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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.
REF ER E NC ES
[1] “Lokomat,” http://www.hocoma.com/.
[2] R. J. Farris, H. A. Quintero, and M. Goldfarb, “Preliminary evaluation
of a powered lower limb orthosis to aid walking in paraplegic
individuals,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 19, no.
6, pp. 652–659, 2011.
[3] S. K. Banala, S. Kim, S. K. Agrawal, and J. P. Scholz, “Robot assisted
gait training with active leg exoskeleton (ALEX),” IEEE Trans. Neural
Syst. Rehabil. Eng., vol. 17, no. 1, pp. 2–8, 2009.
[4] “Vicon Bonita,” http://www.vicon.com/.
[5] “Xsens MVN,” http://www.xsens.com/.
[6] X. Zhang, X. Chen, Y. Li, V. Lantz, K. Wang, and J. Yang, “A
framework for hand gesture recognition based on accelerometer and
emg sensors,” IEEE Trans. Syst., Man, Cybern. A, vol. 41, no. 6, pp.
1064–1076, 2011.
[7] I. Pappas, T. Keller, S. Mangold, M. Popovic, V. Dietz, and M. Morari,
“A reliable gyroscope-based gait-phase detection sensor embedded in
a shoe insole,” IEEE Sensors J., vol. 4, no. 2, pp. 268–274, 2004.
[8] H. Varol, F. Sup, and M. Goldfarb, “Multiclass real-time intent
recognition of a powered lower limb prosthesis,” IEEE Trans. Biomed.
Eng., vol. 57, no. 3, pp. 542–551, 2010.
[9] G. Wu and S. Xue, “Portable preimpact fall detector with inertial
sensors,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 16, no. 2, pp.
178–183, 2008.
[10] W. Zhang, M. Tomizuka, and J. Bae, “Time series prediction of knee
joint movement and its application to a network-based rehabilitation
system,” in Proc. 2014 American Control Conf., 2014, pp. 4810–4815.
[11] W. Zhang, X. Zhu, S. Han, N. Byl, A. K. Mok, and M. Tomizuka,
“Design of a network-based mobile gait rehabilitation system,” in
Proc. Int’l Conf. on Robotics and Biomimetics, 2012, pp. 1773–1778.
[12] G. Box, G. M. Jenkins, and G. C. Reinsel, Time Series Analysis:
Forecasting and Control, Wiley. com, 2013.
[13] F. Ding and T. Chen, “Performance bounds of forgetting factor least-
squares algorithms for time-varying systems with finite measurement
data,” IEEE Trans. Circuits Syst. I, vol. 52, no. 3, pp. 555–566, 2005.
