impulse response of the system is observed (Fig. 8).From the SVD of the H matrix generated with the obtainedmeasurements, it is observed that the system’s rank r = 2. Thecalculated system matrices are

A =

[0.8605 0.1844−0.2944 0.3492

], B =

[0.0334 0.14740.0740 0.2090

]

C =

[0.1087 −0.05000.1498 0.1459

], D =

[−0.0501 −0.04900.0160 0.0416

](12)

10 20 30 40 50 60

Time (s)

-0.05

0

0.05

(ra

d)

10 20 30 40 50 60

Time (s)

-0.05

0

0.05

(ra

d/s

)

10 20 30 40 50 60

Time (s)

0

0.01

0.02

(ra

d)

10 20 30 40 50 60

Time (s)

0

0.02

0.04

0.06

(ra

d/s

)

Impulse given to input u2

Impulse given to input u1

Impulse given to input u2Impulse given to input u

1

Figure 8: Impulse Response of the Identified System

The system identification is followed by a discrete timefeedback controller design. For the wheelchair to traverse thestairs safely, the system is operated at relatively low velocities.Thus, the cost of the R matrix is kept such that the desiredinputs, i.e., the velocities of the left and right tracks, do notexceed the maximum speed, i.e., 1m/s. For the two-inputtwo-output system with a sampling time Ts = 1s, Q and R

matrices are set as in (13), and the dlqr function in MATLABis used for obtaining the feedback gain matrix K.

Q =

[1 00 1

], R =

[0.05 0

0 0.05

](13)

K =

[0.0239 0.28681.6589 0.9201

](14)

C. Verification during Staircase traversal

The performance of the system is verified in the Gazebosimulator for system parameters given in Table I.

Table I: Simulation Parameters

Parameters ValueMass of the system 23.4 kgMass of the payload 50 kg

Coefficient of friction∗ µ = 0.62, µ2 = 0.5Staircase angle (β) 25◦

vsystem(max), ωsystem(max) 1 m/s, 3.07 rad/sfreqimu, freqcamera, freqcontrol 100 Hz, 10 Hz, 4 Hz

∗considered between rubber and wooden surface

The controller’s efficacy is verified by considering thefollowing cases.

1) Heading Control (θinitial = 0◦): For stair traversal withθinitial = 0◦ for a time duration of 300s, θ is compared in thecase of open-loop and closed-loop operation. Fig. 9 showsthat θ increases gradually for the open-loop system. This isbecause, in case of any disturbance or non-ideal operation,even a small heading increases progressively with time due toadded gravitational torque experienced by the system. Thus,the system requires a closed-loop operation, for which theheading is maintained at near zero angles.

Figure 9: Heading during stair traversal for θinitial = 0◦

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2) Heading Control (θinitial = ±30◦): In case the systemis at a non-zero heading angle on the staircase, it is trackedto θ ≈ 0◦ to ensure a safe traversal. Fig. 10 shows the closed-loop control for θinitial = ±30◦, for a staircase angle (β =28◦). Table II shows the root mean square heading error fordifferent staircase angles (β).

20 40 60 80

Time (s)

-10

0

10

20

30

(degre

es)

20 40 60 80

Time (s)

-30

-20

-10

0

10

(degre

es)

(a) (b)

Figure 10: (a) Control in anticlockwise direction (θinitial =30◦) (b) Control in clockwise direction (θinitial = -30◦)

Table II: Heading Error

β 25◦ 28◦ 30◦ Avg.Anticlockwise 2.83◦ 1.89◦ 4.95◦ 3.22◦

Clockwise 3.23◦ 1.37◦ 1.29◦ 1.96◦

3) Seat Inclination Control: The seat inclination control istested for the hardware system (Fig. 1) on a stair with β = 28◦.Fig. 11 depicts that the seat achieves horizontal inclinationwith respect to the ground using proportional control.

10 20 30 40

Time (s)

0

10

20

30

Pitch (

degre

es)

Figure 11: Seat inclination control for hardware model

VII. CONCLUSION

This study proposes a novel approach for modeling atrack-based stair-climbing wheelchair. Unlike the conventionalapproach governed by the laws of physics, we model thesystem using a data-driven approach. We design a low-costsystem employing affordable sensors. A comfortable and safetraversal is ensured by designing a heading correction strategyand seat inclination control mechanism. The correction strat-egy integrates vision-based localization, data-driven systemidentification using OKID/ERA, and an optimal LQR headingcontroller. A ROS-based architecture is implemented, and thesystem is tested for different staircase inclinations (upto 30◦)in the Gazebo environment. It is found that the proposedapproach is successful in maintaining the wheelchair’s headingperpendicular to the stairs throughout the stair traversal. The

data-driven approach discussed in this work can also beextended to other applications involving track-based systemssuch as search and rescue operations for navigation in un-known environments.

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