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Real-time Kinematic Modeling and Prediction of Human Joint Motion

in a Networked Rehabilitation System

Wenlong Zhang, Xu Chen, Joonbum Bae, and Masayoshi Tomizuka

Abstract— In this paper, a networked-based rehabilitation

system is introduced for lower-extremity tele-rehabilitation. In

order to enable high-level motion planning of the rehabilitation

robot in real-time for enhanced safety and appropriate human-

robot interactions, a time series model is proposed to capture

the kinematics of knee joint rotations. A major challenge in such

a system is that measurement data might be delayed or lost due

to wireless communication. With a delay and loss compensation

mechanism, a modiﬁed recursive least square (mRLS) algorithm

is applied for real-time modeling and prediction of knee joint

rotations in the sagittal plane, and convergence of the proposed

algorithm is studied. Simulation and experimental results are

presented to verify the performance of the proposed algorithm.

I. INTRODUCTION

In view of the rapidly increasing number of patients and

elderly people who need physical therapy, there is a large

demand for gait rehabilitation and assistive devices. Many

companies and researchers have developed various kinds

of assistive robots to facilitate the rehabilitation treatment

and patients’ daily life [1]–[3]. As rehabilitation systems

become more sophisticated, mobility becomes an important

issue, which motivates us to integrate network media into a

rehabilitation system, as is shown in this paper.

In order to achieve smart rehabilitation treatment, it is

important to understand the user’s joint movement. This has

motivated intensive studies of human motion capture and

analysis with optical sensors [4], inertial sensors [5], and

electromyography (EMG) sensors [6]. For lower-extremity

motion analysis, joint kinematic model is often employed to

detect gait phases [7] for identifying gait abnormality. It is

also frequently used to estimate human motion intention [8]

for trajectory planning of the assistive robot. Joint kinematics

is also particularly useful in fall prediction to enhance the

safety of a rehabilitation robot [9].

Motivated by the importance of joint kinematic modeling,

an autoregressive integrated (ARI) model was built based

on time series analysis to predict knee joint rotation in a

network-based rehabilitation system [10]. While the pro-

posed technique can provide a reliable human knee joint

model for motion prediction, it is difﬁcult to implement

This work was supported by National Science Foundation under Grant

CMMI-1013657.

W. Zhang and M. Tomizuka are with the Department of Mechanical

Engineering, University of California, Berkeley, CA 94720 USA (e-mail:

wlzhang @berkeley.edu;tomizuka@me.berkeley.edu)

X. Chen is with the Department of Mechanical Engineering, University

of Connecticut, Storrs, CT 06269 USA (e-mail: xchen@engr.uconn.edu)

J. Bae is with the School of Mechanical and Nuclear Engineering, Ulsan

National Institute of Science and Technology, Ulsan 689-798, Korea (e-mail:

jbbae@unist.ac.kr)

the algorithm online due to its computational complexity.

It would be ideal if an online adaptive human motion model

can be built based on new measurements from the user. Such

a model is able to capture the change of walking dynamics

more accurately and provide more insights to the user.

In this paper, an online adaptive knee joint rotation model

is built based on a modiﬁed recursive least square (mRLS)

algorithm. Since measurement data are transmitted over the

wireless network, network-induced constraints, such as time

delay and packet loss, need to be handled in the modeling

process. A delay and loss compensator is thus proposed with

the proof of convergence in this paper. Simulation and exper-

imental results are demonstrated to verify the effectiveness of

the proposed algorithm. The inﬂuences of time delay, packet

loss, and forgetting factors in the algorithm are analyzed

based on the simulation and experimental results.

The remainder of this paper is organized as follows. In

Section II, the networked rehabilitation system is brieﬂy in-

troduced and joint angle measurement for one healthy subject

is presented. Section III proposes the mRLS algorithm for

joint rotation modeling and illustrates how it can handle

the network-induced challenges. Convergence analysis is

presented in Section IV. Section V shows simulation results

and analyzes the performance of the proposed approach.

Experimental results with another healthy subject are shown

in Section VI. Conclusion and future work are given in

Section VII.

II. JOINT ANGLE MEASUREMENT IN A

REHABILITATION SYSTEM

In our previous work [11], a network-based rehabilitation

system was proposed for improved mobility and in-home

tele-rehabilitation. The system consists of a wireless body

sensor network and a rehabilitation robot controlled over

a high-speed wireless network. A computer at the patient’s

home wirelessly connects to all sensors and design the proper

control signals for the robot. The proposed algorithm will be

implemented in the local computer.

In the proposed system, human joint rotation in three

dimensions can be captured by several inertial sensors.

A wireless inertial measurement unit (IMU) is shown in

Fig. 1. An IMU node consists of a three degrees of freedom

accelerometer, magnetometer, and gyroscope. The measure-

ment data can be transmitted to the local computer wirelessly.

The IMU node is powered by a Li-Po battery and it can work

continuously for 90 minutes. The dimension of one wireless

IMU node is 2 inches ×1.4 inches ×0.6 inches and its

weight is around 0.15 lbs including the battery. The wireless

Fig. 1: An inertial measurement unit (IMU) and a human

subject walking on a treadmill with IMUs

IMU can achieve a sampling rate up to 100Hz and it is very

convenient to be attached to a subject’s lower extremities

using velcros, as is shown in Fig. 1.

To examine the performance of the proposed algorithm,

several sets of knee joint rotation measurements were col-

lected. With two IMUs attached to his left thigh and shank,

a 25-year old male user was asked to walk on a treadmill at

1 mph in the Mechanical Systems Control Laboratory at the

University of California, Berkeley. The user had no known

walking abnormalities. The measurement data were recorded

at 50 Hz and representative knee joint angles in the sagittal

plane is shown in Fig. 2. The measurement data will be used

in the modeling and simulation study.

III. HUMAN MOTION MODELING WITH A

MODIFIED RECURSIVE LEAST SQUARE METHOD

A. Problem Formulation and Algorithm Design

In this paper, the following time-varying linear kinematic

model is built

y(k) = ΦT(k)θ(k) + v(k),(1)

where y(k)is the measured human joint angle and Φ (k) =

y(k−1) y(k−2) · · · y(k−n)T∈Rnis the re-

gressor vector that stores the previous measurement of the

joint angles, θ(k) = θ1(k)θ2(k)· · · θn(k)T∈Rn

is a time-varying parameter vector that needs to be estimated.

v(k)is white noise with zero mean and variance σ2

v, and it

is independent of the current and previous regressors, i.e.,

E[v(k)] = 0, E v2(k)=σ2

v<∞,(2)

E[v(k)v(j)] = 0,∀k=j, (3)

E[Φ (k−i)v(k)] = 0,∀i≥0.(4)

The proposed model (1) is an autoregressive model, and its

capability of capturing human joint kinematics was validated

in [10]. In order to estimate the parameter vector, the

following least square cost function is considered:

J(k) = min

ˆ

θ(k)

1

2

k

j=1

λk−jy(j)−ΦT(j)ˆ

θ(k)2

.(5)

0 10 20 30 40 50 60 70 80

0

20

40

60

Time (sec)

Knee joint angle (deg)

Fig. 2: Knee joint angle measurement

This problem can be solved recursively as follows:

eo(k) = y(k)−ΦT(k)ˆ

θ(k),(6)

F(k) = 1

λF(k−1) −F(k−1)Φ(k)ΦT(k)F(k−1)

λ+ΦT(k)F(k−1)Φ(k),(7)

ˆ

θ(k+ 1) = ˆ

θ(k) + F(k) Φ (k)eo(k),(8)

where eo(k)is the a-priori estimation error and F(k)is

the adaptation gain matrix. Note that (7) is equivalent to

F−1(k) = λF −1(k−1) + Φ (k) ΦT(k).λis a constant

forgetting factor that needs to be selected prior to the

modeling process. Furthermore, it is assumed that the true

parameters follow a random walk process as follows:

θ(k+ 1) = θ(k) + w(k),(9)

where w(k)is a white noise with zero mean and variance

σ2

w, and it is independent of the true parameters, current and

previous regressors, and process noise, i.e.,

E[w(k)] = 0, E wT(k)w(k)=σ2

w<∞,(10)

Ew(k)wT(j)=0,∀k=j, (11)

EΦ (k−i)wT(k)=0,∀i≥0,(12)

E[w(k)v(i)] = 0,∀i. (13)

The random walk assumption of the true model parameters

follows the observation that the gait pattern of a subject is

consistent between steps with the same walking speed and

road condition. However, there must be some minor differ-

ences of walking behaviors between steps. The differences

are small for healthy subjects but large for patients with gait

abnormalities. Therefore, σ2

wis typically small for healthy

subjects and large for patients.

B. Model Structure Selection

In this paper, the model to be built has the form (1), and the

order of the model is ﬁrstly determined. It is shown in [10]

that one can decide the order of the model using Box-Jenkins

approach [12]. Applying this approach to the data shown in

Fig. 2 suggests a ninth-order autoregressive model with a

ﬁrst-order integration action. Thus, the order of the proposed

model is picked as 10, i.e., Φ (k)∈R10 and θ(k)∈R10.

C. Time Delay and Packet Loss

In a network-based rehabilitation system, sensing packet is

transmitted to the estimator over the wireless network. Time

delay may happen during data transmission, making the most

recent sensing packet unavailable for parameter update. A

Sensor

Estimator

Time

s

kT

(

)

1

s

k T

+

(

)

2

s

k T

+

(

)

0

s

k n T

+

(

)

0

1

s

k n T

+ +

k

τ

1

k

τ

+

2

k

τ

+

0

k n

τ

+

Fig. 3: Timing diagram of the measurement packet

representative timing diagram of the sensing packet delivery

is shown in Fig. 3, where τkis the delay of the kth

measurement packet and Tsis the sampling interval of the

system. Similarly, packet loss of sensing packets can also

happen randomly, which leads to irregular parameter update

and makes the convergence analysis challenging.

Time delay and packet loss have two major negative effects

to the parameter estimation. On one hand, the measurement

y(k)might be unavailable for calculating eo(k)using (6).

On the other hand, delayed or lost packet might need to be

used in the regressor Φ (k)to update model parameters and

the adaption gain.

In order to deal with varying time delay during net-

work transmission, the following compensation mechanism

is used. If an output measurement y(k)is not available, the

predicted output ˆy(k)from the identiﬁed model is used, and

the parameter will not be updated for this step. The identiﬁed

model will also be used to predict missing elements in the

regressor Φ (k). The mRLS algorithm can be expressed as

eo(k) = z(k)−ΦT

e(k)ˆ

θ(k),(14)

F(k) = 1

λF(k−1) −F(k−1)Φe(k)ΦT

e(k)F(k−1)

λ+ΦT

e(k)F(k−1)Φe(k),(15)

ˆ

θ(k+ 1) = ˆ

θ(k) + F(k) Φe(k)eo(k),(16)

where z(k) = γ(k)y(k) + [1 −γ(k)] ˆy(k),ˆy(k) =

ΦT

e(k)ˆ

θ(k), and γ(k)∼Bernoulli (γ)is an indicator of

successful transmission of the measurement packet y(k).

Φe(k) = s(k−1) s(k−2) · · · s(k−n)T∈Rn,

where

s(k−i) = βk−i(k)y(k−i) + [1 −βk−i(k)] ˆy(k−i).

βk−i(k)is an availability indicator of measurement packet

y(k−i)at the kth time step. Note that a packet will be

available in the regressor several steps later if it is delayed,

but it will not be available if it is lost without retransmission.

IV. CONVERGENCE ANALYSIS

In this section, performance of the proposed mRLS algo-

rithm is examined analytically by deriving the upper bound

of the parameter estimation errors. Based on system model

(1), assumptions for v(k)and w(k)in (2)-(4) and (10)-

(13), and mRLS algorithm (14)-(16), the following further

assumptions are made for derivation of the error bounds [13]:

(A1) For some constants 0< α ≤β < ∞and an integer

N≥n, the following strong persistent excitation

condition holds:

αIn≤1

N+ 1

N

i=0

Φe(k+i) ΦT

e(k+i)≤βIn,(17)

where Inis an n-dimensional identity matrix.

(A2) The estimator gain matrix is initialized as F(−1) =

p0In, where 1−λ

(N+1)β≤p0≤1−λ

α.

(A3) The parameter estimate is initialized as ˆ

θ(0) = c01n,

where c0is a small positive number and 1nis an n-

dimensional vector with all elements equal to 1. In

addition, ˆ

θ(0) is independent of v(k).

(A4) Parmeter estimation error is deﬁned as

˜

θ(k) = ˆ

θ(k)−θ(k),(18)

where ˜

θ(0) satisﬁes E˜

θT(0) ˜

θ(0)=δ0<∞.

The assumptions above lead to the following lemma:

Lemma 1: For the system described in (1) and mRLS

algorithm expressed in (14)-(16), if assumption (A1) holds,

then for k≥Nand 0<λ<1the adaption gain matrix

F(k)satisﬁes

λNα

1−λIn+λk+1 F−1(−1) −α

1−λIn≤F−1(k)≤(19)

(N+ 1) β

1−λIn+λk+1 F−1(−1) −(N+ 1) β

1−λIn.

Proof: The proof is similar to that of Lemma 1 in [13]

and is therefore omitted.

Note from Lemma 1 that if both (A1) and (A2) are

satisﬁed, the following inequality is satisﬁed

1−λ

(N+ 1) βIn≤F(k)≤1−λ

λNαIn(20)

for k≥N. Now the theorem that addresses the upper bounds

of parameter estimate errors can be provided.

Theorem 1: Consider the system described in (1), mRLS

algorithm expressed in (14)-(16), and v(k)and w(k)satis-

fying (2)-(4) and (10)-(13), respectively. If the assumptions

(A1)-(A4) are satisﬁed, the expected norm of estimation error

˜

θ(k)satisﬁes the following upper bound for all N≤k < ∞.

E

˜

θ(0),

γ(1),··· ,γ(k),

β0(1),··· ,βk(k)

[

˜

θ(k)

2]≤α−2p2

0λ2(k−N+1) (1 −λ)2δ0(21)

+1−λ

λNαnB (k) + n(1−λ)

αλN−1σ2

v+(N+1)2β2

α2λ2(N−1)(1−λ)2σ2

w.

where αand βare deﬁned in (A1), and B(k)depends on

the performance of network transmission.

Proof: Based on deﬁnition in (18) and mRLS algorithm

(14)-(16), the following equations are obtained:

˜

θ(k+ 1) = ˆ

θ(k+ 1) −θ(k+ 1) (22)

=ˆ

θ(k+ 1) −[θ(k) + w(k)]

=˜

θ(k) + F(k) Φe(k)v(k)−F(k) Φe(k) ΦT

e(k)˜

θ(k)

+F(k) Φe(k) [Φ (k)−Φe(k)]Tθ(k)−w(k).

By deﬁning ∆ (k) = Φe(k)−Φ (k), the equation above can

be written as

˜

θ(k+ 1) = ˜

θ(k)−w(k) + F(k) Φe(k)v(k)(23)

−F(k) Φe(k) ΦT

e(k)˜

θ(k)−F(k) Φe(k) ∆T(k)θ(k)

=In−F(k) Φe(k) ΦT

e(k)˜

θ(k) + F(k) Φe(k)v(k)

−F(k) Φe(k) ∆T(k)θ(k)−w(k).

Note (15) yields Φe(k) ΦT

e(k) = F−1(k)−λF −1(k−1).

Thus, the equation above can be rewritten as

˜

θ(k+ 1) = λF (k)F−1(k−1) ˜

θ(k) + F(k) Φe(k)v(k)(24)

−F(k) Φe(k) ∆T(k)θ(k)−w(k).

Induction of (15) and (24) yields

F−1(k−1) = λF −1(k−2) + Φe(k−1) ΦT

e(k−1) ,(25)

˜

θ(k) = λF (k−1) F−1(k−2) ˜

θ(k−1) (26)

+F(k−1) Φe(k−1) v(k−1) −w(k−1)

−F(k−1) Φe(k−1) ∆T(k−1) θ(k−1) .

Plugging in F−1(k−1) and ˜

θ(k)into (24) results in the

following equation

˜

θ(k+ 1) = λ2F(k)F−1(k−2) ˜

θ(k−1) (27)

−F(k) Φe(k) ∆T(k)θ(k)

−λF (k) Φe(k−1) ∆T(k−1) θ(k−1)

+λF (k) Φe(k−1) v(k−1) + F(k) Φe(k)v(k)

−λF (k)F−1(k−1) w(k−1) −w(k).

Continuing the induction yields

˜

θ(k+ 1) = λk+1F(k)F−1(−1) ˜

θ(0) (28)

−

k

i=0

λk−iF(k) Φe(i) ∆T(i)θ(i)

−

k

i=0

λk−iF(k)F−1(i)w(i)

+

k

i=0

λk−iF(k) Φe(k)v(i)

=α1(k) + α2(k) + α3(k) + α4(k),

where the upper bounds of expected 2-norm of α1(k),

α3(k), and α4(k)can be similarily calculated as shown in

[13]. For the bounds of α2(k), the following transformation

is performed:

α2(k) = −

k

i=0

λk−iF(k) Φe(i) ∆T(i)θ(i)(29)

=−F(k)GT(k)D(k),

where

G(k) =

ΦT

e(k)

µΦT

e(k−1)

.

.

.

µkΦT

e(0)

∈R(k+1)×n, µ := √λ,

D(k) =

∆T(k)θ(k)

µ∆T(k−1) θ(k−1)

.

.

.

µk∆T(0) θ(0)

∈Rk+1.

The following assumption is made

E

βk−1(k),··· ,βk−n(k)D(k)DT(k)≤B(k)

since both ∆ (k)and θ(k)are ﬁnite as long as k < ∞.

Note that F−1(k) = GT(k)G(k) + λk+1F−1(−1). This

equation yields

tr F(k)GT(k)G(k)=tr In−λk+1F(k)F−1(−1)(30)

≥tr 1−(1 −λ)λk+1

λNαp0In

=n1−α−1p−1

0λk−N+1 (1 −λ)

based on (20). Similarly, one can get

tr F(k)GT(k)G(k)=tr In−λk+1F(k)F−1(−1)(31)

=tr In−λk+1F(k)

p0≤n.

In this case, the following upper bound of E

α2(k)

2

can be derived (the subscript of expectation is omitted)

E

α2(k)

2=Etr F(k)GT(k)D(k)DT(k)G(k)F(k)

(32)

≤tr EF(k)GT(k)G(k) 1−λ

λNαB(k)

≤1−λ

λNαnB (k).

Therefore the estimation error is bounded. Proof is complete.

It is clear that the performance of parameter inference

depends on the bound B(k), which is a function of the time

delay and packet loss. In the next section, simulations will

be conducted to examine the inﬂuence of networked-induced

constraints to parameter adaption and motion prediction.

V. SIMULATION STUDY

A. Simulation Results without Network-induced Constraints

In this subsection, the proposed mRLS algorithm is imple-

mented without any network-induced constraints. The knee

joint rotation data shown in Fig. 2 were used to build a

10th order linear model and the parameter adaption result

with a forgetting factor λ= 1 is shown in Fig. 4, from

which one can observe that the coefﬁcient for the most recent

measurement (θ1)dominates. Moreover, the parameters did

not converge even at the end of the simulation. In order

to achieve a faster convergence of the model parameters,

forgetting factors are introduced into parameter adaption.

Parameter adaption result with a forgetting factor λ= 0.997

is shown in Fig. 5, which conﬁrms the convergence of the

model parameters. Comparing with Fig. 4, one can conclude

that introducing forgetting factors into the mRLS algorithm

leads to faster parameter convergence with ﬂuctuations.

Using the identiﬁed model shown in Fig. 5, a 5-step

prediction is performed to examine the accuracy of the model

and prediction errors are shown in Fig. 6. It is clear that

the prediction error is less than 1 degree, which veriﬁes the

performance of the identiﬁed model. As a baseline algorithm,

the simplest way of 5-step prediction is to use the current

measurement as a rough estimate, which is equivalent to

Max. Delay γλ

0.993 0.995 0.997 0.999 1

0 100% 0.1875 0.1891 0.1918 0.1985 0.2078

2Ts95% 0.6422 0.6368 0.6369 0.6475 0.6732

2Ts90% 1.0002 0.9088 0.8653 0.8165 0.8017

4Ts90% 2.1186 2.1713 2.0135 1.8206 1.7627

TABLE I: Root-mean-square (RMS) errors of 5-step predic-

tion with different forgetting factors (deg)

0 10 20 30 40 50 60 70 80

-2

-1

0

1

Time (sec)

Parameter values

θ

1

θ

2

θ

3

θ

4

θ

5

(a) Adaption of the ﬁrst ﬁve model parameters

0 10 20 30 40 50 60 70 80

-0.2

-0.1

0

0.1

0.2

Time (sec)

Parameter values

θ

6

θ

7

θ

8

θ

9

θ

10

(b) Adaption of the last ﬁve model parameters

Fig. 4: Adaption of model parameters without forgetting

factors

a zero-order hold (ZOH) and it yields a root-mean-square

(RMS) error of 5.0146 degrees. In order to further examine

the effectiveness of the forgetting factors, the 5-step predic-

tion errors of the identiﬁed model with different forgetting

factors are shown in the ﬁrst row of Table I, which veriﬁes the

performance improvement brought by the forgetting factors.

It is also veriﬁed that the prediction errors become smaller

as forgetting factors decrease.

It is noticed that the forgetting factors are chosen to be

very close to 1 in the simulations, which is due to the nature

of the human walking pattern. Since walking patterns in each

gait cycle cannot be the same, the true kinematic model

is also varying. Moreover, there are inevitably unexpected

walking patterns and measurement noises in the parameter

adaption process. Thus, although choosing small forgetting

factors might make the parameters converge faster, it will

make the algorithm aggressive and result in large oscillations

in the parameter adaption. Applying a low-pass ﬁlter to

the measurement data might reduce such oscillations, but

it might eliminate useful information in the measurements.

B. Simulation Results with Time Delay and Packet Loss

This section shows the simulation results of model identi-

ﬁcation with time delay and packet loss. We chose the suc-

cessful transmission rate γto be 90% and 95%, respectively,

and the time delay of the kth packet τk∼unif (0,4Ts)and

τk∼unif (0,2Ts)respectively. Parameter adaption result of

0 10 20 30 40 50 60 70 80

-2

0

2

4

Time (sec)

Parameter values

θ

1

θ

2

θ

3

θ

4

θ

5

(a) Adaption of the ﬁrst ﬁve model parameters

0 10 20 30 40 50 60 70 80

-0.4

-0.2

0

0.2

0.4

Time (sec)

Parameter values

θ

6

θ

7

θ

8

θ

9

θ

10

(b) Adaption of the last ﬁve model parameters

Fig. 5: Adaption of model parameters with a forgetting factor

λ= 0.997

0 10 20 30 40 50 60 70 80

-1

-0.5

0

0.5

1

Time (sec)

Prediction error (deg)

Fig. 6: Performance of 5-step prediction using identiﬁed

linear model (λ= 0.997)

the proposed algorithm with γ= 95%,τk∼unif (0,2Ts),

and λ= 0.997 is shown in Fig. 7. It is veriﬁed that the

parameters converge even with packet loss and time delay,

but the identiﬁed model parameters are quite different.

The effectiveness of forgetting factors is a bit more com-

plicated in the cases with network-induced constraints, as is

shown in Table I. When the time delay and packet loss are

not very signiﬁcant (the case with a maximum of two-step

delay and γ= 95%), reducing forgetting factors may lead

to an improvement of the prediction accuracy. However in

this case, when the forgetting factor is reduced to 0.993, the

prediction error is even larger than the case with a forgetting

factor of 0.995. Moreover, when the packet loss and time

delay become more severe, reducing forgetting factors will

lead to large ﬂuctuations of the adapted parameters and

prediction errors may be even larger than the case without

forgetting factors. To summarize, larger forgetting factors

need to be chosen with network-induced constraints.

VI. EXPERIMENTAL RESULTS

In this section, the proposed system is implemented in

LabVIEW to build the knee joint kinematic model in real-

time. The wireless IMU sensors were used to measure knee

joint rotation angles in the sagittal plane. The sampling rate

0 10 20 30 40 50 60 70 80

-2

-1

0

1

2

3

Time (sec)

Parameter values

θ

1

θ

2

θ

3

θ

4

θ

5

(a) Adaption of the ﬁrst ﬁve model parameters

0 10 20 30 40 50 60 70 80

-1

-0.5

0

0.5

1

Time (sec)

Parameter values

θ

6

θ

7

θ

8

θ

9

θ

10

(b) Adaption of the last ﬁve model parameters

Fig. 7: Adaption of model parameters with a maximum two-

step delay, γ=95%, and λ= 0.997

0 10 20 30 40 50 60 70 80

-2

0

2

4

6

Time (sec)

Parameter values

θ

1

θ

2

θ

3

θ

4

θ

5

θ

6

(a) Adaption of the ﬁrst six model parameters

0 10 20 30 40 50 60 70 80

-1

0

1

2

Time (sec)

Parameter values

θ

7

θ

8

θ

9

θ

10

θ

11

θ

12

(b) Adaption of the last six model parameters

Fig. 8: Adaption of model parameters in the experiment with

a forgetting factor λ= 0.995

was chosen to be 50Hz and there might be time delay and

packet loss due to the wireless transmission of the sensor

signals. In order to verify the performance of the proposed

algorithm, a different subject was selected (a 21-year old

male without known walking abnormalities) and he was

asked to walk on a treadmill at 2 mph. A pre-trial was

conducted for 10 seconds and the recorded data were used to

determine the order of the model. Based on the techniques

presented in Section III-B, the order was determined to be

12. In the formal experiment the subject was asked to walk

on the treadmill with the same speed.

The identiﬁed model parameters for the ﬁrst 80 seconds of

formal experiment are shown in Fig. 8. The forgetting factor

was chosen as 0.995. It is evident that all parameters ﬁnally

converge. The identiﬁed model was used to achieve 5-step

prediction, and the RMS prediction error of the proposed

algorithm is 2.4027 degrees. The 5-step RMS prediction

error from the baseline algorithm is 11.2674 degrees due

to the increased speed, from which the effectiveness of the

proposed algorithm is conﬁrmed.

VII. CONCLUSION

In this paper, a networked rehabilitation system was

introduced for lower-extremity rehabilitation. In order to

enable high-level motion planning for enhanced safety and

appropriate human-robot interactions, a time series model

was proposed to describe the knee joint rotation. In view

of time delay and packet loss of the measurement packets,

a modiﬁed recursive least square (mRLS) algorithm was

proposed for real-time modeling of the knee joint rotation

and convergence of the proposed algorithm was studied.

Effectiveness of the time delay, packet loss, and forgetting

factors was veriﬁed in simulations and experiments.

The proposed algorithm has many potentially important

applications, one of which is to achieve fall prediction based

on a fast change of identiﬁed model parameters. Another

application is to achieve high-level trajectory planning of the

rehabilitation robot based on the predicted human motion.

One of the ongoing work is to get the data from stroke

and Parkinson’s disease patients with walking abnormalities.

Model orders and parameters of the patients will be com-

pared and analyzed.

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