cleveland state university, estimation in prosthetic legs

Seyed FakoorianDepartment of Electrical Engineering

and Computer Science,

Cleveland State University,

Cleveland, OH 44115

e-mail: [email protected]

Vahid AzimiSchool of Electrical and Computer Engineering,

Georgia Institute of Technology,

Atlanta, GA 30313

Mahmoud MoosaviDepartment of Electrical Engineering



Cleveland, OH 44115

Hanz RichterDepartment of Mechanical Engineering,


Cleveland, OH 44115


Dan SimonDepartment of Electrical Engineering



Cleveland, OH 44115


Ground Reaction ForceEstimation in Prosthetic LegsWith Nonlinear Kalman FilteringMethodsA method to estimate ground reaction forces (GRFs) in a robot/prosthesis system is pre-sented. The system includes a robot that emulates human hip and thigh motion, alongwith a powered (active) transfemoral prosthetic leg. We design a continuous-timeextended Kalman filter (EKF) and a continuous-time unscented Kalman filter (UKF) toestimate not only the states of the robot/prosthesis system but also the GRFs that act onthe foot. It is proven using stochastic Lyapunov functions that the estimation error of theEKF is exponentially bounded if the initial estimation errors and the disturbances aresufficiently small. The performance of the estimators in normal walk, fast walk, and slowwalk is studied, when we use four sensors (hip displacement, thigh, knee, and ankleangles), three sensors (thigh, knee, and ankle angles), and two sensors (knee and ankleangles). Simulation results show that when using four sensors, the average root-mean-square (RMS) estimation error of the EKF is 0.0020 rad for the joint angles and 11.85 Nfor the GRFs. The respective numbers for the UKF are 0.0016 rad and 7.98 N, which are20% and 33% lower than those of the EKF. [DOI: 10.1115/1.4036546]

1 Introduction

Recent advances in microelectronics and robotic technologieshave enabled the development of powered prosthetic legs [1] thatcan help amputees walk up stairs and slopes because of the legs’net power contribution to gait; they are also able to adapt theirbehavior to various environmental conditions. The company€Ossur has a lower limb prosthesis called the Power Knee, which isthe first commercially available knee to generate power during thegait cycle [2]. Recent developments in powered knee and ankleprostheses are reported in Refs. [3–8].

Bulky load cells are often employed in robots and prostheticlegs to capture external forces (GRFs) and moments during walk-ing [9]. These data are used as feedback measurements to controlthe robot or prosthesis with force or impedance controls. Evenwhen not used for feedback, force sensing is important for moni-toring and evaluation of prosthesis performance and safety. How-ever, there are several drawbacks to the use of load cells: (i) loadcells are expensive; (ii) a 250-lbf load cell weighs about 1 lb, witha length of about 3 in, and thus does not easily fit in a prostheticleg; (iii) load cell measurements tend to drift, need to be fre-quently offset, and are noisy and need significant signal condition-ing; (iv) load cells can get damaged easily from overloading oroff-axis loading; and (v) load cells consume electrical power,which is an important consideration in prosthetics.

The above-mentioned problems do not arise with angle sensorsbecause high-resolution encoders are accurate, reliable, and inex-pensive. Nevertheless, they do not measure velocity, and velocity

calculated by numerical differentiation is challenging because ofthe difficult compromise between noise rejection and bandwidth.

There have been several methods to reduce the number offorce sensors in robotics. In Ref. [10], a robot compliance con-troller based on a disturbance observer is presented, where thedisturbance observer is used to estimate the external reactionforces. In Ref. [11], a method for force estimation of the end-effector of a Selective Compliance Assembly Robot Arm(SCARA) robot is presented, where servomotor currents andposition information are used to estimate forces. In Ref. [12], theexternal force on the end-effector of a four degrees-of-freedom(DOF) robot manipulator is estimated with a combination oftime delay estimation and input estimation. In Ref. [13], amodel-based observer is used to estimate the external forces act-ing on a rigid body. All the aforementioned techniques dependon either the accuracy of the robot model or servomotor currentdata. Thus, if the accuracy of the robot model or motor currentmeasurement degrades, force estimation can deviate from trueforce. In Ref. [12], the force estimation does not require a pre-cise robot dynamic model, but the accuracy of estimated forcemay degrade in the presence of noise, since the external force isconsidered as an unknown input.

In this paper, we treat GRF as an unknown input with knowndynamic properties and known bounds, and we use an EKF toestimate GRFs along with the states of the robot/prosthesis sys-tem. Although EKF is efficient in many applications, it has twoimportant potential drawbacks. First, the derivation of the Jaco-bian matrix for linearization can be complex and can causenumerical implementation difficulties. Second, linearization canlead to cumulative errors which may affect the accuracy of thestate estimation. To overcome these limitations, other nonlinearestimators could be used, such as the UKF or particle filter, where

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedDecember 6, 2016; final manuscript received March 10, 2017; published online July20, 2017. Assoc. Editor: Dumitru I. Caruntu.

Journal of Dynamic Systems, Measurement, and Control NOVEMBER 2017, Vol. 139 / 111004-1Copyright VC 2017 by ASME

Downloaded From: https://dynamicsystems.asmedigitalcollection.asme.org/pdfaccess.ashx?url=/data/journals/jdsmaa/936354/ on 07/20/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

the state estimations are obtained without the need for derivativesand Jacobian calculations [14–16].

In this research, a powered (active) prosthetic leg is consideredfor transfemoral amputees. The prosthetic leg attached to a robotichip/thigh emulator. The combined system includes four degrees-of-freedom: vertical hip displacement, thigh angle, knee angle,and ankle angle. This paper is an extended version of Ref. [17],where we designed an EKF for the online estimation of joint coor-dinates, velocities, and GRFs.

In this paper, we compare EKF and UKF performance in our4DOF robot/prosthesis system. We study the EKF because it isthe most commonly used nonlinear state estimator due to its ver-satility and simplicity of implementation. We study the UKFbecause it generally provides better performance than the EKF.One of the questions we study in this paper is whether theimproved performance of the UKF relative to the EKF is worththe increased computational effort.

We also study the effect of various sensor sets on estimationerror at different walking speeds. We start with the four mainstates of the robot/prosthesis system, which are hip displacement,thigh, knee, and ankle angles, as system observations to get thebaseline estimation accuracy. Then, we investigate how muchestimation accuracy degrades if fewer measurements are used,and we find a tradeoff between the estimation error accuracy andthe number of measurements. In general, we prefer to use fewermeasurements because fewer sensors mean a simpler system, lesscomplexity, and lower cost.

Finally, we consider the stability of the EKF and boundednessof the estimation error, which are very important for prosthesiscontrol. The stability of the EKF under various values of initialestimation error and noise covariances is explored. We analyti-cally show that the estimation error remains bounded under cer-tain conditions in our 4DOF robot/prosthesis system and confirmthe analysis with simulation results.

This research involves the mechanical integration of a prosthe-sis and a test robot so that we can test the prosthesis withouthuman trials. We will eventually need to use the Kalman filter ona prosthesis that is attached to an amputee. We develop the Kal-man filter for the robot/prosthesis system [18] in this paper forlater implementation on an amputee/prosthesis system.

The paper is organized as follows: In Sec. 2, the model of therobotic system and prosthetic leg is presented. In Sec. 3, the EKFfor state estimation and GRF estimation is discussed. In Sec. 4,the EKF is analyzed by mathematically deriving its stability con-ditions. Section 5 compares performance between the EKF andUKF when different measurement sensors are used; also the con-vergence of the EKF is tested with different initial estimationerrors and noise magnitudes. Section 6 concludes the paper andsuggests future research.

2 System Model

Robotic testing of transfemoral prostheses is presented here,where motion is limited to the sagittal plane and transverse motionis not considered. Typically, only the sagittal plane is consideredin transfemoral prosthesis research. Although the transverse planeis ignored here, sagittal motion captures the essential dynamics ofhuman walking [19]. The model of the test robot/prosthesis isbased on the standard robotic framework. Figure 1 shows a dia-gram of the hip robot and prosthesis combination [20,21]. A gen-eral dynamic model for the system is given as follows:

D qð Þ€q þ C q; _qð Þ _q þ B q; _qð Þ þ JeTFe þ g qð Þ ¼ u (1)

where q ¼ q1; q2; q3; q4½ �T is the vector of joint displacements(q1 is vertical hip displacement, q2 is thigh angle, q3 is knee angle,and q4 is ankle angle), D qð Þ is the inertia matrix, C q; _qð Þ is amatrix accounting for centripetal and Coriolis effects, B q; _qð Þ is anonlinear damping vector, Je is the kinematic Jacobian relative to

the point of application of the external forces Fe, g qð Þ is the grav-ity vector, and u is the four-element vector of control signals [22].The kinematic and dynamic models of the robot/prosthesis combi-nation are given in Refs. [23–25], where a mixed tracking/impedance controller based on passivity methods is designed [22].The control signals consist of hip force, and thigh, knee, and ankletorques. As Fig. 1 shows, a triangular foot with two points ofground contact is assumed. Horizontal and vertical GRFs areapplied to contact points at the toe and heel. The GRFs aredenoted as Fxh, Fzh, Fxt, Fzt, which represent the horizontal andvertical GRFs at the heel and toe. Thus, the external force vectorFe in Eq. (1) comprises these four GRFs.

We assume the robot walks along the x-axis. A treadmill pro-vides the walking surface of the robot/prosthesis system. Thebelt stiffness is modeled to calculate reaction forces during con-tact between the heel and toe with the treadmill [22,23]. TheGRFs are entirely determined by kinematics and are given as fol-lows [23]:

zh ¼ �aH sin q2 þ q3 þ q4 þp2þ cos�1 ah

aH

� �� þl3 sin q2 þ q3ð Þ þ l2 sin q2ð Þ þ q1 (2)

zt ¼ aT sin q2 þ q3 þ q4 þp2� cos�1 ah

aT

� �� þl3 sin q2 þ q3ð Þ þ l2 sin q2ð Þ þ q1 (3)

Fzh ¼ �kb zh � szð Þ1þ sign zh � szð Þ

2

� �(4)

Fzt ¼ �kb zt � szð Þ1þ sign zt � szð Þ

2

� �(5)

Fxh ¼ bFzh (6)

Fig. 1 The robotic model of the hip robot/prosthesis system. Itis desired to eliminate the load cells on the foot and insteadestimate forces with an EKF.

111004-2 / Vol. 139, NOVEMBER 2017 Transactions of the ASME


Fxt ¼ bFzt (7)

where kb is the belt stiffness, sz is the treadmill standoff (the dis-tance between the origin of the world coordinate system and thebelt when the leg is fully extended), l2 and l3 are the lengths oflink 2 (thigh) and link 3 (shank), respectively, aH and aT are thedistance from the ankle joint to the heel and the toe, respectively,ah is the height of the ankle joint above the sole of the foot, and bis the friction coefficient between the treadmill belt and the foot.The vertical positions of the toe and heel in the world coordinatesystem are shown in Fig. 1 as zt and zh, respectively. We thus havefour states for the positions and four states for the velocities of thejoint displacements.

The main objective of this research is to estimate the GRFswhich comprise Fe in Eq. (1). We, therefore, augment the externalforces to the state vector. The augmented state vector includes theeight original states of the robot and the four GRFs and is given asfollows:

x ¼ q1; q2; q3; q4; q1:; q2:; q3:; q4:;Fxt;Fzt;Fxh;Fzh

� �T(8)

It should be noted that the forces are not states of the original sys-tem model, but are augmented here to the state vector, since weneed to estimate them with a state estimator. The 12-element vec-tor of Eq. (8) will be estimated by the state estimator.

Equations (2)–(7) are a preliminary approximation to GRF andare accurate only if the amputee (or robot in our case) walks in ahighly controlled environment with a known walking surface stiff-ness. In future work, it will be important to study the robustnessof the Kalman filter to modeling errors and particularly to errorsin these GRF equations.

We should mention that there are at least two other possibleapproaches to obtain GRF estimates. In the first alternativeapproach, GRF could be considered as an unknown input with noprior assumptions about its model. In that case, the EKF wouldhave difficulty estimating GRF, since its effect on the systemwould be indistinguishable from process noise. In the secondalternative approach, state estimation of the robot/prosthesis sys-tem could be performed by a nonlinear observer, such as the EKF,and then the estimated states of the robot could be substituted intothe GRF Eqs. (2)–(7) to obtain the GRF estimates. However, thisapproach would be less flexible than the approach that we areusing. Augmenting the GRF states onto the original state as in Eq.(8) allows us additional tuning flexibility in the artificial processnoise that is included in the augmented states, as we will see inSec. 5.

3 Extended Kalman Filtering for Robot/Prosthesis

State Estimation

The Kalman filter applies directly only to linear systems. How-ever, we can linearize a nonlinear system and then use linear esti-mation techniques. Over the last few decades, the EKF hasbecome one of the most popular estimation techniques in nonlin-ear systems. The EKF applies the standard linear Kalman filtermethodology to a linearization of the nonlinear system [26].According to the continuous time-EKF equations [27], the systemand measurement equations are assumed as

_x ¼ f x; u; tð Þ þ G tð Þw tð Þ (9)

y ¼ h x; tð Þ þ D tð Þv tð Þ (10)

where the state x(t), the input u(t), and the output y(t) are inRq; Rp and Rm, respectively. Moreover, the noise terms w(t) andv(t) are in Rl and Rk and are uncorrelated, zero-mean white noiseprocesses with identity covariances. G(t) and D(t) are time-varying matrices of size q� l and m� k. If the nonlinear functionsf and h are sufficiently smooth in x, Taylor series expansions can

be performed. The following Jacobian matrices are used to linea-rize the system:

A tð Þ ¼ @f

@x

��x̂

(11)

C tð Þ ¼ @h

@x

��x̂

(12)

The initial state x(0) is a random vector with covariance P(0)

x̂ 0ð Þ ¼ E x 0ð Þ½ �

P 0ð Þ ¼ E x 0ð Þ � x̂ 0ð Þð Þ x 0ð Þ � x̂ 0ð Þð ÞTh i

and x(0) is assumed to be uncorrelated with w(t) and v(t). TheEKF equations are given as

_̂x ¼ f x̂; u; tð Þ þ K y� h x̂; tð Þ½ � (13)

K tð Þ ¼ P tð ÞCT tð ÞR�1 tð Þ (14)

_P ¼ A tð ÞP tð Þ þ P tð ÞAT tð Þ þ Q tð Þ � P tð ÞCT tð ÞR�1 tð ÞC tð ÞP tð Þ(15)

where K(t) and P(t) are the Kalman gain and estimation errorcovariance matrix. Covariance matrices Q(t) and R(t) are q� qand m�m, respectively, and are defined as follows:

Q tð Þ ¼ G tð ÞGT tð Þ (16)

R tð Þ ¼ D tð ÞDT tð Þ (17)

The state-space equations of the robotic/prosthetic system can beobtained from Eq. (1) and the GRF Eqs. (2)–(7) must be differen-tiated to obtain the state-space model of the GRF. However, dis-continuities of Eqs. (4) and (5) can be problematic. There are twoways to handle this problem. First, we can use a smooth approxi-mation of the sign functions using hyperbolic tangent functions;but, this approach requires a lot of computational effort for theJacobian calculation and often results in EKF divergence [16]. Inthe second approach, which is used in this paper, the sign functionis not considered when we take the derivative of Eqs. (2)–(7) toinclude the GRFs in the state-space model. The sign function isre-introduced in the computed Jacobian matrix after taking thederivative. The state-space equations of the GRFs are given asfollows:

_Fzh ¼ �kb _zh

1þ sign zh � szð Þ2

� �(18)

_Fzt ¼ �kb zt: 1þ sign zt � szð Þ

2

� �(19)

_Fxh ¼ b _Fzh (20)

_Fxt ¼ b _Fzt (21)

where _zh ; _zt represent the derivatives of Eqs. (2) and (3).

4 The Stability of the Extended Kalman Filter

The analysis of the stability properties of continuous-timeextended Kalman filters is complex and has been treated only forespecial cases. Safonov and Athans [28] considered the stabilityof the constant and modified gain EKF. The boundedness of theEKF estimates were investigated when used as a parameter

Journal of Dynamic Systems, Measurement, and Control NOVEMBER 2017, Vol. 139 / 111004-3


estimator for linear systems [29,30]. Reif et al. [31,32] determinedthat the estimation error in both the discrete and continuous-timeEKF is exponentially bounded under certain conditions. Theyshowed that the estimation error remains bounded if the initialestimation error and disturbances are sufficiently small and thenonlinear system satisfies a detectability rank condition [33,34].Here, we extend Reif’s work [31] to obtain the stability conditionsof the state estimator of the robot/prosthesis system. The mainpurpose of this section is to derive the stability of the estimator interms of initial condition errors and disturbances.

The nonlinear functions f and h from Eqs. (9) and (10) can beexpanded to first-order as follows:

f x; u; tð Þ � f x̂; u; tð Þ ¼ A tð Þ x tð Þ � x̂ tð Þ½ � þ u x; x̂; u; t½ � (22)

h x; uð Þ � h x̂; uð Þ ¼ C tð Þ x tð Þ � x̂ tð Þ½ � þ X x; x̂; u½ � (23)

where u and X are the remaining terms of the Taylor series. Theestimation error n tð Þ ¼ x tð Þ � x̂ tð Þ can be written recursively asfollows:

dn tð Þ ¼ A tð Þ � K tð ÞC tð Þ½ �n tð Þ þ q tð Þ þ r tð Þ dw tð Þdv tð Þ

� �(24)

where r tð Þ and q tð Þ are defined by

r tð Þ ¼ G tð Þ � K tð ÞD tð Þ½ � (25)

q tð Þ ¼ u x; x̂; u; t½ � � K tð ÞX x; x̂; t½ � (26)

In stability theory, there are two commonly used methods to proveboundedness: supermartingales for stochastic differential equa-tions, and Lyapunov functions for deterministic differential equa-tions. Supermartingales can be regarded as the stochasticequivalent of Lyapunov functions in some cases [35]. In order toanalyze the error dynamics of the EKF, we first present some pre-liminary results [31,35]. Note that k:k represents the spectralnorm of a matrix or the Euclidian norm of a vector.

LEMMA 4.1. Suppose there is a stochastic process V n tð Þ; t½ � withn tð Þ 2 Rq governed by Eq. (24) and there exist real numbersv1; v2; l; # � 0 such that

v1kn tð Þk2 � V n tð Þ; t½ � � v2kn tð Þk2(27)

and

LV n tð Þ; t½ � � �#V n tð Þ; t½ � þ l (28)

where LV n tð Þ; t½ � is the differential generator for the stochasticprocess and is defined as

LV n tð Þ; t½ � ¼ @V

@tn; tð Þ þ @V

@nn; tð Þ A tð Þ � K tð ÞC tð Þ½ �n tð Þð

þq tð ÞÞ þ 1

2

Xq

i¼1

Xq

j¼1

@2V n; tð Þ@ni@nj

r tð ÞrT tð Þ� �

ij (29)

Then, for every t � 0 the estimation error, n tð Þ is exponentiallybounded in mean square [36,37]

E kn tð Þk2� �

� v2

v1

kn 0ð Þk2e�#t þ lv1#

(30)

Proof of Lemma 4.1. See Appendix A.THEOREM 4.1. Consider the continuous-time nonlinear dynamic

system Eqs. (9) and (10) and the extended Kalman filter Eqs.(13)–(15). Suppose that the following conditions hold:

(i) There exist p1, p2, c1, c2, q1, q2, r1, r2 such that the follow-ing bounds hold for every t � 0:

p1I � P tð Þ � p2I (31)

c1 � kC tð Þk � c2 (32)

q1I � Q tð Þ � q2I (33)

r1I � R tð Þ � r2I (34)

(ii) There exist eu, ex, Ku; Kx � 0 such that the nonlinear func-tions in Eq. (26) are bounded by

ku x; x̂; uð Þk � Kukx� x̂k2(35)

kX x; x̂ð Þk � Kxkx� x̂k2(36)

for x, x̂ 2 Rq and u 2 Rp with kx� x̂k � eu andkx� x̂k � ex, respectively.

Then, the estimation error n tð Þ is exponentially bounded in meansquare, if the initial estimation error satisfies

kn 0ð Þk � e (37)

Fig. 2 State estimation of joint displacements of the robot withthe use of EKF and four measurements during normal gaitspeed. Note that the initial estimate quickly converges to thetrue state. (a) The estimate of the hip displacement, (b) the esti-mate of the thigh angle, (c) the estimate of the knee angle, and(d) the estimate of the ankle angle.



where e is a small positive number. Note that the covariance mat-rices of the noise terms are bounded as in Eqs. (33) and (34), ifthere is a real number d � 0 such that the following bounds aresatisfied for every t � 0:

G tð ÞGT tð Þ � dI (38)

D tð ÞDT tð Þ � dI (39)

Proof of Theorem 4.1. See Appendix B.To calculate Ku and Kx, we can consider a compact subset

N 2 Rq; the constants Ku, Kx can be obtained by

Ku ¼ maxi

supx2NkHess fi x; uð Þk (40)

Kx ¼ maxi

supx2NkHess hi xð Þk (41)

If f and h are twice differentiable with respect to x for everyx 2 N, the spectral norm of the Hessian matrices of fi and hi arebounded, where fi and hi are the components of f and h, respec-tively (see Refs. [31] and [38, Chap. 7]). In the given robot/prosthesis system, the Hessian matrix of f is not a function of con-trol input u.

Note that the nonlinear system has to be uniformly detectablein order to satisfy the boundedness condition of Eq. (31) [39,40].The detectability of the nonlinear system results in the bounded-ness of the estimation error bounds for the solution P(t) of the Ric-cati differential equation (15) based on Eq. (31). We present thefollowing definition for the detectability of nonlinear stochasticsystems [39].

DEFINITION 4.1. The pair @f=@xð Þ x; uð Þ; @h=@xð Þ xð Þ� �

; x 2Rq; u 2 Rp is uniformly detectable if there exists a boundedmatrix valued function K xð Þ such that

gT @f

@xx; uð Þ þ K xð Þ @h

@xxð Þ

g � �jkmjkgk2

(42)

holds for all x, g 2 Rq, where km is the maximum eigenvalue of@f=@xð Þ x; uð Þ þ K xð Þ @h=@xð Þ xð Þ

� �.

Values for e and d in Eqs. (B21) and (B24) are obtained inAppendix B. In Sec. 5.2, the stability of the EKF for the state esti-mation of the robot/prosthesis system is considered via simulationand related to Theorem 4.1.

5 Simulation Results

The performance of the robot/prosthesis system during one stepof normal walking, which is approximately 1 s, is considered. Thereference data are provided by able-bodied research participantsat the Motion Studies Laboratory of the Cleveland Department ofVeterans Affairs Medical Center (VAMC) [41]. In the prostheticleg, we have 12 states that are going to be estimated so the Amatrix in Eq. (11) is 12� 12. For our first simulation, we use fourstates as the measurements: vertical hip position, thigh angle,knee angle, and ankle angle. Therefore, the dimension of the mea-surement matrix C in Eq. (12) is 4� 12. The initial value of thestate vector and estimate, and the diagonal covariance matrices Qand R for the process and measurement noise are given as

x 0ð Þ ¼ 0:019; 1:130; 0:090;½�2:246; 0:093; 0:770; 1:410; 2:890; 0; 0; 0; 0�T

x̂ 0ð Þ ¼ 0:040; 0:901; 0:292;½�2:440; 0:293; 0:430; 1:112; 2:620; 0; 0; 0; 0�T

Q ¼ Qd=Dt ¼ diagf0:0005; 0:0005; 0:0005; 0:0005;

0:0002; 0:0002; 0:0002; 0:0002; 0; 0; 0; 0g=Dt

Fig. 3 State estimation of the robot velocities with the use of EKF and four measurementsduring normal gait speed



R ¼ RdDt ¼ diag 10�3; 10�3; 10�3; 10�3� �

Dt

where x̂0 differs from x(0) due to random initial state estimationerrors, Dt denotes the discretization step size with Dt ¼ 0:5 ms, Q

and R represent continuous-time noise covariances, and Qd and Rd

represent discrete-time noise covariances [27, Chap. 8]. The pro-cess noise represents small values of unmodeled dynamics andparameter uncertainties. The Q matrix is chosen based on prior

Fig. 4 Horizontal and vertical GRFs estimations with the use of EKF and four measurementsduring normal gait speed

Table 1 Root-mean-square error (RMSE) for EKF state estimates of the robot/prosthesis system with different numbers of meas-urements and nonzero initial estimation errors

Gait modes No: measurements x1 (m) x2 (rad) x3 (rad) x4 (rad) x5 (m/s) x6 (rad/s) x7 (rad/s) x8 (rad/s) x9 (N) x10 (N) x11 (N) x12 (N)

Normal walking 4 0.003 0.002 0.003 0.005 0.022 0.051 0.084 0.322 3.145 9.758 4.247 21.0153 0.008 0.003 0.003 0.005 0.057 0.091 0.214 0.524 5.015 16.247 9.365 38.1462 0.036 0.009 0.004 0.004 0.085 0.352 0.314 0.558 9.524 23.287 15.174 49.561

Fast walking 4 0.005 0.001 0.002 0.002 0.013 0.055 0.0946 0.218 3.124 10.015 4.647 22.1443 0.009 0.002 0.004 0.005 0.048 0.137 0.376 0.933 8.715 22.014 15.247 46.9842 0.042 0.007 0.004 0.006 0.092 0.445 0.765 1.234 14.845 28.956 21.568 53.175

Slow walking 4 0.007 0.001 0.001 0.001 0.015 0.036 0.065 0.159 2.399 8.995 3.512 18.6213 0.008 0.004 0.003 0.006 0.058 0.143 0.275 0.856 5.067 19.324 13.189 42.8722 0.035 0.007 0.005 0.006 0.058 0.355 0.586 1.058 10.188 25.357 18.957 50.143

Table 2 RMSE for UKF state estimates of the robot/prosthesis system with different numbers of measurements and nonzero initialestimation errors

Gait modes No: measurements x1 (m) x2 (rad) x3 (rad) x4 (rad) x5 (m/s) x6 (rad/s) x7 (rad/s) x8 (rad/s) x9 (N) x10 (N) x11 (N) x12 (N)

Normal walking 4 0.003 0.002 0.002 0.002 0.015 0.025 0.08 0.184 2.547 7.985 3.521 18.6873 0.006 0.003 0.002 0.005 0.066 0.089 0.252 0.604 4.170 14.178 8.115 36.0272 0.035 0.009 0.003 0.004 0.079 0.362 0.264 0.452 10.015 23.057 16.185 47.111

Fast walking 4 0.004 0.001 0.002 0.002 0.011 0.015 0.0614 0.205 2.552 8.095 4.012 19.5743 0.007 0.002 0.004 0.005 0.033 0.139 0.289 0.721 7.075 20.584 13.449 44.1592 0.037 0.007 0.003 0.006 0.078 0.415 0.611 1.002 11.899 24.152 20.222 49.788

Slow walking 4 0.005 0.001 0.001 0.001 0.008 0.022 0.054 0.160 2.005 6.778 3.444 16.5503 0.007 0.004 0.003 0.005 0.044 0.186 0.201 0.699 4.997 18.875 12.854 39.9212 0.028 0.007 0.005 0.006 0.048 0.368 0.502 0.917 9.187 24.179 17.045 47.854



experience with the accuracy of system dynamic modeling; thelast four elements of the diagonal matrix Q are zero since the GRFmodel is assumed to be perfectly known. The R matrix is usuallystraightforward to determine on the basis of our knowledge of theaccuracy of the measurement system.

The results for the state estimation of the robot/prosthesis sys-tem are shown in Figs. 2–4. Although significant initial estimationerrors are imposed on the joint displacements and velocities, theEKF converges to the true states quickly. After toe-off there is nofoot contact and the GRF is zero.

5.1 Reduced Sensor Sets and Unscented Kalman Filtering.In a real-world scenario, it would not be practical to use a hip dis-placement sensor on an amputee because of its invasiveness. Theother sensors—thigh angle, knee angle, and ankle angle—can eas-ily be used in the real world because they can be mounted on theprosthesis. In this paper, we use a hip displacement sensor to pro-vide a baseline scenario for the reduced sensor sets, which westudy later in this section.

More observations result in more accurate state estimation.However, our goal is to use the fewest possible number of sensorsfor estimation, which result in a reduction of complexity and costin the prosthesis hardware. Here, we remove the hip displacementsensor and use only three measurements: the thigh, knee, andankle angles. For our final test in this section, we will use onlytwo measurements: the knee and ankle angles. Moreover, we willalso compare the performance of the EKF and UKF for the stateand GRF estimation. We use the same initial values for state andestimated state as used in the EKF for normal walking. We alsotest the performance of the filters with different gait speeds, andwe will see that no extra filter tuning is needed for fast and slowwalking.

The original discrete-time form of the UKF has been widelyused for state estimation of discrete-time systems. However, the

discrete-time UKF cannot be directly applied to continuous-timefiltering problems, in which the process and measurement equa-tions are modeled as continuous-time stochastic systems. In Ref.[15], the continuous-time UKF was derived from the discrete-timeUKF in matrix form, and this is the algorithm that we implementhere. The tuning parameters in the unscented transforms werechosen to be a ¼ 0:9, b¼ 2, and j ¼ �3.

Table 1 compares the accuracy of EKF state estimation in termsof the RMSE in three different gait modes (fast walk, normalwalk, slow walk). The EKF works well for different walkingspeeds with a reduced measurement set. It can be seen, asexpected, that for all three walking speeds, the estimation errorswith four measurements are smaller than with three or twomeasurements.

Table 2 shows that the UKF achieves smaller RMS estimationerrors than the EKF for almost all measurement sets and walkingspeeds. Table 3 summarizes the average performance of the EKFand UKF. We see that the UKF achieves 30% improvement in theaverage RMSE of GRFs with four measurements. Although theUKF performs better than the EKF, its computational effort isbetween two and three times that of the EKF in terms of the num-ber of multiplications and additions [15].

5.2 Stability of the Extended Kalman Filter. It has beenshown that the estimation error in the continuous-time Kalman filterremains bounded under certain conditions. Small initial estimationerrors and small noise magnitudes are required conditions to obtainerror bounds. In this section, we study the stability of the state esti-mates of the robot/prosthesis system. First, we need to check that theJacobian matrices in the robot/prosthesis system satisfy the uniformdetectability condition of Definition 4.1. To find an appropriate K xð Þ,we assume that @f=@xð Þ x; uð Þ þ K xð Þ @h=@xð Þ xð Þ ¼ �Iq�q

� �and

solve for K xð Þ¼ �I� @f=@xð Þð Þ @h=@xð ÞT @h=@xð Þð @h=@xð ÞTÞ�1.

Note that with our measurement system, ð @h=@xð Þ @h=@xð ÞTÞ�1is an

identity matrix and is thus invertible. Therefore, the uniform

Table 3 Comparison of average RMSE between EKF and UKFwith different numbers of measurements

No:measurements Filter

Ave. RMSEangular

positions (rad)

Ave. RMSEangular

velocities (rad/s)

Ave.RMSE

GRFs (N)

4 EKF 0.0020 0.1205 11.8525UKF 0.0016 0.0896 7.9792

3 EKF 0.0039 0.3943 19.8650UKF 0.0037 0.3533 18.7003

2 EKF 0.0058 0.6297 26.7279UKF 0.0055 0.5437 25.0579

Table 4 Initial values and covariance matrices noise terms

Small initial errorand small noise

Small initial errorand large noise

Large initial errorand small noise

x̂ð0Þ x̂goodð0Þ x̂goodð0Þ x̂poorð0ÞQ 10�5I12�12 10�2I12�12 10�5I12�12

R 10�4I4�4 I4�4 10�4I4�4

Error behavior Bounded Divergent DivergentFig. 5 Fig. 6 Fig. 7

Fig. 5 The simulation results show the boundedness of the estimation error with small initialerror and small noise terms. The sixth state is used here for illustration purposes, but similarresults hold for all of the other states as well. (a) The estimate of x6(t) and (b) the estimationerror n6(t).



detectability condition of Definition 4.1 is satisfied in the robotic/prosthesis system. To test the stability of the robotic/prosthesis EKF,

the initial value x̂ 0ð Þ and the measurement and process noise cova-riances R, Q are chosen as shown in Table 4, where

x 0ð Þ ¼ 0:019; 1:130; 0:090;½�2:246; 0:093; 0:770; 1:410; 2:890; 0; 0; 0; 0�T

x̂good 0ð Þ ¼ 0:027; 1:091; 0:112;½�2:350; 0:193; 0:539; 1:3122:797; 0; 0; 0; 0�T

x̂poor 0ð Þ ¼ 0:181; 1:991; 0:712;½�2:714; 0:393; 1:291; 2:1123:440; 0; 0; 0; 0�T

x̂good 0ð Þ is a reasonably accurate initial condition for the state esti-mate, and we will see that it is good enough to ensure conver-gence of the EKF. x̂bad 0ð Þ is a significantly worse initial conditionfor the state estimate, and we will see that it is not good enough toensure convergence of the EKF. The R and Q values in Table 4have been chosen to demonstrate conditions that, respectively,ensure, or do not ensure, EKF convergence. These values forx̂good 0ð Þ; x̂bad 0ð Þ, and R and Q were chosen by trial and error todemonstrate their effect on EKF convergence.

The simulation results are presented in Figs. 5–7, where theunknown state x6 tð Þ in Eq. (8) (angular velocity of thigh), the esti-mated state x̂6 tð Þ, and its estimation error n6 tð Þ are plotted. We

can see in Fig. 5 that for small initial estimation error and smallnoise, the estimation error is bounded. However, Figs. 6 and 7show that for large initial estimation errors or large noise magni-tudes, the estimation error is no longer bounded.

Extensive simulation tests show that boundedness of the esti-

mation error is obtained in the numerical simulation if kn 0ð Þk�e¼0:35;G tð ÞGT tð Þ�d¼2�10�2 and D tð ÞDT tð Þ�d¼2�10�7.The theoretical results for e and d via Eqs. (B21) and (B24) from

Appendix B yield smaller bounds for stability: e�6:5�10�5 and

d�8:7�10�13 with N¼ x2Rq;kxk�1000� �

. Since the estima-

tion error does not diverge in practice even with larger initial con-dition errors and noise terms than those given by the theorem, weconclude that the bounds are very conservative in this system.However, we note that it is not possible to test all possible condi-tions that exceed the theoretical error bounds; therefore, we natu-rally expect the theoretical results to yield more conservativestability bounds than the simulation results.

We want the state to be estimated as accurately as possible foreventual implementation in a state-feedback controller. The con-troller that we used in this research is robust to estimation errors[22], although more accurate state estimates are always desirablein order to reduce controller errors. We have not explored the rela-tionship between estimation error and controller error in this paperbut leave it as an important area for future research.

6 Conclusion and Future Work

We designed an EKF and an UKF to estimate not only thestates of a robot/prosthesis system but also the external forces

Fig. 6 The simulation results show the divergence of the estimation error with small initialerror and large noise terms. The sixth state is used here for illustration purposes, but similarresults hold for all of the other states as well. (a) The estimate of x6(t) and (b) the estimationerror n6(t).

Fig. 7 The simulation results show the divergence of the estimation error with large initialerror and small noise terms. The sixth state is used here for illustration purposes, but similarresults hold for all of the other states as well. (a) The estimate of x6(t) and (b) the estimationerror n6(t).



acting on the prosthetic foot. This approach removes the need forheavy and bulky load cells that are otherwise needed for GRF esti-mation. We achieved satisfactory estimation errors in various gaitspeeds for the robot/prosthesis system using four, three, and twomeasurements. The average RMS estimation errors of the EKF forthe thigh, knee, and ankle angles in normal walking with fourmeasurements is 0.0033 rad, in comparison with that the UKFwhich is 0.0020 rad. Although the UKF outperforms the EKF, itrequires more computational effort than the EKF, which will be aconsideration for real-time implementation.

We proved mathematically that the estimation error in the EKFis exponentially bounded if the initial estimation errors and distur-bances are sufficiently small and if the nonlinear system satisfies adetectability rank condition. In simulation tests, we verified thatthe estimation errors remained bounded for small initial estima-tion errors and small disturbances. However, the filter is unstableif the initial estimation errors or disturbances are too large.

As far as we know, the present research is the first time thatGRF has been estimated for prostheses using state estimationtechniques. As we noted in the introduction, some other methodshave been used for external force estimation in robotics. Forinstance, [12] achieved a 1.72% estimation error for a 4DOFrobot, compared to our results, which show a 1.85% error whenusing the EKF with four sensors during normal walking. Ourresults are, therefore, quantitatively similar to Ref. [12], althoughthe comparison may not be fair since the robotic systems in thetwo approaches are much different. Other publications in the areaof external force estimation do not include quantitative results[10,11,13].

Future work will include increasing and verifying the robust-ness of the filters, and experimental implementation and verifica-tion of the EKF and UKF, first on robotic hardware [21] and thenin human trials.

Funding Data

• NSF Grant No. 1344954.

Appendix A

Proof of Lemma 4.1. Let n tð Þ be the unique solution to the sto-chastic differential in Eq. (24) and let V n tð Þ; t½ � be a stochastic pro-cess which is a non-negative and twice differentiable. Supposethat V n tð Þ; t½ � satisfies the conditions in Eqs. (27) and (28). Byapplying Ito’s formula [42] we can derive

dV n tð Þ; t½ � ¼ LV n tð Þ; t½ � þ @V n; tð Þ@t

r tð Þd ~w tð Þ (A1)

where LV n tð Þ; t½ � is the differential generator given in Eq. (29)and ~w tð Þ is a Wiener process [35]. We take the integral of bothsides of Eq. (A1)

V n tð Þ; t½ � � V n 0ð Þ;0� �

¼ðt

0

LV n sð Þ; s½ � dsþðt

0

@V n; sð Þ@t

r sð Þ d ~w sð Þ

(A2)

Since ~w tð Þ is a Wiener process (Gaussian noise with zero mean),the expectation of the last integral in the preceding equation iszero because of the properties of Ito’s formula [28]. Therefore

E V n tð Þ; t½ ��

¼ E V n 0ð Þ; 0� ��

þðt

0

E LV n sð Þ; s½ ��

ds (A3)

Equation (A3) implies the continuity of E V n tð Þ; t½ ��

, so differen-tiating and using Eq. (28) yields

d E V n sð Þ; s½ ��

ds¼ E LV n sð Þ; s½ �

� �� #E V n sð Þ; s½ �

� �þ l (A4)

Using Ito’s formula, we can write Eq. (A4) as

d exp #sð ÞE V n sð Þ; s½ ��

ds� l exp #sð Þ (A5)

Integrating both sides of Eq. (A5) from 0 to t yields

exp #tð ÞE V n tð Þ; t½ ��

� E V n 0ð Þ; 0� ��

� l#

exp #tð Þ � 1� �

(A6)

Multiplying exp �#tð Þ on both sides of Eq. (A6) results in

E V n tð Þ;t½ ��

�exp �#tð ÞE V n 0ð Þ;0� ��

þl#

1�exp �#tð Þ� �

�exp �#tð ÞE V n 0ð Þ;0� ��

þl#

(A7)

Finally, applying the expectation operator to Eq. (27) and substi-tuting the result into Eq. (A7) result in the conclusion that n tð Þ isexponentially bounded in the mean square sense as shown inEq. (30).

Appendix B

Proof of Theorem 4.1. To prove this theorem, we need to con-struct a stochastic Lyapunov function to satisfy the conditions ofLemma 4.1. The stochastic Lyapunov function is chosen asfollows:

V n tð Þ; t½ � ¼ nT tð ÞP�1 tð Þn tð Þ (B1)

For notational convenience, we denote P�1 tð Þ as p tð Þ where P(t)is positive definite with probability one. Therefore, from Eq. (31),it follows that

1

p2

I � p tð Þ � 1

p1

I (B2)

so that Eq. (B1) we can write

1

p2

kn tð Þk2 � V n tð Þ; t½ � � 1

p1

kn tð Þk2(B3)

The first condition of Lemma 4.1, which is Eq. (27), is satisfied.For the second condition, we need to construct the differentialgenerator LV n tð Þ; t½ � of the stochastic process in Eq. (B1). The dif-ferential generator can be defined by Eq. (29). The second part ofEq. (29) can be written as follows [31]:

Xq

i¼1

Xq

j¼1

@2V n;tð Þ@ni@nj

r tð ÞrT tð Þ� �

ij¼tr r tð ÞrT tð ÞHessV n tð Þ;t½ ��

(B4)

Therefore, from Eqs. (24), (25), (B1), and (B4), we can write thedifferential generator as

LV n tð Þ; t½ � ¼ nT tð Þ dp tð Þdt

n tð Þ þ 2nT tð Þp tð Þ A tð Þ � K tð ÞC tð Þ½ �n tð Þð Þ

þ2nT tð Þp tð Þq tð Þ þ tr G tð ÞGT tð Þ��

þK tð ÞD tð ÞDT tð ÞKT tð Þ�p tð Þ

�(B5)

From Eqs. (14), (31), (32), and (34) we have

kK tð Þk ¼ kP tð ÞCT tð ÞR�1 tð Þk� kP tð ÞkkCT tð ÞkkR�1 tð Þk

� p2c2

r1

(B6)



We define d ¼ p2c2=r1 which allows us to write Eq. (B6) as

kK tð Þk � d (B7)

Based on Eqs. (38) and (39), one can obtain

tr G tð ÞGT tð Þ� �

� d tr I½ � ¼ qd (B8)

tr D tð ÞDT tð Þ� �

� d tr I½ � ¼ md (B9)

where q and m are the number of rows of G(t) and D(t), respec-tively. Moreover, by using Eqs. (31) and (B6)–(B9), we obtain

tr G tð ÞGT tð Þ þ K tð ÞD tð ÞDT tð ÞKT tð Þ� �

p tð Þ� �¼ tr G tð ÞGT tð Þp tð Þ

� �� þ tr D tð ÞDT tð ÞK tð ÞKT tð Þp tð Þ

� �� 1

p1

tr G tð ÞGT tð Þ� �

þ d2

p1

tr D tð ÞDT tð Þ� �

� Kdd (B10)

where Kd is defined as

Kd ¼qþ d2m

p1

(B11)

By substituting for q tð Þ from Eq. (26) into the fourth term of Eq.(B5), it can be shown that

k2nT tð Þp tð Þq tð Þk ¼ k2nT tð Þp tð Þ u x; x̂; u; t½ � � K tð ÞX x; x̂; t½ ��

k� k2nT tð Þp tð Þu x; x̂; u; t½ �kþk2nT tð Þp tð ÞK tð ÞX x; x̂; t½ �k (B12)

We can use the fact that p tð ÞP tð Þ ¼ I and substitute Eq. (14) intoEq. (B12) to obtain

k2nT tð Þp tð Þq tð Þk � k2nT tð Þp tð Þu x; x̂; u; t½ �kþk2nT tð ÞCT tð ÞR�1 tð ÞX x; x̂; t½ �k (B13)

From Eqs. (31)–(36), it follows that

k2nT tð Þp tð Þq tð Þk � 2kn tð ÞkKu

p1

kn tð Þk2 þ 2kn tð Þk c2Kx

r1

kn tð Þk2

(B14)

We simplify Eq. (B14) to obtain

k2nT tð Þp tð Þq tð Þk � Knkn tð Þk3(B15)

where Kn ¼ 2Ku=p1

� �þ 2c2Kx=r1ð Þ. Substituting Eqs. (14),

(B10), and (B15) into Eq. (B5) yields

LV n tð Þ; t½ � � nT tð Þ dpdtþ AT tð Þp tð Þ þ p tð ÞA tð Þ

�2CT tð ÞR�1 tð ÞC tð Þ�

n tð Þ þ Knkn tð Þk3 þ Kdd

(B16)

We can find dp tð Þ using p tð ÞP tð Þ ¼ I

dp tð Þ ¼ �p tð ÞdP tð Þp tð Þ (B17)

where dP(t) is calculated by the Riccati differential equationwhich is given in Eq. (15). So, if we substitute Eq. (B17) intoEq. (B16), we obtain

LV n tð Þ; t½ � � �nT tð Þ p tð ÞQ tð Þp tð Þ þ CT tð ÞR�1 tð ÞC tð Þ� �

n tð Þþ Knkn tð Þk3 þ Kdd (B18)

We can use Eqs. (31)–(34) to write Eq. (B18) as

LV n tð Þ; t½ � � � q1

p22

þ c21

r2

" #kn tð Þk2 þ Knkn tð Þk3 þ Kdd (B19)

We can also use the upper bound of Eq. (B3) to obtain

LV n tð Þ; t½ � � � q1r2 þ c21p2

2 � p22r2Knkn tð Þk

p22r2

!p1V n tð Þ; t½ � þ Kdd

(B20)

Now, we just need to show that q1r2 þ c21p2

2 � p22r2Kn

��kn tð ÞkÞ=p2

2r2Þp1 � 0. Since the real numbers p1; p2; c1; r2 are posi-

tive, we require n tð Þ � q1r2 þ c21p2

2

� �=p2

2r2K2

� �to satisfy this con-

dition. Now, according to Theorem 4.1, we can obtain the initialestimation error bound e as

e ¼ min eu; ex;q1r2 þ c2

1p22

2p22r2K2

!(B21)

Finally, it can be seen that the second condition Eq. (28) ofLemma 4.1 is satisfied if

LV n tð Þ; t½ � � � q1p1r2 þ c21p1p2

2

2p22r2

!V n tð Þ; t½ � þ Kdd (B22)

In summary, the assumptions of Lemma 4.1 Eqs. (27) and (28) are

satisfied by Eqs. (B3) and (B22), where 1=p2ð Þ ¼ v1; 1=p1ð Þ ¼v2; q1p1r2 þ c2

1p1p22

� �=2p2

2r2

� �¼ # and Kdd ¼ l. We conclude

that the estimation error is exponentially bounded in mean squareunder the conditions in Eqs. (37)–(39).

To meet these conditions, we use e from Eq. (B21) and calcu-late d as follows. We need to take care that ~e � kn tð Þk � e withsome ~e � e satisfies the inequality in Eq. (B22)

LV n tð Þ; t½ � � � q1p1r2 þ c21p1p2

2

2p22r2

!V n tð Þ; t½ � þ Kdd � 0 (B23)

So, to guarantee the boundedness of the estimation error, wechoose d on the basis of Eq. (B23) and the lower bound ofEq. (B3)

d ¼ q1p1r2 þ c21p1p2

2

� �~e2

2p32r2Kd

(B24)

References[1] Sub, F., Varol, H. A., and Goldfarb, M., 2010, “Upslope Walking With a Pow-

ered Knee and Ankle Prosthesis: Initial Results With an Amputee Subject,”IEEE Trans. Neural Syst. Rehabil. Eng., 19(1), pp. 71–78.

[2] Harvey, Z., Potter, B. K., Vandersea, J., and Wolf, E., 2011, “ProstheticAdvances,” J. Surg. Orthop. Adv., 21(1), pp. 58–64.

[3] Martinez-Villalpando, E. C., Weber, J., Elliott, G., and Herr, H., 2008, “Designof an Agonist-Antagonist Active Knee Prosthesis,” IEEE/RAS-EMBS Interna-tional Conference on Biomedical Robotics and Biomechatronics, Scottsdale,AZ, Oct. 19–22, pp. 529–534.

[4] Au, S. K., Herr, H., Weber, J., and Martinez-Villalpando, E. C., 2007, “PoweredAnkle-Foot Prosthesis for the Improvement of Amputee Ambulation,” 29thAnnual IEEE Conference on Engineering in Medicine and Biology Society(EMBS), Lyon, France, Aug. 22–26, pp. 3020–3026.

[5] Au, S. K., Weber, J., and Herr, H., 2007, “Biomechanical Design of a PoweredAnkle Foot Prosthesis,” IEEE International Conference on RehabilitationRobotics (ICORR), Noordwijk, The Netherlands, June 13–15, pp. 298–303.



http://dx.doi.org/10.1109/TNSRE.2010.2087360

http://www.armdynamics.com/caffeine/uploads/files/21-1-9.pdf

http://dx.doi.org/10.1109/BIOROB.2008.4762919

http://dx.doi.org/10.1109/IEMBS.2007.4352965

http://dx.doi.org/10.1109/ICORR.2007.4428441

[6] Zhao, H., Kolathaya, S., and Ames, A. D., 2014, “Quadratic Programming andImpedance Control for Transfemoral Prosthesis,” International Conference onRobotics and Automation (ICRA), Hong Kong, China, May 31–June 7, pp.1341–1347.

[7] Gregg, R. D., Lenzi, T., Hargrove, L. J., and Sensinger, J. W., 2014, “VirtualConstraint Control of a Powered Prosthetic Leg: From Simulation to Experi-ments With Transfemoral Amputees,” IEEE Trans. Rob., 30(6), pp. 1455–1471.

[8] Young, A. J., Simon, A. M., and Hargrove, L. J., 2014, “A Training Method forLocomotion Mode Prediction Using Powered Lower Limb Prostheses,” IEEETrans. Neural Syst. Rehabil. Eng., 22(3), pp. 671–677.

[9] Gonzalez, M., 2014, “Biomechanical Analysis of Gait Kinetics Resulting FromUse of a Vacuum Socket on a Transtibial Prosthesis,” Master’s thesis, Univer-sity of Nevada, Las Vegas, NV.

[10] Murakami, T., Nakamura, R., Yu, F., and Ohnishi, K., 1993, “Force SensorlessImpedance Control by Disturbance Observer,” IEEE Power Conversion Confer-ence, Yokohama, Japan, Apr. 19–21, pp. 352–357.

[11] Simpson, J. W. L., Cook, C. D., and Li, Z., 2002, “Sensorless Force Estimationfor Robots With Friction,” Australasian Conference on Robotics and Automa-tion, Auckland, New Zealand, Nov. 27–29, pp. 94–99.

[12] Phong, L. D., Choi, J., Lee, W., and Kang, S., 2015, “A Novel Method for Esti-mating External Force: Simulation Study With a 4-DOF Robot Manipulator,”Int. J. Precis. Eng. Manuf., 16(4), pp. 755–766.

[13] Hacksel, P. J., and Salcudean, S. E., 1994, “Estimation of Environment Forcesand Rigid-Body Velocities Using Observers,” IEEE International Conferenceon Robotics and Automation (ICRA), San Diego, CA, May 8–13, pp. 931–936.

[14] Rigatos, G. G., 2012, “A Derivative-Free Kalman Filtering Approach to StateEstimation-Based Control of a Class of Nonlinear Systems,” IEEE Trans. Ind.Electron., 59(10), pp. 3987–3997.

[15] Sarkka, S., 2007, “On Unscented Kalman Filtering for State Estimation ofContinuous-Time Nonlinear Systems,” IEEE Trans. Autom. Control, 52(9),pp. 1631–1641.

[16] Julier, S. J., Uhlmann, J. K., and Durrant-Whyte, H. F., 1995, “A NewApproach for Filtering Nonlinear Systems,” American Control Conference(ACC), Seattle, WA, June 21–23, pp. 1628–1632.

[17] Fakoorian, S. A., Simon, D., Richter, H., and Azimi, V., 2016, “Ground Reac-tion Force Estimation in Prosthetic Legs With an Extended Kalman Filter,”IEEE International Systems Conference (SysCon), Orlando, FL, Apr. 18–21,pp. 338–343.

[18] Richter, H., and Simon, D., 2014, “Robust Tracking Control of a ProsthesisTest Robot,” ASME J. Dyn. Syst. Meas. Control, 136(3), p. 031011.

[19] van den Bogert, A. J., Geijtenbeek, T., Even-Zohar, O., Steenbrink, F., and Har-din, E. C., 2013, “A Real-Time System for Biomechanical Analysis of HumanMovement and Muscle Function,” Med. Biol. Eng. Comput., 51(10), pp.1069–1077.

[20] Azimi, V., Simon, D., and Richter, H., 2015, “Stable Robust Adaptive Imped-ance Control of a Prosthetic Leg,” ASME Paper No. DSCC2015-9794.

[21] Richter, H., Simon, D., Smith, W. A., and Samorezov, S., 2015, “DynamicModeling, Parameter Estimation and Control of a Leg Prosthesis Test Robot,”Appl. Math. Modell., 39(2), pp. 559–573.

[22] Mohammadi, H., and Richter, H., 2015, “Robust Tracking/Impedance Control:Application to Prosthetics,” American Control Conference (ACC), Chicago, IL,July 1–3, pp. 2673–2678.

[23] Warner, H., 2015, “Optimal Design and Control of a Lower-Limb ProsthesisWith Energy Regeneration,” Master’s thesis, Cleveland State University, Cleve-land, OH.

[24] Azimi, V., Simon, D., Richter, H., and Fakoorian, S. A., 2016, “Robust Com-posite Adaptive Transfemoral Prosthesis Control With Non-Scalar BoundaryLayer Trajectories,” American Control Conference (ACC), Boston, MA, July6–8, pp. 3002–3007.

[25] Warner, H., Simon, D., Mohammadi, H., and Richter, H., 2016, “SwitchedRobust Tracking/Impedance Control for an Active Transfemoral Prosthesis,”American Control Conference (ACC), Boston, MA, July 6–8, pp. 2187–2192.

[26] van der Merwe, R., and Wan, E. A., 2001, “The Square-Root Unscented Kal-man Filter for State and Parameter-Estimation,” IEEE International Conferenceon Acoustics, Speech, and Signal Processing (ICASSP’01), Salt Lake City, UT,May 7–11, pp. 3461–3464.

[27] Simon, D., 2006, Optimal State Estimation: Kalman, H Infinity, and NonlinearApproaches, Wiley, Hoboken, NJ.

[28] Safonov, M., and Athans, M., 1978, “Robustness and Computational Aspects ofNonlinear Stochastic Estimators and Regulators,” IEEE Trans. Autom. Control,23(4), pp. 717–725.

[29] Ljung, L., 1979, “Asymptotic Behavior of the Extended Kalman Filter as aParameter Estimator for Linear Systems,” IEEE Trans. Autom. Control, 24(1),pp. 36–50.

[30] Ursin, B., 1980, “Asymptotic Convergence Properties of the Extended KalmanFilter Using Filtered State Estimates,” IEEE Trans. Autom. Control, 25(6),pp. 1207–1211.

[31] Reif, K., Gunther, S., Yaz, E., and Unbehauen, R., 2000, “Stochastic Stabilityof the Continuous-Time Extended Kalman Filter,” IEEE Proc. Control TheoryAppl., 147(1), pp. 45–72.

[32] Reif, K., Gunther, S., Yaz, E., and Unbehauen, R., 1999, “Stochastic Stabilityof the Discrete-Time Extended Kalman Filter,” IEEE Trans. Autom. Control,44(4), pp. 714–728.

[33] Fitts, J. M., 1972, “On the Observability of Nonlinear Systems With Applica-tions to Nonlinear Regression Analysis,” Inf. Sci., 4(2), pp. 129–156.

[34] Sontag, E. D., 1979, “On the Observability of Polynomial Systems—I: Finite-Time Problems,” SIAM J. Control Optim., 17(1), pp. 139–151.

[35] Gard, T. C., 1988, Introduction to Stochastic Differential Equations, M.Dekker, New York.

[36] Zakai, M., 1967, “On the Ultimate Boundedness of Moments Associated WithSolutions of Stochastic Differential Equations,” SIAM J. Control, 5(4),pp. 588–593.

[37] Shanying, Z., Chen, C., Li, W., Yang, B., and Guan, X., 2013, “DistributedOptimal Consensus Filter for Target Tracking in Heterogeneous SensorNetworks,” IEEE Trans. Cybern., 43(6), pp. 1963–1976.

[38] Tønne, K. K., 2007, “Stability Analysis of EKF-Based Attitude Determinationand Control,” Master’s thesis, Norwegian University of Science and Technol-ogy, Trondheim, Norway.

[39] Baras, J. S., Bensoussan, A., and James, M. R., 1988, “Dynamic Observersas Asymptotic Limits of Recursive Filters: Special Cases,” SIAM J. Appl.Math., 48(5), pp. 1147–1158.

[40] Gauthier, J. P., Hammouri, H., and Othman, S., 1992, “A Simple Observer forNonlinear Systems Applications to Bioreactors,” IEEE Trans. Autom. Control,37(6), pp. 875–880.

[41] Khademi, G., Mohammadi, H., Hardin, E. C., and Simon, D., 2015,“Evolutionary Optimization of User Intent Recognition for TransfemoralAmputees,” IEEE Biomedical Circuits and Systems Conference (BioCAS),Atlanta, GA, Oct. 22–24, pp. 1–4.

[42] Mao, X., 2007, Stochastic Differential Equations and Applications, Elsevier,Cambridge, UK.



http://dx.doi.org/10.1109/ICRA.2014.6907026

http://dx.doi.org/10.1109/TRO.2014.2361937



http://digitalscholarship.unlv.edu/cgi/viewcontent.cgi?article=1014amp;context=honors_theses

http://dx.doi.org/10.1109/PCCON.1993.264158

http://dx.doi.org/10.1109/PCCON.1993.264158

https://pdfs.semanticscholar.org/17ad/b33f0e6f6ceaa05dbecbe5781232709967b0.pdf

https://pdfs.semanticscholar.org/17ad/b33f0e6f6ceaa05dbecbe5781232709967b0.pdf

http://dx.doi.org/10.1007/s12541-015-0100-7

http://dx.doi.org/10.1109/ROBOT.1994.351233

http://dx.doi.org/10.1109/TIE.2011.2159954

http://dx.doi.org/10.1109/TIE.2011.2159954

http://dx.doi.org/10.1109/TAC.2007.904453

http://dx.doi.org/10.1109/ACC.1995.529783

http://dx.doi.org/10.1109/SYSCON.2016.7490563

http://dx.doi.org/10.1115/1.4026342

http://dx.doi.org/10.1115/DSCC2015-9794

http://dx.doi.org/10.1016/j.apm.2014.06.006


http://engagedscholarship.csuohio.edu/cgi/viewcontent.cgi?article=1678&context=etdarchive



http://dx.doi.org/10.1109/ICASSP.2001.940586




http://dx.doi.org/10.1049/ip-cta:20000125

http://dx.doi.org/10.1049/ip-cta:20000125

http://dx.doi.org/10.1109/9.754809

http://dx.doi.org/10.1016/S0020-0255(72)80008-2

http://dx.doi.org/10.1137/0317011

http://dx.doi.org/10.1137/0305037

http://dx.doi.org/10.1109/TSMCB.2012.2236647

http://folk.ntnu.no/tomgra/Diplomer/Tonne.pdf

http://dx.doi.org/10.1137/0148068

http://dx.doi.org/10.1137/0148068

http://dx.doi.org/10.1109/9.256352

http://dx.doi.org/10.1109/BioCAS.2015.7348280

cleveland state university, estimation in prosthetic legs

Documents