Adaptive Motion Control of Nonholonomic Intelligent Walker-Human Systems

12th 12th IFAC IFAC International International Workshop Workshop on on 12th Workshop on Adaptation and in 12th IFAC IFAC International International Workshop onand Adaptation and Learning Learning in Control Controlon and Signal Signal Processing Processing 12th IFAC International Workshop Adaptation and Learning in Control and Signal Processing June 29 July 1, 2016. Eindhoven, The Netherlands Available online at www.sciencedirect.com Adaptation and Learning in Control and Signal June 29 July 1, 2016. Eindhoven, The Netherlands Adaptation and1,Learning in ControlThe andNetherlands Signal Processing Processing June 29 July 2016. Eindhoven, June 29 July 1, 2016. Eindhoven, The Netherlands June 29 - July 1, 2016. Eindhoven, The Netherlands

ScienceDirect

IFAC-PapersOnLine 49-13 (2016) 312–317

Adaptive Motion Control of Nonholonomic Adaptive Adaptive Motion Motion Control Control of of Nonholonomic Nonholonomic Intelligent Walker-Human Systems Intelligent Walker-Human Intelligent Walker-Human Systems Systems

Halit Zengin, Zengin, Nursefa Nursefa Zengin, Zengin, Baris Baris Fidan Fidan ∗∗∗ Halit Halit Zengin, Nursefa Zengin, Baris Fidan ∗∗ Halit Zengin, Nursefa Zengin, Baris Halit Zengin, Nursefa Zengin, Baris Fidan Fidan ∗ ∗ Department of Mechanical and Mechatronics Engineering ∗ Department of Mechanical and Mechatronics Engineering ∗ of Mechanical and Mechatronics Engineering ∗ Department Department and Mechatronics Engineering University of Waterloo Department of of Mechanical Mechanical and Mechatronics Engineering University of Waterloo University of Waterloo University of Waterloo Waterloo, ON, Canada, N2L 3G1,+1 519 888 4567 University of Waterloo Waterloo, ON, Canada, N2L 3G1,+1 519 888 4567 Waterloo, ON, Waterloo, ON, Canada, Canada, N2L N2L 3G1,+1 3G1,+1 519 519 888 888 4567 4567 (e-mail: {hzengin,nyarbasi,fidan}@uwaterloo.ca). Waterloo, ON, Canada, N2L 3G1,+1 519 888 4567 (e-mail: {hzengin,nyarbasi,fidan}@uwaterloo.ca). (e-mail: {hzengin,nyarbasi,fidan}@uwaterloo.ca). (e-mail: (e-mail: {hzengin,nyarbasi,fidan}@uwaterloo.ca). {hzengin,nyarbasi,fidan}@uwaterloo.ca). Abstract: This This paper paper focuses focuses on on motion motion control control of of active active intelligent intelligent walkers walkers robust robust to to Abstract: Abstract: This paper focuses on motion control of active intelligent walkers robust to Abstract: This paper focuses on motion control of active intelligent walkers robust to system parameter variations and uncertainties. It presents aa new realistic control-oriented Abstract: This paper focuses on motion control of active intelligent walkers robust to system parameter variations and uncertainties. It presents new realistic control-oriented system parameter variations and uncertainties. It presents adesign new realistic control-oriented system parameter variations and uncertainties. It new control-oriented model and, based on this model, an adaptive motion control to generate appropriate system parameter variations and an uncertainties. It presents presents new realistic realistic control-oriented model and, and, based on on this model, model, an adaptive motion motion controlaadesign design to generate generate appropriate model based this adaptive control to appropriate model and, based on this model, an adaptive motion control design to generate appropriate torques to keep the i-walker in front of the user at the desired distance. Our control model and, based on this model, an adaptive motion control design to generate appropriate torques to keep the i-walker in front of the user at the desired distance. Our control torques to the i-walker front the at desired distance. Our torques to keep keep the kinematics i-walker in in equations front of of derived the user user at aathe the desiredkinematic distance. model Our control control design utilizes utilizes inverse kinematics equations derived from two-body kinematic model and to to torques to keep the i-walker in front of the user at the desired distance. Our control design inverse from two-body and design utilizes inverse kinematics equations derived from a two-body kinematic model and to design utilizes inverse kinematics equations derived from aa two-body kinematic model and to adaptively generate the reference velocities for maintaining ideal relative position of i-walker design utilizes inverse kinematics equations derived from two-body kinematic model and to adaptively generate the reference velocities for maintaining ideal relative position of i-walker adaptively generate the reference velocities for maintaining ideal relative position of i-walker adaptively the velocities for maintaining relative of i-walker with respect to the user. The torques to track the reference velocities are generated using an adaptively generate the reference reference velocities for the maintaining ideal relative position of using i-walker with respect respectgenerate to the the user. user. The torques torques to track track the reference ideal velocities are position generated using an with to The to reference velocities are generated an with respect to the user. The torques to track the reference velocities are generated using an adaptive proportional−integral−derivative control scheme, which is robust to unknown torque with respect to the user. The torques to track the reference velocities are generated using an adaptive proportional−integral−derivative control scheme, which is robust to unknown torque adaptive proportional−integral−derivative controlbased scheme, which linearization is robust robust to unknown unknown torque adaptive proportional−integral−derivative control scheme, which is disturbances, combined with with a a computed computed torque torque feedback unit and and torque a high high adaptive proportional−integral−derivative controlbased scheme, which linearization is robust to to unknown torque disturbances, combined based feedback linearization unit disturbances, combined with a computed torque feedback unit and aaa high disturbances, combined with a computed torque based feedback linearization unit and high gain observer to estimate the wheel velocities. The designed control scheme is formally analyzed disturbances, combined with a computed torque based feedback linearization unit and a high gain observer to estimate the wheel velocities. The designed control scheme is formally analyzed gain observer totested estimate the wheel velocities. The designed control scheme is formally analyzed gain observer estimate wheel velocities. designed control scheme and simulation for both symmetric and asymmetric gait patterns. gain observer to totested estimate the wheel velocities. The designed gait control scheme is is formally formally analyzed analyzed and simulation simulation tested for the both symmetric andThe asymmetric gait patterns. and for both symmetric and asymmetric patterns. and simulation simulation tested tested for for both both symmetric symmetric and and asymmetric asymmetric gait gait patterns. patterns. and © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: adaptive control, physical human-walker interaction, least squares based parameter Keywords: adaptive adaptive control, control, physical physical human-walker human-walker interaction, interaction, least least squares squares based based parameter parameter Keywords: Keywords: adaptive control, physical human-walker interaction, least squares based parameter identifier, nonholonomic i-walker, feedback linearization, high gain observer Keywords: adaptive control, physical human-walker interaction, least squares based parameter identifier, nonholonomic i-walker, feedback linearization, high gain observer identifier, identifier, nonholonomic nonholonomic i-walker, i-walker, feedback feedback linearization, linearization, high high gain gain observer observer identifier, nonholonomic i-walker, feedback linearization, high gain observer 1. INTRODUCTION kinematic models as in Huang et al. [2014], we use the 1. INTRODUCTION INTRODUCTION kinematic models models as as in in Huang Huang et et al. al. [2014], [2014], we we use use the the 1. kinematic 1. INTRODUCTION kinematic models as in Huang et al. [2014], we use the system dynamic model to estimate the wheel velocities. 1. INTRODUCTION kinematic models as in Huang et al. [2014], we use the system dynamic model to estimate the wheel velocities. system Safety and stable motion are two required properties for system dynamic dynamic model model to to estimate estimate the the wheel wheel velocities. velocities. Safety and and stable stable motion motion are are two two required required properties properties for for system dynamic model to estimate the wheel velocities. Safety As the high-level layer of the control scheme, an inAs the high-level layer of the control scheme, an an ininSafety and stable motion are two required properties for intelligent walkers (i-walkers). In this regard, significant Safety and stable motion are two required properties for intelligent walkers (i-walkers). In this regard, significant As the high-level layer of the control scheme, intelligent walkers (i-walkers). In this regard, significant As the high-level layer of the control scheme, an inverse kinematic controller in Cifuentes et al. [2014] As the high-level layer of the control scheme, an inverse kinematic controller in Cifuentes et al. [2014] intelligent walkers (i-walkers). In this regard, significant research has been conducted on motion control of various intelligent walkers (i-walkers). this regard, research has has been conducted onIn motion control significant of various various verse kinematic controller in Cifuentes et research been on motion control of verse kinematic controller in i-walker Cifuentesvelocities et al. al. [2014] [2014] is utilized to produce desired based kinematic controller in Cifuentes et al. [2014] is utilized utilized to produce produce desired i-walker velocities based research has been conducted conducted onal. motion control of various types of i-walkers in Hirata et [2005, 2007], Ko et al. research been conducted on motion types of of has i-walkers in Hirata Hirata et et al. [2005,control 2007], of Kovarious et al. al. verse is to desired i-walker velocities based types i-walkers in al. [2005, 2007], Ko et is utilized to motion producerelative desiredto i-walker velocities based on the user the i-walker, which is is utilized to produce desired i-walker velocities based on the user motion relative to the i-walker, which is types of i-walkers in Hirata et al. [2005, 2007], Ko et al. [2013], Silva Jr. and Sup [2013], Cifuentes et al. [2014], types of i-walkers in Hirata et al. [2005, 2007], Ko et al. [2013], Silva Jr. and Sup [2013], Cifuentes et al. [2014], on the user motion relative to the i-walker, which is [2013], Silva Jr. and Sup [2013], Cifuentes et al. [2014], on the the useruser motion relative to the theTo i-walker, which is called motion intention. generate the reon the user motion relative to i-walker, which is called the user motion intention. To generate the re[2013], Silva Jr. and Sup [2013], Cifuentes et al. [2014], Tan et al. [2011], Wang et al. [2012]. A set of kinematic [2013], Jr. and Supet Cifuentes [2014], called Tan et et Silva al. [2011], Wang et[2013], al. [2012]. A set et of al. kinematic the user motion intention. To generate the Tan Wang al. A of kinematic called the user for motion intention. To generate the rere-aa quired torques tracking these desired velocities, called user motion intention. generate the required the torques for tracking these To desired velocities, Tan et al. al. [2011], [2011], Wang et been al. [2012]. [2012]. A set setfor of active kinematic and dynamic models have developed and Tan et al. [2011], Wang et al. [2012]. A set of kinematic and dynamic models have been developed for active and quired torques for tracking these desired velocities, a and dynamic models have been developed for active and quired torques for tracking these desired velocities, proportional−integral−derivative (PID) control scheme is quired torques for tracking these desired velocities, proportional−integral−derivative (PID) (PID) control control scheme scheme is isaa and dynamic models have been developed developed for active active and passive types i-walkers in Hirata et al. [2005, 2007], Ko and dynamic have been for passive types models i-walkers in Hirata Hirata et al. al. [2005, [2005, 2007], and Ko proportional−integral−derivative passive types i-walkers in et 2007], Ko proportional−integral−derivative (PID) control scheme is designed such that loop system is robust to conscheme is designed such such that that the the closed closed loop loop (PID) systemcontrol is robust robust to conconpassive types Silva i-walkers in Hirata et al. al. [2005, [2005, 2007], Ko et al. [2013], Jr. and Sup [2013], Cifuentes et al. passive types i-walkers et 2007], et al. al. [2013], [2013], Silva Jr. in andHirata Sup [2013], [2013], Cifuentes et Ko al. proportional−integral−derivative designed system et Silva Jr. and Sup Cifuentes et al. designed suchdisturbances that the the closed closed loop system is is friction robust to to constant torque due to unknown forces, designed such that the closed loop system is robust to constant torque disturbances due to unknown friction forces, et al. [2013], Silva Jr. and Sup [2013], Cifuentes et al. [2014], without considering the dynamic effects such as et al. [2013], Jr. and the Sup dynamic [2013], Cifuentes et al. [2014], withoutSilva considering the dynamic effects such such as stant torque disturbances due to unknown friction forces, [2014], without considering effects as stant torque disturbances due to load unknown friction forces, i-walker CG displacement and changes. The PID torque unknown friction i-walker CG disturbances displacementdue andto load changes. Theforces, PID [2014], without considering the dynamic effects suchand as stant center of gravity (CG) shifts, load changes, friction, [2014], without considering the dynamic effects such as center of gravity (CG) shifts, load changes, friction, and i-walker CG displacement and load changes. The PID center of gravity (CG) shifts, load changes, friction, and i-walker CG displacement and load changes. The PID control scheme is combined with feedback linearization i-walker CG displacement and load changes. The PID control scheme is combined with feedback linearization center of uncertainties gravity (CG) (CG) due shifts, load changes, changes, friction, and dynamic to partial weight support. To center of gravity shifts, load friction, and dynamic uncertainties due to partial weight support. To control scheme is with feedback linearization dynamic uncertainties due weight To control scheme is combined combined withcontrol feedback linearization unit based on computed torque approach, and scheme is combined with feedback linearization unit based based on computed computed torque control approach, and aa dynamic uncertainties due to to partial partial weight support. support. Toaa control address such dynamic variation and uncertainty issues, dynamic uncertainties due to partial weight support. address such such dynamic variation variation and uncertainty uncertainty issues,To unit on torque control approach, and address dynamic and issues, a unit based on computed torque control approach, and a high gain observer (HGO) to estimate the wheel velocities, unit based on computed torque control approach, and aa high gain observer (HGO) to estimate the wheel velocities, address such dynamic variation and uncertainty issues, a limited number of adaptive controllers have been proposed address such dynamic variation and uncertainty issues, a high limited number number of adaptive adaptive controllers have been been proposed proposed gain observer (HGO) to estimate the wheel velocities, limited of controllers have high gain observer (HGO) to estimate the wheel velocities, robustly to sensor noises. Lastly, an adaptive version of gain to observer estimate wheel version velocities, robustly to sensor (HGO) noises. to Lastly, an the adaptive version of limited number ofomni-directional adaptive controllers controllers have been proposed high for i-walkers with wheels in the limited number of adaptive have proposed for i-walkers i-walkers with omni-directional wheels inbeen the literature, literature, robustly sensor Lastly, an of for with wheels in robustly todynamic sensor noises. noises. Lastly, an adaptive adaptive version of developed controller is proposed, utilizing an onrobustly to sensor noises. Lastly, an adaptive version of developed dynamic controller is proposed, utilizing an onfor i-walkers with omni-directional omni-directional wheels in the the literature, literature, Tan et al. [2011], Wang et al. [2012]. However, all these for i-walkers with omni-directional wheels in the literature, Tan et al. [2011], Wang et al. [2012]. However, all these developed dynamic controller is utilizing an Tan et [2011], Wang et [2012]. However, all developed dynamicbased controller is proposed, proposed, utilizing an ononline least squares parameter identifier estimating dynamic controller is proposed, utilizing an online least least squares squares based parameter identifier estimating Tan et al. al. [2011], Wang et al. al.the [2012]. However, all these these developed studies neglect the fact that physical human-walker Tan et al. [2011], et al. [2012]. However, all these studies neglect theWang fact that that the physical human-walker line based parameter identifier estimating studies neglect the fact the physical human-walker line least squares based parameter identifier estimating the unknown system parameters. The proposed adaptive least squares based parameter identifier the unknown unknown system parameters. The proposedestimating adaptive studies neglect the fact fact thatonly the dependent physical human-walker human-walker interaction (pHWI) is not on the user studies neglect the that the physical interaction (pHWI) is not not only dependent on the the user user line the system parameters. The proposed adaptive interaction (pHWI) is only dependent on the unknown system parameters. The proposed adaptive control scheme is simulation tested and compared for the unknown system parameters. The proposed adaptive control scheme is simulation tested and compared for both both interaction (pHWI) is not only dependent on the user weight but also the user gait dynamics and characteristics. interaction (pHWI) is not only dependent on the user weight but but also also the the user user gait gait dynamics dynamics and and characteristics. characteristics. control scheme weight control scheme is is simulation simulation tested tested and and compared compared for for both both users. control scheme is simulation tested and compared for both users. weight but also the user gait dynamics and characteristics. weight but also the user gait dynamics and characteristics. users. The main of are development users. The main main contributions contributions of of this this paper paper are are development development users. The The paper is organized as follows. Section is dedicated The paper paper is is organized organized as as follows. follows. Section Section 222 is is dedicated dedicated The main contributions contributions of this this paper paper are model development of aa nonholonomic i-walker-human system taking The main contributions of this paper are development of nonholonomic i-walker-human system model taking The of a nonholonomic i-walker-human system model taking The paper is organized as follows. Section 2 is dedicated to analysis of the effect of pHWI on the i-walker dynamics. The paper is organized as follows. Section 2 is dedicated to analysis of the effect of pHWI on the i-walker dynamics. of a nonholonomic i-walker-human system model taking into account users with symmetric and asymmetric gait of a nonholonomic i-walker-human system model taking into account users with symmetric and asymmetric gait to analysis of the effect of pHWI on the i-walker dynamics. into account users with and asymmetric gait to analysis of3, the effect of of pHWI pHWI on on the i-walker i-walker dynamics. In Section control-oriented kinematic and dynamic effect the In analysis Section of 3,the control-oriented kinematic anddynamics. dynamic into account users with symmetric symmetric andthe asymmetric gait to dynamics and characteristics, and thus model is more into account with symmetric and asymmetric gait dynamics andusers characteristics, and thus thus the model is is more more In Section 3, control-oriented kinematic and dynamic dynamics and characteristics, and the model In Section 3, control-oriented kinematic and dynamic models of the i-walker-human user system are derived. In Section 3, control-oriented kinematic and dynamic models of the i-walker-human user system are derived. dynamics and characteristics, and thus the model is more comprehensive and realistic for human-walker motion, and dynamics and characteristics, thus the model is more comprehensive and realistic realistic for forand human-walker motion, and models of the i-walker-human user system are derived. comprehensive and human-walker motion, and models of the i-walker-human user system are derived. Designs of base robust motion control scheme, state models of the i-walker-human user system are derived. Designs of base robust motion control scheme, state comprehensive and realistic realistic forcontrol human-walker motion, and design of an adaptive motion scheme to make the comprehensive and for human-walker motion, and design of an adaptive motion control scheme to make the Designs of the parameter base motion scheme, design of an adaptive motion control scheme to make the Designs of the base robust robustestimation, motion control control scheme, state state estimator, and adaptive PID Designs of base robust motion control scheme, state estimator, the parameter estimation, and adaptive PID design of an adaptive motion control scheme to make the system robust to load changes and the user steering chardesign of an adaptive motion control scheme to make the system robust to load changes and the user steering charestimator, the parameter estimation, and adaptive PID system robust to load changes and the user steering charestimator, the parameter parameter estimation, and adaptive adaptive PID design scheme are presented in Sections 4, 5, and 6, estimator, the estimation, and PID design scheme are presented in Sections 4, 5, and 6, system robust to load changes and the user steering characteristic for providing safe, stable, and efficient humansystem robust to load changes and the user steering char- design acteristic for providing providing safe, stable, stable, and efficient humanscheme are presented in Sections 4, 5, and 6, acteristic for safe, and efficient humandesign scheme are presented in Sections 4, 5, and 6, respectively. Section 7 presents the simulation results to design scheme are presented in Sections 4, 5, and 6, respectively. Section 7 presents the simulation results to acteristic for providing safe, stable, and efficient humanwalker motion. To design a practically implementable acteristic for providing safe, aastable, and efficient human- respectively. walker motion. motion. To design practically implementable Section 7 analyze presents the simulation to walker To practically implementable respectively. Section presentstheir the characteristics. simulation results results to verify the designs and Final, Section presents the simulation results to verify the the designs designs and77 analyze analyze their characteristics. Final, walker motion. To design design practically implementable controller, different from the existing literature, where walker motion. To design aa practically implementable controller, different from the the existing literature, literature, where respectively. verify and their characteristics. Final, controller, different from existing where verify the designs designs and analyze theirin characteristics. Final, conclusions of the paper are given Section 8. verify the and analyze their characteristics. Final, conclusions of the paper are given in Section 8. controller, different from the existing literature, where wheel velocities of the i-walker are directly obtained using controller, different the are existing literature, wheel velocities velocities of thefrom i-walker are directly obtainedwhere using conclusions wheel of obtained using conclusions of of the the paper paper are are given given in in Section Section 8. 8. of the paper are given in Section 8. wheel velocities of the the i-walker i-walker are directly directly obtained using conclusions tachometers or estimated by observers based on i-walker wheel velocities of the i-walker are directly obtained using tachometers or estimated by observers based on i-walker tachometers or estimated by observers based on i-walker tachometers or or estimated estimated by by observers observers based based on on i-walker i-walker tachometers Copyright © 2016 1 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2016, 2016 IFAC IFAC 1 Hosting by Elsevier Ltd. All rights reserved. Copyright © 2016 IFAC 1 Copyright © 2016 IFAC 1 Peer review under responsibility of International Federation of Automatic Copyright © 2016 IFAC 1 Control. 10.1016/j.ifacol.2016.07.982

IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Halit Zengin et al. / IFAC-PapersOnLine 49-13 (2016) 312–317

3. STRUCTURE AND DYNAMIC MODELLING OF INTELLIGENT WALKER

2. HUMAN EFFECT ON THE INTELLIGENT WALKER DYNAMICS

The i-walker configuration studied in this paper consists of a support frame with controller, battery and LRF, two differential powered wheels and two free casters as shown in Fig.1.

I-walker physically interacts with the upper limbs of the user as seen in Fig. 1. This pHWI affects the dynamics of the i-walker due to the applied forces by the user on the i-walker handles during human-walker motion. The pHWI in the sagittal plane (x-z plane) can be described as Fx = Rx − mh x ¨h , Fz = mh (¨ zh − g) − Rz , Fz c = Rz b

313

3.1 Kinematic Modelling

(1)

‫ܦ‬

where xh , zh denote the location of the human CG in the x and z axes, respectively. mh refers to human mass. Human CG location is the function of the human joint angles. Rx and Rz are the total ground reaction forces (GRFs) in the directions of x and z axes. b and c are the distances of human CG to GRFs and to the walker handles in x-axis, respectively. (1) implies that in addition

݈

߱௛ ‫ܪ‬

ܻ

߰ ‫ݒ‬ ௛ ߙ௛

߱௪

߶ ‫ݒ‬௪ ߙ௪

‫ݕ‬ ‫ݔ‬



‫ܮ‬ௐ஽

݀

ܹ

ܹ

ߚ

ߙ௪

ܻ ܺ

(a) Human-walker motion kinematic model.

(b)ܺ Proposed dynamic model of the i-walker.

Fig. 2. Kinematic and dynamic models of the i-walker.

݉௛ ݃

‫ܨ‬௫

‫ܨ‬௫

Battery ‫ܨ‬௭ ݉௙ ݃

ܼ

ܴ௫ ܺ

ܴ௭

The kinematic model of human-walker motion is based on that in Cifuentes et al. [2014], and illustrated in Fig. 2a. The nonholonomic i-walker motion is driven by two independent motors mounted at rear wheels of radius r. These wheels are separated by distance 2L. LW D is the distance between the point W and the point D.

Controller

‫ܨ‬௭

Differential Wheel with servo motor

Laser Range Finder (LRF)

The variables, v, ω, and α represent linear velocity, angular velocity, and orientation, respectively. The parameters for the human are described with the subscript h, whereas the walker parameters are denoted by the subscript w. Kinematic parameters l, ψ, and φ of human-walker relative motion are defined as the distance between the points D and H, the angle between vh and l, the angle between l and the longitudinal axis of i-walker, respectively.

Free caster

Fig. 1. Free body diagrams of human and i-walker in the sagittal plane.

The location of the i-walker in the reference frame is defined by the vector pw = [xw , yw , αw ]T where (xw , yw ) and αw are the coordinates of point W and the heading T angle, respectively. Choosing q = [xw yw αw θR θL ] as the generalized coordinate vector, the forward kinematic model of the i-walker is obtained as follows:  T 1 1 1 r cos α sin α 1 0 r r w w   2 2L q˙ = S(q)η, ˙ S(q) =  21  1 1 r cos αw r sin αw −r 0 1 2 2 2L (2) T where η = [θR θL ] is the rotational displacements of the right and left wheels with the radius of r. There exist three constraint equations in the kinematic model of the i-walker, which are based on two assumptions: (i) no lateral slip, and (ii) pure rolling motion. The constraint equations, detailed in Sarkar et al. [1992], can be written in the matrix form as follows:   − sin αw cos αw 0 0 0 A(q)q˙ = 0, A(q) = cos αw sin αw L −r 0 . (3) cos αw sin αw −L 0 −r

to the user body weight and body pose, the total vertical force Fz is also dependent on the user CG motion and GRFs which have periodic nature. It can be concluded that while the total load on the i-walker is periodically changing with human CG motion in the sagittal plane, the load distribution between the handles shows periodic changes with the horizontal motion of human CG in the frontal plane (in y-axis) and human gait phase during walking as in Alwana et al. [2007], Abellanas et al. [2010]. Houglum and Bertoti [2012] states that the human CG displacement is closely related to the gait parameters, such as swing phase, stance phase, stride length, step width and walking speed. This implies that the user gait pattern determines the load distribution on the i-walker during walking. In this paper, the i-walker CG displacements and the load changes on the i-walker due to pHWI based on gait dynamics and characteristics are considered to achieve more comprehensive dynamic model. This dynamic model can be used to take into account both types of users in controller designs for i-walker-human systems to better generate the required torques for stable human-walker motion.

The time derivative of (2) is ˙ η˙ + S(q)¨ q¨ = S(q) η. 2

(4)

IFAC ALCOSP 2016 314 Halit Zengin et al. / IFAC-PapersOnLine 49-13 (2016) 312–317 June 29 - July 1, 2016. Eindhoven, The Netherlands

¯ = S T (q)M (q)S(q), M ˙ + S T (q)C(q, q)S(q), ˙ C¯ = S T (q)M (q)S(q)

We assume that a full rank matrix S(q) is formed by a set of smooth and linearly independent vector fields spanning the null space of A(q), which satisfies the equation

¯ = S (q)B(q). B

(5) S T (q)AT (q) = 0. The kinematic equations of the i-walker human relative motion illustrated in Fig. 2a, are derived using polar coordinates. Kinematic model of the human-walker relative motion is derived as follows:        cos φ −L −vh cos ψ W D sin φ l˙ vw cos φ sin ψ , = sin φ + ωw LW D −ωh + vh ψ˙ l l l (6) where φ = αw − αh − ψ from the geometry of the system.

Applying (11), the reduced order matrices are obtained as     ¯ = m11 m12 , τ = τR , M m21 m22 τL   2 mc r d cos β   ωw  bm 1 0  ¯ 2L , B = , C¯ =   0 1 mc r2 d cos β ωw − bm 2L where bm is the viscous damping on the motor, and the ¯ are parameters of the matrix M 2 mr2 Ir2 mc dr sin β + + + Iw , m11 = − 2L 4 4L2 2 2 Ir mr m12 = m21 = − , 4 4L2 2 2 mr Ir2 mc dr sin β m22 = + + + Iw . 2L 4 4L2

3.2 Dynamic Modelling The Lagrange formalism is used to derive the dynamic equation of non-holonomic differential wheeled mobile platforms to eliminate the constraint forces:   d ∂L ∂L = F − AT (q)λ, (7) − dt ∂ q˙i ∂qi where L, F , and λ denote the Lagrangian function, generalized force vector, the vector of the Lagrange multipliers, respectively. The system potential energy is considered as zero since the system motion is planar. The total kinetic energy of the system is  1  2 + mc dωw [y˙ w cos(αw + β) T = m x˙ 2w + y˙ w 2 (8)  1 1  2 2 2 + θ˙L , + Iωw −x˙ w sin(αw + β)] + Iw θ˙R 2 2 ˙

(11)

T

4. CONTROLLER DESIGN The control objective in this paper is to generate the required control torques, τR and τL , to keep the i-walker in front of the user such that the angle ψ is regulated to 0, and the distance l is maintained at value ld , where ld is desired human distance from the point D at which a sensor like LRF is placed to feed inverse kinematic controller in Fig.2a. In this regard, an inverse kinematic control scheme proposed in Cifuentes et al. [2014] is utilized to produce the desired linear and angular velocities of the i-walker, vwd , ωwd , based on the human motion and orientation. These desired velocities are mapped to the desired wheel angular  T velocities, η˙ d = θ˙Rd θ˙Ld . Then, a dynamic controller is designed to generate appropriate motor torques in order to track the desired wheel positions and velocities. To this end, a PID control scheme is used as the underlying dynamic controller. This dynamic controller is combined with a feedback linearization unit based on computed torque control approach, whose details can be found in Zengin [2015a]. The block diagram of the overall algorithm is given in Fig.3.

˙

θL )r where ωw = α˙ w = (θR − , m = mc + 2mw and I = Ic + 2L 2 2 mc d + 2mw L + 2Im . mc represents the mass of the support frame including human effect. mw is the mass of each driven wheel with a motor. Ic is the moment of inertia of the support frame about the vertical axis through the CG, Iw and Im are the moment of inertia of each powered wheel and a motor about the wheel axis, respectively. vwR and vwL are the linear velocities of the powered wheels.

Different from the existing literature, the parameters β, d, which defined the location of the i-walker CG in the local frame, and mc are considered in this paper to be dependent on user weight, body pose, gait dynamics and gait characteristics in Fig.2b. Thus, all these parameters are periodically changing during walking. Substituting (8) in (7), the dynamic model of the i-walker is obtained as follows: M (q)¨ q + C(q, q) ˙ q˙ = B(q)τ − AT (q)λ, (9) where M (q), C(q, q), ˙ B(q), and τ are the symmetric positive definite inertia matrix, the centripetal and Coriolis matrix, the input matrix, and the torque vector, respectively. M (q), C(q, q) ˙ and B(q) matrices are explicitly written in Zengin [2015a]. (9) can be rewritten for simulation and control purposes by eliminating the constraint term AT (q)λ using (5) as in Dhaouadi and Hatab [2013]. Substituting (2) and (4) into (9), and then multiplying both sides from left by S T (q), the reduced dynamic model is obtained as ¯ η¨ + C¯ η˙ = Bτ, ¯ M (10) where

4.1 Human Intention Based Inverse Kinematic Controller An inverse kinematic control scheme for the model in Section III is designed to control the angle ψ and the distance l, modifying the inverse kinematic control design of Cifuentes et al. [2014] . The designed controller aims at maintaining desired human-walker distance and making ψ exponentially converged to zero, which keeps the i-walker in front of human according to human relative position and orientation during walking. In other words, human intention is detected based on human kinematic data, and thus the desired i-walker angular and linear velocities are produced. For our control purpose, (6) is rewritten as       −vh cos ψ cos φ −LW D sin φ  v  ˜l˙ w cos φ sin ψ , + = sin φ ωw LW D −ωh + vh ψ˙ l l l (12) 3

IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Halit Zengin et al. / IFAC-PapersOnLine 49-13 (2016) 312–317

where ˜l = l − ld . The inverse kinematic control law is obtained using the inverse kinematics of human-walker motion in (12) as follows:     cos φ l sin φ   −k ˜l + v cos ψ l h vw sin φ cos φ = sin ψ , ωw l − −kψ ψ + ωh − vh LW D LW D l (13) where kl , kψ are positive control gains. For implementation of the control law (13), the length l, the angles φ and ψ, and the velocities vw , ωw , vh and ωh can be continuously obtained by LRF and techniques in Cifuentes et al. [2014].

315

With the feedback linearization transform (16), the computed torque control law is written as follows: ¯ (¨ τ =M ηd − u) + C¯ η. ˙ (18)  T T T  T T T We define X = X1 X2 = η η˙ to have a general system model, then we can obtain the following equations X˙ 1 = X2 , ¯ −1 C¯ (X2 ) X2 + M ¯ −1 u, (19) X˙ 2 = −M y = X1 . Note that C¯ matrix depends on the state X2 . 4.3 PID Controller In this section, u(t) is picked as PID feedback control input. u = −Kv e˙ − Kp e − Ki ε (20) ε˙ = e Substituting (20) in (18), the computed torque control law is expressed as follows: ¯ (¨ ˙ (21) τ =M ηd + Kv e˙ + Kp e + Ki ε) + C¯ η. The closed-loop system is given in the following state-space form        ε 0 ε˙ 0 I 0 e + 0  e˙ = 0 0 I (22) I e¨ −Ki −Kp −Kv e˙   

Fig. 3. Overall system block diagram. 4.2 Feedback Linearization The main idea of feedback linearization is to express the nonlinear system as an equivalent linear system to compute the required torques. (10) is redefined considering the torque disturbance τd due to the unknown frictions as follows: ¯ η¨ + C¯ η˙ + τd = τ. M (14) A desired trajectory is selected as ηd = [θRd θLd ]T , and tracking error is defined as e = ηd − η. Substituting tracking error, the following equation is obtained ¯ −1 (C¯ η˙ − τ + τd ). (15) e¨ = η¨d + M The control input function and the disturbance function are defined as mc [kg]

40

K

The closed-loop characteristic polynomial is (23) ∆c (s) = |s3 I + Kv s2 + Kp s + Ki |, where Kv = diag(kvi ), Kp = diag(kpi ), and Ki = diag(kii ) are the control gains defined in Lewis et al. [2003]. We can find the following equation utilizing these gains: n  (s3 I + kvi s2 + kpi s + kii ). (24) ∆c (s) = i=1

The Routh stability criteria is satisfied for k v i k p i > kii . 5. STATE ESTIMATION

sym asym

30 0

50

100

150

200

250

0

50

100

150

200

250

0

50

100

150 time [s]

200

250

Velocity measurements mostly have noise as compared to position measurements. Thus, a state estimator, which is HGO, is needed for reliable velocity information, details can be found in Zengin [2015b]. To design an HGO, consider the system model given in (19). In order to estimate the state X2 , an HGO is designed as ˆ 2 − h1 X ˜1, ˆ˙ 1 = X X   (26) ¯ −1 C¯ X ¯ −1 τ, ˆ2 + M ˜1 − M ˆ2 X ˆ˙ 2 = −h2 X X

β [rad]

2 1 0

d [m]

-1

(25)

0.2 0.1

˜1 = X ˆ 1 −X1 where y = X1 , and h1 , h2 are positive gains. X ˆ and X2 is the estimate ofX2 . Additionally, h1 and h2 are  −h1 1 chosen such that is Hurwitz. −h2 0

Fig. 4. Time varying parameters for different gait patterns. ¯ −1 (C¯ η˙ − τ ), u = η¨d + M (16) ¯ −1 τd . =M T  Defining a state as E ∈ R4 , E = eT e˙ T , tracking error dynamics are written in the form       0I 0 0 ˙ E= E+ u+ . (17) 0 0 I I

6. PARAMETER ESTIMATION AND ADAPTIVE PID CONTROLLER DESIGN In this section, a parameter estimator identifier based on an on-line least squares is developed for unknown 4

IFAC ALCOSP 2016 316 Halit Zengin et al. / IFAC-PapersOnLine 49-13 (2016) 312–317 June 29 - July 1, 2016. Eindhoven, The Netherlands

parameter estimators. Adaptive PID scheme can be reˆ ¯ , C¯ with their estimates M ¯ , Cˆ ¯ as designed, replacing M ˆ¯ (¨ ˆ2. (31) τ =M ηd + Kv eˆ˙ + Kp eˆ + Ki εˆ) + Cˆ¯ X

parameters of i-walker-human system. Considering (10), we have two unknown parameters, mc and β. It should be noted that d is also unknown but it can be derived by mc and β. In order to estimate these parameters, parameter identification are implemented.

This design guarantees stable, safe and efficient motion of the i-walker-human system by generating required torques for keeping the i-walker in front of the user.

6.1 Parameter Identification

7. SIMULATIONS

First step of the parameter identification based on Ioannou and Fidan [2006] is to define a parametric model . Each ¯ and C¯ matrices is defined as unknown component of M parameters of the system. Then, dynamic equation (10) is rewritten as  ∗ ∗        θ1 θ2 θ¨R bm θ4∗ ωw θ˙R τ + = R . (27) θ2∗ θ3∗ θ¨L −θ4∗ ωw bm τL θ˙L

In this section, the proposed system model and control scheme are simulated for users with different gait characteristics in the MATLAB/Simulink environment. Two users are defined via two different sets of parameters. mc = 36 + 5 cos(2πt) kg is used to simulate the timevarying load on the i-walker during motion. The periodic oscillation in the load represents the pHWI based on gait dynamics. Time histories of the parameters, mc , d, and β during walking, for both types of users are given in Fig. 4.

θR and θL are available for measurement, θ˙R and θ˙L are estimated by HGO. The second derivations can be obtained via estimated values. These equations can be rewritten by collecting all unknown parameters in one side. Unknown parameters are collected into a vector, then static parametric models (SPM) are formed as follows: T T ˙ z1 = Θ∗1 Φ1 = τR − bm θˆR , Θ∗1 = [θ1∗ θ2∗ θ4∗ ] , T T ˙ z2 = Θ∗2 Φ2 = τL − bm θˆL , Θ∗2 = [θ3∗ θ2∗ θ4∗ ] , T  (28) Φ1 = θ¨R θ¨L ωw θˆ˙L ,  T Φ2 = θ¨L θ¨R −ωw θˆ˙R ,

The physical parameters of the i-walker are taken as r = 0.15 m, L = 0.27 m, LW D = 0.65 m, bm = 0.4 kg.m2 /s, Ic = 5.2 kg.m2 , Iw = 0.015 kg.m2 , Im = 0.002 kg.m2 , mw = 1.5 kg. The control parameters are selected as kl = 0.3, kψ = 0.4, Kp = diag(8.5), Kv = diag(6) and Ki = diag(2). The desired distance of human from the point W , ld is 0.20 m. The generated torques, τR and τL are bounded with saturation ±3N m. The constant torque and observer disturbances are assumed to be 0.2. HGO gains are set to h1 = 3, h2 = 2. Initial states of human and i-walker are (x0 , y0 , α0 )h = (0, 0, 0) and (x0 , y0 , α0 )w = (0.3, 0, 0), respectively. A figure-8 trajectory is defined for the users to follow xd = R sin(2ωt), yd = R cos(ωt) − R, (32) x˙ d = 2Rω cos(2ωt), y˙ d = −Rω sin(ωt). According to simulation results, for both users, the desired distance of 0.2 m is successfully maintained through out the trajectory (32). The relative orientation of the user with respect to the i-walker follows a similar pattern except the transient region. Estimation results in Fig. 5c,5f verify that proposed observer accurately estimates the wheel velocities for both users. Figs 5b,5d,5e demonstrate that the control goals are well-achieved.

where z1 , z2 ∈ R and Φ1 , Φ2 ∈ Rn are signals available for measurement, and Θ∗ ∈ Rn is the vector with all unknown parameters. After having a suitable parametric model for the system, now estimation model and error can be obtained. Estimation models and errors are designed by using the parametric models. ˜ T Φ1 T z1 − zˆ1 −Θ = , zˆ1 = Θ1 Φ1 , ε1 = m2s1 m2s1 (29) ˜ T Φ2 T z2 − zˆ2 −Θ zˆ2 = Θ2 Φ2 , ε2 = = m2s2 m2s2 ˜ = where zˆ1 , zˆ2 are the estimation of z1 , z2 , respectively.Θ Θ − Θ∗ , m2si = 1 + αi ΦTi Φi are normalizing signals that guarantee Φi /msi bounded for any αi ≥ 1. Leastsquares (LS) algorithm is implemented to the system as an adaptive law to analyse the unknown parameters in static parametric model. Recursive LS algorithm with forgetting factor can be written as ˙ = P1 ε1 Φ1 + P2 ε2 Φ2 , θ(0) = θ0 , Θ Φ1 ΦT P˙ 1 = βP1 − P1 2 1 P1 , P1 (0) = P10 = Q−1 0 , (30) ms1 T

Φ2 Φ P˙2 = βP2 − P2 2 2 P2 , ms2

8. CONCLUSIONS In this paper, a more comprehensive and realistic controloriented model for a nonholonomic i-walker-human system is proposed by considering the pHWI based on the user gait dynamics and characteristics. Based on this model an adaptive PID control scheme is designed for a safe and stable human-walker motion regardless of the user weight, gait dynamics and characteristic. In this regard, a feedback linearization and an inverse kinematic control based PID control scheme is developed for the proposed i-walker-human system. Due to sensor noises, an HGO is developed to estimate the wheel velocities of i-walker. Also, parameter estimation is performed because of the changes in system parameters. Simulations are conducted for the user having different gait characteristics. The simulation results demonstrate the validity of the proposed system model, and the effectiveness and robustness of designed adaptive control scheme. For the future work, real-time application of this work will be performed.

P2 (0) = P20 = Q−1 0 ,

where P1 , P2 are covariance matrices, Q0 = QT0 > 0 and Φ1 ΦT1 , Φ2 ΦT2 are positive definite. 6.2 Adaptive PID Controller Design Unknown states and parameters of the i-walker-human system are obtained by using the proposed state and 5

IFAC ALCOSP 2016 June 29 - July 1, 2016. Eindhoven, The Netherlands Halit Zengin et al. / IFAC-PapersOnLine 49-13 (2016) 312–317

-5 -10 -15

3 des sym asym

Human-walker dist. [m]

0

Y [m]

0.3

Reference sym asym

τR τL

2 1

0.25 τ [Nm]

5

317

0 1.5 -1

0.2

1 0.5

-2

0

-20

0

-3 -4

-2

0

2

4

0.15

6

0

50

100

X [m]

(a) Figure-8 tracking.

150 time [s]

200

(b) Figure-8 tracking error. 0.06

3 τR

0

50

5

10 100

15

150 time [s]

200

2 1 Actual Estimated

0.02

ψ [rad]

0

-0.04

0.5 0 0

-3

5

10

-0.05 0

-0.06

15

0 0

50

100

150 time [s]

200

50

10 100

20 150

200

250

time [s]

250

(d) Torques for asymmetric gait pattern

0

50

100

150

200

250

0

50

100

150 time [s]

200

250

0.05 0

1

-2

0 -0.02

1.5

-1

0

θ˙L [rad/s]

τ [Nm]

1

250

(c) Torques for symmetric gait pattern

des sym asym

0.04

τL

2

250

θ˙R [rad/s]

-25 -6

(e) Relative orientation of users to iwalker.

2 1 0

(f) Estimation results for the user with asymmetric gait pattern.

Fig. 5. Figure-8 tracking results for symmetric and asymmetric gait patterns. REFERENCES

Ko, C., Young, K., Huang, Y., and Agrawal, S. (2013). Active and passive control of walk–assist robot for outdoor guidance. IEEE/ASME Transactions on Mechatronics, 18, 1211–1220. Lewis, F., Dawson, D., and Abdallah, C. (2003). Robot Manipulator Control Theory and Practice. CRC Press. Sarkar, N., Yun, X., and Kumar, R. (1992). Control of mechanical systems with rolling constraints: Application to dynamic control of mobile robots. Technical report, University of Pennsylvania. Silva Jr., A. and Sup, F. (2013). Design and control of a two–wheeled robotic walker for balance enhancement. IEEE International Conference on Rehabilitation Robotics, 1–6. Tan, R., Wang, S., Jiang, Y., Ishida, K., and Fujie, M. (2011). Motion control of omni–directional walker for walking support. IEEE/ICME International Conference on Complex Medical Engineering, 633–636. Wang, Y., Wang, S., Tan, R., Jiang, Y., Ishida, K., and Fujie, M. (2012). Adaptive control method for a walking support machine considering center of gravity shifts and load changes. IEEE International Conference on Advanced Mechatronic Systems, 684–689. Zengin, H. (2015a). Control Oriented System Modelling of Intelligent Walker–Human Systems. Master’s thesis, University of Waterloo. Zengin, N. (2015b). Adaptive Motion Estimation and Control Of Intelligent Walkers. Master’s thesis, University of Waterloo.

Abellanas, A., Frizera, A., Ceres, R., and Gallego, J. (2010). Estimation of gait parameters by measuring upper limb–walker interaction forces. Sensors and Actuators A: Physical, 162(2), 276–283. Alwana, M., Ledoux, A., Wasson, G., Sheth, P., and Huang, C. (2007). Basic walker–assisted gait characteristics derived from forces and moments exerted on the walkers handles: results on normal subjects. Medical Engineering & Physics, 29(3), 380–389. Cifuentes, C., Rodriguez, C., Frizera, A., Bastos, T., and Carelli, R. (2014). Multimodal human–robot interaction for walker–assisted gait. IEEE Systems Journal, 1–11. Dhaouadi, R. and Hatab, A. (2013). Dynamic modelling of differential–drive mobile robots using lagrange and newton–euler methodologies: A unified framework. Advances in Robotics and Automation, 2. Hirata, Y., Hara, A., and Kosuge, K. (2007). Motion control of passive intelligent walker using servo brakes. IEEE Transactions on Robotics, 23, 981–990. Hirata, Y., Oscar Jr, C., Hara, A., and Kosuge, K. (2005). Human adaptive motion control of active and passive type walking support system. IEEE Workshop on Advanced Robotics and its Social Impacts, 139–144. Houglum, P. and Bertoti, D. (2012). Brunnstrom’s Clinical Kinesiology. F.A. Davis Company, Philadelphia. Huang, J., Wen, C., Wang, W., and Jiang, Z. (2014). Adaptive output feedback tracking control of a nonholonomic mobile robot. Automatica, 50(3), 821–831. Ioannou, P. and Fidan, B. (2006). Adaptive Control Tutorial. Society for Industrial and Applied Mathematics, Philadelphia. 6