# Enhancing the Trajectory Generation of a Stair-Climbing Mobility System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental System

#### 2.1. Mechanical Description

#### 2.2. Operating Modes

#### 2.3. Laser Distance Sensors for Environment Recognition

## 3. Kinematics and Dynamics Modeling

#### 3.1. Kinematics Modeling

**f**(${\theta}_{4}$) and

**f**(${\theta}_{3}$), where ${\theta}_{4}$ is the movement of the front wheels and ${\theta}_{3}$ is the movement of the rear wheels (the driving wheels). The positions of the front and rear sliding supports are denoted as ${z}_{1}$ and ${z}_{2}$. They form the angles ${\delta}_{1}$ and ${\delta}_{2}$ with the imaginary axis, respectively. ${\theta}_{1}$, ${\theta}_{2}$, ${\theta}_{3}$ and ${\theta}_{4}$ are rotational DOFs and ${z}_{1}$ and ${z}_{2}$ are translational DOFs. The actuated DOFs can be joined to the vector $\mathbf{q}={\left[{\theta}_{1},{\theta}_{2},{\theta}_{3},{z}_{1},{z}_{2}\right]}^{T}$. The reference trajectories for the vectors $\mathbf{p}$ and $\mathbf{q}$ are defined as ${\mathbf{p}}^{*}={\left[{\mathbf{P}}_{g}^{*T},{\gamma}^{*}\right]}^{T}$ and ${\mathbf{q}}^{*}={\left[{\theta}_{1}^{*},{\theta}_{2}^{*},{\theta}_{3}^{*},{z}_{1}^{*},{z}_{2}^{*}\right]}^{T}$, respectively.

#### 3.2. Dynamics Modeling

**q**rather than

**r**, i.e.,

## 4. Definition of the Trajectory Generator

- A function $Tran\left(\mathit{\Im}\right(t\left)\right)$ is used to indicate whether it is possible to change the configuration of the SCMS:$$Tran\left(\mathit{\Im}\right(t\left)\right)=true$$
- Bounds on the SCMS configurations:$${\mathit{\Im}}_{min}\le \mathit{\Im}\left(t\right)\le {\mathit{\Im}}_{max}$$These bounds make it possible for the SCMS to surpass stairs of different sizes and to maintain an inclination with regard to the direction of gravity of its chassis ($\gamma =0$).
- Bounds on the actuator velocities and accelerations:$${\dot{{\theta}_{i}}}_{min}\le \dot{{\theta}_{i}}\left(t\right)\le {\dot{{\theta}_{i}}}_{max},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}3$$$${\ddot{{\theta}_{i}}}_{min}\le \ddot{{\theta}_{i}}\left(t\right)\le {\ddot{{\theta}_{i}}}_{max},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}3$$$${\dot{{z}_{i}}}_{min}\le \dot{{z}_{i}}\left(t\right)\le {\dot{{z}_{i}}}_{max},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}2$$$${\ddot{{z}_{i}}}_{min}\le \ddot{{z}_{i}}\left(t\right)\le {\ddot{{z}_{i}}}_{max},\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}2$$
- Bounds on the comfort velocities and accelerations:$${\dot{\gamma}}_{min}\le \dot{\gamma}\left(t\right)\le {\dot{\gamma}}_{max}$$$${\ddot{\gamma}}_{min}\le \ddot{\gamma}\left(t\right)\le {\ddot{\gamma}}_{max}$$
- A function $Col\left(\mathit{\Im}\right(t\left)\right)$ is also used to indicate whether or not a given wheel or sliding support of the SMCS is accidentally colliding with an obstacle (Information regarding the environment is obtained by using the strategy designed to estimate the size and shape of the stairs, which was presented in Section 2.3. In this case, knowledge about the environment allows the proposed prototype to determine whether or not a collision will occur):$$Col\left(\mathit{\Im}\right(t\left)\right)=false$$

- Generate $\mathbf{P}\left(\upsilon \right)$, $\upsilon \in [0,\phantom{\rule{3.33333pt}{0ex}}1]$ (Figure 13a) by means of a quintic B-spline model. The fifth degree ensures the continuity of the second order derivatives of angles ${\theta}_{1},{\theta}_{2},{\theta}_{3},{z}_{1},$ and ${z}_{2}$. This B-spline is built by using a set of ${N}_{p}$ control points (five points at least), and is generated within the admissible workspace defined by (37) and by accounting for constraints (33) and (44).
- Build the motion profile $\upsilon \left(t\right)$ on the interval $[0,\phantom{\rule{3.33333pt}{0ex}}T]$ using a quintic B-spline model (Figure 13b), generated by ${N}_{m}$ control points that are uniformly distributed throughout the time scale, and by accounting for the following boundary conditions:$$\upsilon \left(0\right)=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dot{\upsilon}\left(0\right)=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\ddot{\upsilon}\left(0\right)=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{\u20db}{\upsilon}\left(0\right)=0$$$$\upsilon \left(T\right)=1\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\dot{\upsilon}\left(T\right)=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\ddot{\upsilon}\left(T\right)=0\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{\u20db}{\upsilon}\left(T\right)=0\phantom{\rule{-2.84526pt}{0ex}}$$

#### Optimization and the SCMS’s Control Scheme

## 5. Experimental Results

#### 5.1. Experimental Setup

#### 5.2. Results

#### 5.2.1. First Experiment

#### 5.2.2. Second Experiment

## 6. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Wellman, P.; Krovi, V.; Kumar, V.; Harwin, W. Design of a wheelchair with legs for people with motor disabilities. IEEE Trans. Rehabil. Eng.
**1995**, 3, 343–353. [Google Scholar] [CrossRef] - Hashimoto, K.; Hosobata, T.; Sugahara, Y.; Mikuriya, Y.; Sunazuka, H.; Kawase, M.; Lim, H.O.; Takanishi, A. Realization by biped leg-wheeled robot of biped walking and wheel-driven locomotion. ICRA
**2005**, 2970–2975. [Google Scholar] [CrossRef] - Chen, C.-H.; Pham, C.H. Design and fabrication of a statically stable stair-climbing robotic wheelchair. Ind. Robot Int. J.
**2009**, 36, 562–569. [Google Scholar] [CrossRef] - Yuan, J.; Paisley, R.; Song, Y.; Zhang, W. Virtual realization of automatic stair-climbing motion by leg-wheeled hybrid mobile robot. IEEE ROBIO
**2010**, 1352–1357. [Google Scholar] [CrossRef] - Lee, C.H.; Lee, K.M.; Yoo, J.; Kim, I.S.; Bang, Y.B. A compact stair-climbing wheelchair with two 3-DOF legs and a 1-DOF base. Ind. Robot Int. J.
**2016**, 43, 181–192. [Google Scholar] [CrossRef] - Tomo Co. Ltd.; Tamagawa University. Freedom. Available online: www.tomo-co.co.jp/free.htm (accessed on 19 January 2017).
- Sugahara, Y.; Yonezawa, N.; Kosuge, K. A novel stair-climbing wheelchair with transformable wheeled four-bar linkages. IEEE/RSJ IROS
**2010**, 3333–3339. [Google Scholar] [CrossRef] - Independence Technology, L.L.C. The ibot Website. Available online: urlm.co/www.ibotnow.com (accessed on 17 September 2017).
- The Company TopChair. Available online: www.topchair.fr/ (accessed on 17 September 2017).
- Galileo Mobility Instruments Ltd. The Galileo Mobility Website. Available online: www.galileomobility.com (accessed on 17 September 2017).
- Quaglia, G.; Nisi, M.; Franco, W.; Bruzzone, L. Dynamic simulation of an electric stair-climbing wheelchair. IJAT
**2017**, 11, 472–480. [Google Scholar] [CrossRef] - Creative DNA Austria. Available online: www.creativednaaustria.com/ (accessed on 17 September 2017).
- Leaman, J.; La, H.M. A comprehensive review of smart wheelchairs: Past, present and future. arXiv, 2017; arXiv:1704.04697. [Google Scholar]
- Sundaram, S.A.; Wang, H.; Ding, D.; Cooper, R.A. Step-climbing power wheelchairs: A literature review. Top. Spinal Cord Inj. Rehabil.
**2017**, 23, 98–109. [Google Scholar] [CrossRef] - Bang, Y.; Lee, C.; Yoo, J.; Lee, K.; Kim, I. Two-legged stair-climbing wheelchair and its stair dimension measurement using distance sensors. In Proceedings of the 2011 11th International Conference on Control, Automation and Systems (ICCAS), Gyeonggi-do, Korea, 26–29 October 2011; pp. 1788–1791. [Google Scholar]
- Yamamoto, T.; Takemori, F.; Itakura, R. Development of stair locomotive wheelchair with adjustable wheelbase. In Proceedings of the 2012 SICE Annual Conference (SICE), Akita, Japan, 20–23 August 2012; pp. 629–633. [Google Scholar]
- Solea, R.; Nunes, U. Robotic wheelchair trajectory control considering user comfort. ICINCO
**2009**, 113–125. [Google Scholar] [CrossRef] - Huang, J.J.; Chiang, H.H.; Lee, T.T.; Kou, K.Y. Riding comfort improvement by considering passenger’s behavior suppression on powered wheelchairs. IEEE ICSSE
**2012**, 131–136. [Google Scholar] [CrossRef] - Chocoteco, J.A.; Morales, R.; Feliu, V. Improving the climbing/descent performance of stair-climbing systems confronting architectural barriers with geometric dusturbances. Mechatronics
**2015**, 30, 11–26. [Google Scholar] [CrossRef] - Morales, R.; Feliu, V.; González, A. Optimized obstacle avoidance trajectory generation for a reconfigurable staircase climbing wheelchair. Robot. Auton. Syst.
**2010**, 58, 97–114. [Google Scholar] [CrossRef] - Chocoteco, J.A.; Morales, R.; Feliu, V.; Sira-Ramírez, H. Robust control for the trajectory tracking of robotic wheelchairs. Robotica
**2015**, 33, 41–59. [Google Scholar] [CrossRef] - Morales, R.; Sira-Ramírez, H.; Somolinos, J.A. Robust control of underactuated wheeled mobile manipulators using GPI disturbance observers. Multibody Syst. Dyn.
**2015**, 32, 511–533. [Google Scholar] [CrossRef] - Morales, R.; Feliu, V.; González, A.; Pintado, P. Coordinated motion of a new staircase climbing wheelchair with increased passenger comfort. In Proceedings of the 2006 IEEE Internacional Conference on Robotics and Automation (ICRA 2006), Orlando, FL, USA, 15–19 May 2006; pp. 3995–4001. [Google Scholar]
- Leishman, F.; Monfort, V.; Horn, O.; Bourhis, G. Driving assistance by deictic control for a smart wheelchair: The assessment issue. IEEE Trans. Hum. Mach. Syst.
**2014**, 44, 66–77. [Google Scholar] [CrossRef] - Morales, Y.; Abdur-Rahim, J.A.; Even, J.; Watanabe, A.; Kondo, T.; Hagita, N.; Ishii, S. Modeling of human velocity habituation for a robotic wheelchair. IEEE/RSJ IROS
**2014**, 3284–3290. [Google Scholar] [CrossRef] - Wu, B.F.; Chen, P.Y.; Lin, C.H. A new criterion of human comfort assessment for wheelchair robots by Q-Learning based accompanist tracking fuzzy controller. Int. J. Fuzzy Syst.
**2016**, 18, 1039–1053. [Google Scholar] [CrossRef] - Wolf, E.; Cooper, R.A.; Pearlman, J.; Fitzgerald, S.G.; Kelleher, A. Longitudinal assessment of vibrations during manual and power wheelchair driving over select sidewalk surfaces. J. Rehabil. Res. Dev.
**2007**, 44, 573–580. [Google Scholar] [CrossRef] [PubMed] - Garcia-Mendez, Y.; Pearlman, J.L.; Boninger, M.L.; Cooper, R.A. Health risks of vibration exposure to wheelchair users in the community. J. Spinal Cord Med.
**2013**, 36, 365–375. [Google Scholar] [CrossRef] [PubMed] - Chénier, F.; Aissaoui, R. Effect of wheelchair frame material on users’ mechanical work and transmitted vibration. Biomed. Res. Int.
**2014**, 2014, 1–13. [Google Scholar] [CrossRef] [PubMed] - Hirata, K.; Murakami, T. Assisted control of step passage motion and its comfort evaluation in two-wheel wheelchair systems. EEJ
**2015**, 192, 50–58. [Google Scholar] [CrossRef] - Hobson, D.A.; Crane, B. State of the science white paper on: Wheelchair seating comfort. In Proceedings of the Conference on Seating Issues for Persons with Disabilities, Orlando, FL, USA, 19–20 February 2001; pp. 29–33. [Google Scholar]
- Geyer, M.J.; Brienza, D.M.; Bertocci, G.E.; Crane, B.; Hobson, D.; Karg, P.; Schmeler, M.; Trefler, E. Wheelchair seating: A state of the science report. Assist. Technol.
**2003**, 15, 120–128. [Google Scholar] [CrossRef] [PubMed] - Shino, M.; Tomokuni, N.; Murata, G.; Segawa, M. Wheeled inverted pendulum type robotic wheelchair with integrated control of seat slider and rotary link between wheels for climbing stairs. IEEE ARSO
**2014**, 121–126. [Google Scholar] [CrossRef] - Dobrzyński, G.; Choromański, W.; Grabarek, I. Analysis of ride comfort on the stairs climbing wheelchair. In Advances in Human Aspects of Transportation: Part II; Springer: Berlin, Germany, 2014; Volume 8, pp. 114–125. [Google Scholar]
- Cox, M.G. The numerical evaluation of B-splines. IMA J. Appl. Math.
**1972**, 10, 134–149. [Google Scholar] [CrossRef] - De Boor, C. A Practical Guide to Splines; Springer: New York, NY, USA, 1978; p. 325. [Google Scholar]
- Balluff Company. Photoelectric Distance Sensor, BOD 63M-LA04-S115. Available online: http://www.balluff.com/balluff/MUS/en/products/product_detail.jsp#/152994 (accessed on 22 October 2017).
- Chocoteco, J.; Morales, R.; Feliu, V.; Sánchez, L. Trajectory planning for a stair-climbing mobility system using laser distance sensors. IEEE Syst. J.
**2016**, 10, 944–956. [Google Scholar] [CrossRef] - Morales, R.; Gonzalez, A.; Feliu, V.; Pintado, P. Environment adaptation of a new staircase-climbing wheelchair. Auton. Robots
**2007**, 23, 275–292. [Google Scholar] [CrossRef] - Morales, R.; Gonzalez, A.; Feliu, V.; Pintado, P. Kinematic model of a new staircase climbing wheelchair and its experimental validation. Int. J. Robot. Res.
**2006**, 25, 825–841. [Google Scholar] [CrossRef] - Gonzalez, A.; Morales, R.; Feliu-Batlle, V.; Pintado, P. Improving the mechanical design of new staircase wheelchair. Ind. Robot Int. J.
**2007**, 34, 110–115. [Google Scholar] [CrossRef] - Morales, R.; Somolinos, J.A.; Cerrada, J.A. Dynamic model of a reconfigurable stair-climbing mobility system and its experimental validation. Multibody Syst. Dyn.
**2012**, 28, 349–367. [Google Scholar] [CrossRef] - Morales, R.; Somolinos, J.A.; Cerrada, J.A. Dynamic control of a reconfigurable stair-climbing mobility system. Robotica
**2013**, 31, 295–310. [Google Scholar] [CrossRef] - Morales, R.; Chocoteco, J.; Feliu, V.; Sira-Ramírez, H. Obstacle surpassing and posture control of a stair-climbing robotic mechanism. Control Eng. Pract.
**2013**, 21, 604–621. [Google Scholar] [CrossRef] - National Instruments Corp.: LabVIEW 2016 Help. B-Spline Fit VI. Available online: zone.ni.com/reference/en-XX/help/371361N-01/gmath/bspline_fit/ (accessed on 17 September 2017).
- National Instruments Corp.: LabVIEW 2016 Help. Constrained Nonlinear Optimization VI. Available online: zone.ni.com/reference/en-XX/help/371361L-01/gmath/constrained_nonlinear_optimization/ (accessed on 17 September 2017).
- Foo, J.L.; Knutzon, J.; Kalivarapu, V.; Oliver, J.; Winer, E. Path planning of unmanned aerial vehicles using B-splines and particle swarm optimization. J. Aerosp. Comput. Inf. Commun.
**2009**, 6, 271–290. [Google Scholar] [CrossRef] - Besada-Portas, E.; De la Torre, L.; Jesus, M.; De Andrés-Toro, B. Evolutionary trajectory planner for multiple UAVs in realistic scenarios. IEEE Trans. Robot.
**2010**, 10, 619–634. [Google Scholar] [CrossRef] - Zhao, S.; Wang, X.; Zhang, D.; Shen, L. Model predictive control based integral line-of-sight curved path following for unmanned aerial vehicle. In Proceedings of the 3rd AIAA Guidance, Navigation, and Control Conference, Grapevine, TX, USA, 9–13 January 2017; p. 1511. [Google Scholar] [CrossRef]
- Connors, J.; Elkaim, G. Manipulating B-Spline based paths for obstacle avoidance in autonomous ground vehicles. ION NTM
**2007**, 5, 1081–1088. [Google Scholar] - Berglund, T.; Brodnik, A.; Jonsson, H.; Staffanson, M.; Soderkvist, I. Planning smooth and obstacle-avoiding B-spline paths for autonomous mining vehicles. IEEE Trans. Autom. Sci. Eng.
**2010**, 7, 167–172. [Google Scholar] [CrossRef] - Gu, T.; Snider, J.; Dolan, J.M.; Lee, J. Focused trajectory planning for autonomous on-road driving. IEEE Intell. Veh. Symp.
**2013**, 547–552. [Google Scholar] [CrossRef] - Elbanhawi, M.; Simic, M.; Jazar, R.N. Continuous path smoothing for car-like robots using B-spline curves. J. Intell. Robot. Syst.
**2015**, 80, 23–56. [Google Scholar] [CrossRef] - Zeng, Q.; Burdet, E.; Rebsamen, B.; Teo, C.L. Evaluation of the collaborative wheelchair assistant system. IEEE ICORR
**2007**, 601–608. [Google Scholar] [CrossRef] - Rebsamen, B.; Burdet, E.; Guan, C.; Zhang, H.; Teo, C.L.; Zeng, Q.; Laugier, C.; Ang, M.H., Jr. Controlling a wheelchair indoors using thought. IEEE Intell. Syst.
**2007**, 22, 18–24. [Google Scholar] [CrossRef] - Zhou, L.; Teo, C.L.; Burdet, E. A nonlinear elastic path controller for a robotic wheelchair. IEEE ICIEA
**2008**, 142–147. [Google Scholar] [CrossRef] - Carlson, T.; Demiris, Y. Collaborative control for a robotic wheelchair: Evaluation of performance, attention, and workload. IEEE Trans. Syst. Man Cybern. B
**2012**, 42, 876–888. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Ammouri, A.H.; Hamade, R.F. On the selection of constitutive equation for modeling the friction stir processes of twin roll cast wrought AZ31B. Mater. Des.
**2014**, 57, 673–688. [Google Scholar] [CrossRef] - Yoon, S.H.; Park, C.S.; Augenbroe, G.; Kim, D.W. Self-calibration and optimal control of a double-skin system. In Proceedings of the 11th IBPSA Conference (International Building Performance Simulation Association), Glasgow, UK, 27–30 July 2009; pp. 80–87. [Google Scholar]
- Kim, J.H.; Woo, L.D.; Cho, K.R.; Jo, S.Y.; Kim, J.H.; Min, C.; Han, D.; Cho, S. Development of an electro-optical system for small UAV. Aerosp. Sci. Technol.
**2010**, 14, 505–511. [Google Scholar] [CrossRef] - Chica, J.A.V.; Torres, A.G.D.; Acosta, D.A. NMPC controller applied to the operation of an internal combustion engine: Formulation and solution of the optimization problem in real time. IJIDeM
**2016**, 1–22. [Google Scholar] [CrossRef]

**Figure 1.**The proposed Stair-Climbing Mobility System (SCMS) is characterized by the fact that it can climb up and down stairs of different sizes and maintain a stable equilibrium on the stair at all times without assistance. It is also characterized by the fact that it can estimate the size of the stair using laser distance sensors and generate safe and comfortable trajectories for the central platform when carrying a user.

**Figure 2.**Kinematic scheme of the SCMS: (1) positioning mechanism, which is responsible for ensuring the posture of the entire vehicle; (2) climbing mechanism, which is responsible for surpassing the obstacle.

**Figure 3.**Actuating sequence of the rear climbing mechanism. . In (

**a**) the wheel makes contact with the ground. In (

**b**) the sliding support slides and makes contact with the step tread and the wheel is off the ground. In (

**c**) the screw-nut mechanism is again actuated and the sliding support continues to slide. In (

**d**) the sliding support is completely deployed. In (

**e**) the screw-nut mechanism is actuated and the wheel moves backwards to its initial point. In (

**f**) the wheel again makes contact with the ground and the sliding support is retracted.

**Figure 6.**Side view of scanning on the staircase surface; (

**a**) before climbing up and (

**b**) before climbing down.

**Figure 12.**Definition of the generalized coordinate variables for each of the configurations of the SCMS.

**Figure 16.**Position profiles of the center of mass of the SCMS used in the (

**a**) first and (

**b**) second experiments.

**Figure 18.**The two main Virtual Instruments (VIs) used to solve the optimization problem; (

**a**) the B-Spline Fit VI and (

**b**) the Constrained Nonlinear Optimization VI.

**Figure 19.**Sequence of climbing process when it surpasses a three-step staircase. All possible configurations of the SCMS are included; Configuration 1 in (

**a**,

**c**,

**e**,

**i**,

**l**); Configuration 2 in (

**b**,

**d**,

**f**); Configuration 3 in (

**h**,

**j**,

**k**); and Configuration 4 in (

**g**). See the video referenced in the Supplementary Materials.

**Figure 20.**Sequence of descent process when it surpasses a three-step staircase. All possible configurations of the SCMS are included; Configuration 1 in (

**a**,

**c**,

**e**,

**i**,

**l**); Configuration 2 in (

**h**,

**j**,

**k**); Configuration 3 in (

**b**,

**d**,

**f**); and Configuration 4 in (

**g**). See the video referenced in the Supplementary Materials.

**Figure 21.**Inclination of the SCMS frame ($\gamma $) when the optimization is carried out with a $\alpha $ equal to zero (—) and when the optimization is not considered (—).

**Figure 22.**Inclination of the SCMS frame ($\gamma $) when the optimization is carried out with a $\alpha $ equal to 0.5 (—) and when the optimization is not considered (—).

**Figure 23.**Inclination of the SCMS frame ($\gamma $) when the optimization is carried out with a $\alpha $ equal to one (—) and when the optimization is not considered (—).

**Figure 24.**Behavior of the angular position of the driving wheels (${\theta}_{3}$) when the optimization is carried out with $\alpha =0$ (—), $\alpha =0.5$ (—) and $\alpha =1$ (—) and when the optimization is not considered (—).

**Figure 25.**Behavior of ${\theta}_{1}$ when the optimization is carried out with $\alpha =0$ (—), $\alpha =0.5$ (—) and $\alpha =1$ (—) and when the optimization is not considered (—).

**Figure 26.**Path of the center of mass (${\mathbf{P}}_{g}$) when the optimization is carried out with $\alpha $ equal to zero (—) and when the optimization is not considered (—).

**Figure 27.**Inclination of the SCMS frame ($\gamma $) when the optimization is carried out with a $\alpha $ equal to zero (—) and when the optimization is not considered (—).

**Figure 28.**Inclination of the SCMS frame ($\gamma $) when the optimization is carried out with a $\alpha $ equal to 0.5 (—) and when the optimization is not considered (—).

**Figure 29.**Inclination of the SCMS frame ($\gamma $) when the optimization is carried out with a $\alpha $ equal to one (—) and when the optimization is not considered (—).

**Figure 30.**Behavior of the angular position of the driving wheels (${\theta}_{3}$) when the path is optimized with $\alpha =0$ (—), $\alpha =0.5$ (—) and $\alpha =1$ (—) and when the optimization is not considered (—).

**Figure 31.**Behavior of the angular position (${\theta}_{1}$) when the path is optimized with $\alpha =0$ (—), $\alpha =0.5$ (—) and $\alpha =1$ (—) and when the optimization is not considered (—).

**Figure 32.**Behavior of the angular position (${\theta}_{2}$) when the path is optimized with $\alpha =0$ (—), $\alpha =0.5$ (—) and $\alpha =1$ (—) and when the optimization is not considered (—).

**Figure 33.**Path of the center of mass (${\mathbf{P}}_{g}$) when the optimization is carried out with $\alpha $ equal to zero (—) and when the optimization is not considered (—).

Configurations | Expressions | |
---|---|---|

1 (see Figure 11a) | ${\mathbf{P}}_{g}=\mathbf{f}\left({\theta}_{3}\right)+{l}_{6}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{6})}+{l}_{4}{\mathrm{e}}^{j(\gamma +\frac{3\pi}{2}-{\theta}_{2})}+{l}_{5}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{5})}$ | (1) |

${\mathbf{P}}_{g}=\mathbf{f}\left({\theta}_{4}\right)+{l}_{1}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{1})}-{l}_{3}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\theta}_{1})}+{l}_{5}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{5})}$ | (2) | |

2 (see Figure 11b) | ${\mathbf{P}}_{g}={\mathbf{P}}_{C2}+{z}_{2}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}-{\delta}_{2})}+{l}_{6}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{6})}+{l}_{4}{\mathrm{e}}^{j(\gamma +\frac{3\pi}{2}-{\theta}_{2})}+{l}_{5}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{5})}$ | (3) |

Equation (2) | ||

3 (see Figure 11c) | Equation (1) | |

${\mathbf{P}}_{g}={\mathbf{P}}_{C1}+{z}_{1}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}-{\delta}_{1})}+{l}_{1}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{1})}-{l}_{3}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\theta}_{1})}+{l}_{5}{\mathrm{e}}^{j(\gamma +\frac{\pi}{2}+{\mu}_{5})}$ | (4) | |

4 (see Figure 11d) | Equation (3) | |

Equation (4) |

**Table 2.**Terms of differential equations of the implicit Jacobian for the four configurations (where $sin(\xb7)\equiv {S}_{(\xb7)}$ and $cos(\xb7)\equiv {C}_{(\xb7)}$).

k | $\frac{\mathit{\partial}{\mathit{F}}_{\mathit{k}}}{\mathit{\partial}\mathit{\gamma}}$ | $\frac{\mathit{\partial}{\mathit{F}}_{\mathit{k}}}{\mathit{\partial}{\mathit{\theta}}_{1}}$ | $\frac{\mathit{\partial}{\mathit{F}}_{\mathit{k}}}{\mathit{\partial}{\mathit{\theta}}_{2}}$ | g${}_{\mathit{k}}$($\mathit{q},\dot{\mathit{q}}$) |
---|---|---|---|---|

1 | ${l}_{1}{S}_{(\gamma +{\mu}_{1})}-{l}_{6}{S}_{(\gamma +{\mu}_{6})}+{l}_{4}{S}_{(\gamma -{\theta}_{2})}-{l}_{3}{S}_{(\gamma +{\theta}_{1})}$ | $-{l}_{3}{S}_{(\gamma +{\theta}_{1})}$ | $-{l}_{4}{S}_{(\gamma -{\theta}_{2})}$ | |

2 | ${l}_{1}{S}_{(\gamma +{\mu}_{1})}-{l}_{6}{S}_{(\gamma +{\mu}_{6})}+{l}_{4}{S}_{(\gamma -{\theta}_{2})}-{l}_{3}{S}_{(\gamma +{\theta}_{1})}-{z}_{2}{S}_{(\gamma -{\delta}_{2})}$ | $-{l}_{3}{S}_{(\gamma +{\theta}_{1})}$ | $-{l}_{4}{S}_{(\gamma -{\theta}_{2})}$ | $+{C}_{(\gamma -{\delta}_{2})}{\dot{z}}_{2}$ |

3 | ${l}_{1}{S}_{(\gamma +{\mu}_{1})}-{l}_{6}{S}_{(\gamma +{\mu}_{6})}+{l}_{4}{S}_{(\gamma -{\theta}_{2})}-{l}_{3}{S}_{(\gamma +{\theta}_{1})}+{z}_{1}{S}_{(\gamma -{\delta}_{1})}$ | $-{l}_{3}{S}_{(\gamma +{\theta}_{1})}$ | $-{l}_{4}{S}_{(\gamma -{\theta}_{2})}$ | $-{C}_{(\gamma -{\delta}_{1})}{\dot{z}}_{1}$ |

4 | ${l}_{1}{S}_{(\gamma +{\mu}_{1})}-{l}_{6}{S}_{(\gamma +{\mu}_{6})}+{l}_{4}{S}_{(\gamma -{\theta}_{2})}-{l}_{3}{S}_{(\gamma +{\theta}_{1})}+{z}_{1}{S}_{(\gamma -{\delta}_{1})}$ | $-{l}_{3}{S}_{(\gamma +{\theta}_{1})}$ | $-{l}_{4}{S}_{(\gamma -{\theta}_{2})}$ | $-{C}_{(\gamma -{\delta}_{1})}{\dot{z}}_{1}$ |

$-{z}_{2}{S}_{(\gamma -{\delta}_{2})}$ | $+{C}_{(\gamma -{\delta}_{2})}{\dot{z}}_{2}$ |

Variable | Value |
---|---|

User’s weight | 70 kg |

Vehicle weight | 60 kg |

Height steps | 150 mm |

Wide steps | 300 mm |

Sampling time | 15 mm |

Variable | Value |
---|---|

${l}_{1}$ | 270 mm |

${l}_{3}$ | 410 mm |

${l}_{4}$ | 420 mm |

${l}_{5}$ | 323 mm |

${l}_{6}$ | 390 mm |

${\delta}_{i}$ | 35 deg |

${r}_{wheel}$ | 100 mm |

Variable and Value |
---|

${\gamma}_{mim}=10\xb0$ |

${\gamma}_{max}=-10\xb0$ |

${\dot{\gamma}}_{max}=\pm 0.22\xb0$/s |

${\ddot{\gamma}}_{max}=\pm 0.1\xb0$/s${}^{2}$ |

${\theta}_{1mim}=1.02$ rad |

${\theta}_{2mim}=0.7$ rad |

${\theta}_{1max}=2.32$ rad |

${\theta}_{2max}=1.97$ rad |

${\dot{\theta}}_{1max}={\dot{\theta}}_{2max}=\pm 0.12$ rad/s |

${\ddot{\theta}}_{1max}={\ddot{\theta}}_{2max}=\pm 0.03$ rad/s${}^{2}$ |

${\dot{\theta}}_{3max}=\pm 4$ rad/s |

${\ddot{\theta}}_{3max}=\pm 0.24$ rad/s${}^{2}$ |

${z}_{1max}={z}_{2max}=260$ mm |

${\dot{z}}_{1max}={\dot{z}}_{2max}=\pm 33$ mm/s |

${\ddot{z}}_{1max}={\ddot{z}}_{2max}=\pm 0.6$ mm/s${}^{2}$ |

$\mathit{\alpha}$ | Climbing Time (s) | Comfort (Std) | Algorithm Execution Time (s) |
---|---|---|---|

0 | 86 | 0.0019 | 2.6 |

0.5 | 80 | 0.0032 | 3.1 |

1 | 74 | 0.0047 | 4.0 |

No optimization | 85 | 0.0053 |

$\mathit{\alpha}$ | Climbing Time (s) | Comfort (Std) | Algorithm Execution Time (s) |
---|---|---|---|

0 | 95.2 | 0.0021 | 1.1 |

0.5 | 90.0 | 0.0033 | 1.9 |

1 | 84.3 | 0.0042 | 3.6 |

No optimization | 97.3 | 0.0062 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chocoteco, J.A.; Morales, R.; Feliu-Batlle, V.
Enhancing the Trajectory Generation of a Stair-Climbing Mobility System. *Sensors* **2017**, *17*, 2608.
https://doi.org/10.3390/s17112608

**AMA Style**

Chocoteco JA, Morales R, Feliu-Batlle V.
Enhancing the Trajectory Generation of a Stair-Climbing Mobility System. *Sensors*. 2017; 17(11):2608.
https://doi.org/10.3390/s17112608

**Chicago/Turabian Style**

Chocoteco, Jose Abel, Rafael Morales, and Vicente Feliu-Batlle.
2017. "Enhancing the Trajectory Generation of a Stair-Climbing Mobility System" *Sensors* 17, no. 11: 2608.
https://doi.org/10.3390/s17112608