Proceedings of the ASME 2015 Dynamic Systems and Control Conference
DSCC2015
October 28-30, 2015, Columbus, Ohio, USA
DSCC2015-9717
TORQUE ESTIMATION IN A WRIST REHABILITATION ROBOT USING A
NONLINEAR DISTURBANCE OBSERVER
Mohammadhossein Saadatzi∗, David C. Long, Ozkan Celik
Department of Mechanical Engineering
Colorado School of Mines
Golden, Colorado 80401
Email: [msaadatz,dalong,ocelik]@mines.edu
ABSTRACT
In this paper, we present implementation of a nonlinear disturbance observer algorithm to estimate disturbance torques on
a wrist rehabilitation robot. The ultimate goal is to enable accurate estimation of user interaction torques without force/torque
sensors. A dynamic model of the Wrist Gimbal two degree-offreedom forearm and wrist exoskeleton is developed. A nonlinear disturbance observer (NDO) algorithm that incorporates the
robot dynamics is implemented. Friction models for both joints
of the robot are identified. The robot dynamic model and the accuracy of the disturbance torque estimations are experimentally
verified under closed-loop trajectory tracking scenarios.
help re-establish part of the lost motor function for both impairments [3–5].
Since their initiation in late 1990s, rehabilitation robots have
found an increasing number of applications in therapy protocols
due to a variety of advantages they offer [6–10]. These advantages include lowering therapy costs, increasing patient motivation and engagement in therapy (thereby improving therapy outcomes), and enabling quantitative and objective measures of motor function improvement level [8–10].
To capitalize on these advantages, a variety of commercial and research rehabilitation devices have been developed and
implemented, including both exoskeleton and end-effector-type
mechanisms designed specifically for upper as well as lower extremities (for a review of various devices and control methods,
see review papers by Marchal-Crespo and Reinkensmeyer [11],
Harwin et al. [12] and Lo and Xie [13]). Our group’s efforts
have focused on devices for rehabilitation of the distal upperlimb, specifically of wrist and forearm movements. The focus on
wrist and forearm is motivated by preliminary findings of potential benefits of distal arm exercises on motor function level of the
whole limb [3,14]. One of the devices that was developed by our
group previously is Wrist Gimbal 3 DOF (degrees-of-freedom),
a compact wrist and forearm exoskeleton [15].
The torque estimation algorithms in the current paper are
implemented and tested on a more recent and refined version:
Wrist Gimbal 2 DOF (WG2). In WG2, one DOF (wrist abduction/adduction) was eliminated in favor of increased robustness/rigidity, lower cost, and more intuitive mapping of limb
movements to a visual interface for therapy games. It is of inter-
1
Introduction
In the United States, stroke and spinal cord injury (SCI) are
major causes of long term adult disability. The majority of the
approximately 795,000 yearly stroke incidents occurs in the elderly population, leading to an estimated direct and indirect cost
of $68.9 billion [1]. Although the annual number of SCI incidents is much lower (about 12,000/yr) the average SCI survivor
is much younger than the average stroke survivor. This leads to
an estimated annual direct and indirect costs of $14.5 billion and
$5.5 billion, respectively, for SCI [2].
Even though the etiologies of stroke and spinal cord injury
are quite different, the significant negative impact of the incident
on motor function and on the daily life of the patients is similar.
Also common is the finding that repetitive movement exercises
∗ Address
all correspondence to this author.
1
Copyright © 2015 by ASME
(a)
(b)
FIGURE 1: (a) WRIST GIMBAL 2 DOF (WG2) REHABILITATION EXOSKELETON SIDE VIEW, (b) WG2 FRONT VIEW WITH
VISUAL INTERFACE ON COMPUTER MONITOR.
est to add accurate torque estimation capability to WG2 since interaction torques between the robot and the patient provide valuable information for both assessment and planning/adjustment
of robot-aided therapy protocols. The golden standard with full
DOF force/torque sensors adds significantly to the cost. Therefore, disturbance observers have been considered in this work as
an alternative solution to estimate the interaction forces/torques
on the robot handle.
Disturbance observers (DO) were proposed by Ohnishi et al.
and Lee et al. for improving closed-loop control performance in
motion control systems [16, 17]. Murakami et al. proposed a
linear DO formulation for sensorless reaction torque estimation
and control in a multi-DOF industrial manipulator [18]. Chen
et al. presented a nonlinear DO formulation that extended the
linear stability proof for the DO provided by Murakami et al.,
to nonlinear multi-DOF systems, but limited to planar kinematic
configurations with revolute joints [19]. Chen et al.’s formulation has been widely adopted for a variety of applications and
purposes, including reducing cost and improving performance of
force-feedback devices.
Gupta and O’Malley adapted and implemented DO for estimation and closed-loop control of user interaction forces in a
single-DOF haptic interface [20]. They showed that the DObased closed-loop control of interaction forces improved the fidelity of the haptic interaction, as quantified by an increase in
transparency bandwidth. In a rehabilitation exoskeleton application, Ugurlu et al. proposed using independent-joint-based disturbance observers in combination with feedforward cancellation
of joint gravity and friction effects, to obtain an estimation of interaction torques between a user and a shoulder-elbow exoskeleton [21, 22]. Ugurlu et al. used a formulation based on that of
Murakami et al., while Gupta and O’Malley used the formulation
introduced by Chen et al. Gupta and O’Malley’s experiments
were limited to a single DOF device that did not include gravitational effects or coupled dynamic effects. Although Ugurlu et
al.’s implementation considered gravitational forces and involved
multiple DOF, the independent joint-based implementation excluded consideration of coupled multi-DOF effects.
More recently, Mohammadi et al. [23] extended Chen et al.’s
nonlinear DO formulation to cover multi-DOF serial manipulators with non-planar configurations as well as with both prismatic
and revolute joints. They pursued an application of this formulation for improving position tracking performance of a multiple
DOF robot. In this paper, we present results from implementation of a nonlinear disturbance observer based on Mohammadi
et al.’s formulation, as a disturbance torque estimation algorithm
which is not reported before. Our implementation takes into account gravitational and coupled multi-DOF effects as part of the
robot dynamic model, and uses joint-based friction models to estimate contribution of friction to total disturbance torques.
The paper is structured as follows: Section 2 provides a brief
description of Wrist Gimbal 2 DOF. Section 3 discusses the implementation of the disturbance observer algorithm for the device. Section 4 summarizes the results of the disturbance observer implementation and includes a discussion of the results.
Section 5 concludes the paper.
2
Device Description
This section summarizes the differences between Wrist
Gimbal 2 DOF (WG2) and its 3 DOF counterpart. Majority of
the technical specifications for WG2 remained unaltered from
2
Copyright © 2015 by ASME
FIGURE 2: BLOCK DIAGRAM FOR TORQUE ESTIMATION AND CONTROL. NOTE THAT ALTHOUGH EQUATION (2) IS
INCLUDED IN THE DIAGRAM, EQUATIONS (3-7) WHICH ELIMINATE THE NEED FOR ACCELERATION WERE USED IN
IMPLEMENTATION AND EXPERIMENTS. SEE TEXT FOR DESCRIPTION OF VARIABLES.
3
Disturbance Observer Design
A block diagram depicting the components involved in user
interaction torque estimation using a disturbance observer is provided in Fig. 2. First, the total disturbance torque applied to the
robot handle, τd , is estimated using a nonlinear disturbance observer (the estimated total disturbance is represented by τˆd ). This
total disturbance torque estimation τˆd includes both the friction at
the joints of the robot (τ f ri ) and the torque applied to the robot by
the patient/user (τint ), which is the interaction torque of interest.
Subtracting the estimation of the friction torque (τ̂ f ri ), based on
a friction model, from the total disturbance, provides the interaction torque. This implementation requires identification and use
of accurate models of friction. Also, the nonlinear disturbance
observer algorithm requires an accurate model representing the
robot dynamics.
In what follows, first the kinematic and dynamic equations
of the robot are introduced. Then the equations of the nonlinear
disturbance observer are stated. Finally, identified friction models for the joints are presented.
the earlier version, details of which can be found in [15].
WG2 exoskeleton differs from the 3 DOF version in that the
wrist adduction/abduction rotation axis was removed, primarily
because its absence reduced device (and visual interface) complexity, weight, and cost without unduly sacrificing utility. In
comparison with its forerunner, which was primarily 3D printed
from ABS plastic and laser cut from acrylic, WG2 is assembled
from aluminum extrusion, polycarbonate, and reinforced printed
parts (see Fig. 1). This change in material has improved the
rigidity of the forearm pronation/supination (PS) and wrist flexion/extension (FE) nested gimbals and the frame which cradles
them. Despite efforts to radially balance both gimbals about their
respective axes, the motor placement necessitated the use of a
counterweight mass, located on the outer frame.
The former design oriented the user’s forearm parallel to the
sagittal plane. WG2 orients the upper arm parallel with the transverse plane at the shoulder level, and the forearm points in the
anterior direction as it did originally, with the exception that the
palm of the user faces the ground. With the device located at the
users side, it is easier to accommodate patients on wheelchairs,
and the upper arm can be supported and restrained more effectively.
All respective motor, amplifier, encoder, and data acquisition hardware used for the 3 DOF version of the device were preserved for the WG2. The PS and FE cable drive gear ratios were
maintained at 15:1 and 16:1, respectively. All Matlab/Simulink
control algorithms were executed at a 1 kHz loop rate.
3.1
Dynamics of the Robot
Fig. 3 depicts a CAD of WG2 together with the coordinate
systems assigned to the robot for analysis. Denavit-Hartenberg
convention has been used to extract the kinematic and dynamic
equations of the robot. Frame {0} represents the ground/base
frame for the mechanism. Frames {1} and {2} are attached to
the ends of the first and second links. D-H parameters for the
3
Copyright © 2015 by ASME
TABLE 1: LINK PARAMETERS FOR WRIST GIMBAL.
Link
ai
αi
di
θi
1
0
- π2
d1
θ1 − π2
2
a2
0
0
−θ2 − π2
once is usually problematic and avoided. To overcome this limitation, Mohammadi et al. proposes the following re-formulation
that makes it possible to eliminate the need for acceleration measurements:
FIGURE 3: SOLIDWORKS MODEL AND COORDINATE
FRAMES FOR WRIST GIMBAL 2 DOF (WG2).
ż = −L(θ , θ̇ )z + L(θ , θ̇ ){Ĉ(θ , θ̇ )θ̇ + Ĝ(θ ) − τ − p(θ , θ̇ )} (3)
robot are provided in Table 1. Dynamics of the mechanism can
be written as
τ = M(θ )θ̈ + C(θ , θ̇ )θ̇ + G(θ ).
(4)
d
p(θ , θ̇ ) = L(θ , θ̇ )M̂(θ )θ̈
dt
(5)
(1)
In this project, we have used the Newton-Euler formulation for
derivation of the dynamic equations [24]. Table 2 gives the mass,
and inertial parameters of the two links of the mechanism which
have been estimated using the SolidWorks model and material
properties. Contribution of rotor inertia of the motors to total
robot inertia was neglected.
The auxiliary variable z, and the vector p(θ , θ̇ ) are introduced to remove the need for the acceleration θ̈ measurement.
Given this final disturbance observer design, the vector p(θ , θ̇ )
and the matrix L(θ , θ̇ ) should be determined to complete the algorithm. The following disturbance observer gain matrix was
proposed [23]:
3.2
Nonlinear Disturbance Observer
For the disturbance observer algorithm, the design method
proposed by Mohammadi et al. [23] has been used. In this section, we provide a brief description of this formulation, and refer
the readers to [23] for a complete formulation and discussion of
this NDO algorithm.
There are multiple sources of disturbance. The goal of the
disturbance observer is to provide a way to estimate a single
lumped disturbance term which will include contributions from
all disturbance sources [23]. The disturbance observer proposed
by Mohammadi et al. takes the following form:
τ̂˙d = −Lτ̂d + L{M̂(θ )θ̈ + Ĉ(θ , θ̇ )θ̇ + Ĝ(θ ) − τ}
τ̂d = z + p(θ , θ̇ )
−1
L(θ ) = X−1 M̂ (θ )
(6)
where X is a constant invertible n × n matrix to be determined.
So by substituting 6 into 5, for p(θ , θ̇ ) one obtains:
p(θ̇ ) = X−1 θ̇
(7)
Hence, eliminating the need for acceleration (θ̈ ) measurement.
Consider the disturbance observer given by equations (3-5)
with the disturbance observer gain matrix L(θ ) defined in equation (6) and the disturbance observer auxiliary vector p defined
in equation (7). The disturbance tracking error τd is globally uniformly ultimately bounded if:
(2)
where L is the observer gain matrix. It can be seen in this formulation that acceleration measurements θ̈ are needed . In robotic
systems, typically only joint angles (displacements) are measured, and various differentiation techniques are used to compute
joint velocities [25]. Differentiation of position signals more than
1. The matrix X is invertible,
4
Copyright © 2015 by ASME
TABLE 2: ESTIMATED MASS AND INERTIA PARAME-
First joint frictional torque
Torque (N.m)
TERS USING SOLIDWORKS.
First Link
Mass
2.176 kg
0.1
0
−0.1
−2
−1
tensor elements
I1xy = −3, I1yz = −29, I1xz = −77 kg.cm2
Second Link
1
1.5
2
1.5
2
0.05
0
−0.05
−0.1
Mass
0.216 kg
Center of mass
c2x = −48.7, c2y = −4.3, c2z = 53.3 mm
Moment of inertia
I2xx = 9, I2yy = 9, I2zz = 12
−2
−1.5
−1
−0.5
0
0.5
1
dθ /dt (rad/sec)
2
FIGURE 4: FRICTION MODELING: DATA POINTS FROM
EXPERIMENTS AND PIECEWISE LINEAR FUNCTION
FITS.
kg.cm2
PD feedback control to track constant velocity trajectories. During the motion of each joint, applied motor torque was recorded,
and Coriolis torque computed using equation (1) was subtracted
from it. Each joint has a limited range of rotation. To ensure consideration of only constant velocity movement of each joint along
this range of motion, the data corresponding to the initiation and
termination of the joint motion were excluded. Blue circles in
Fig. 4 represent average velocity values obtained during the rotation of each joint for different constant velocities. Obtained
friction models are parametrized as piecewise linear functions
(Fig. 4). Based on the patterns in the experimental data, four linear functions were considered for the first joint, and two linear
functions were considered for the second joint.
2. There exists a positive definite and symmetric matrix Γ such
that
(8)
3. The rate of change of the lumped disturbance is bounded,
i.e., ∃κ > 0 such that k τ̇d k< κ for all t > 0.
and also when the robotic manipulator is subject to slow-varying
disturbances, i.e., τ̇d ≈ 0, then the disturbance tracking error converges exponentially to 0 for all ∆τd (0) ∈ ℜn .
According to the above formulation, the disturbance observer design problem reduces to finding a constant invertible
matrix X such that the inequality (8) is satisfied. Define the ma˙ ) k is ζ .
trix Y = X−1 and assume that an upper bound of k M̂(θ
The inequality (8) holds if the following Linear Matrix Inequality
(LMI) is satisfied:
Y + YT − ζ I YT
≥ 0.
Y
Γ−1
0.5
Second joint frictional torque
Torque (N.m)
I1xx = 1161, I1yy = 327, I1zz = 1071
˙ )X ≥ Γ
X + XT − XT M̂(θ
0
0.1
Moment of inertia
I2xy = 0, I2yz = 0, I2xz = −2
−0.5
dθ /dt (rad/sec)
1
c1x = −1.6, c1y = 17.8, c1z = −5.9 mm
Center of mass
tensor elements
−1.5

−0.0036θ̇1 + 0.1382



0.0523θ̇1 + 0.1100
τ̂ f ri,1 =
0.0475θ̇1 − 0.1010



−0.0042θ̇1 − 0.1271
θ̇1 > 0.5
0 < θ̇1 < 0.5
−0.5 < θ̇1 < 0
θ̇1 < −0.5
(10)
(9)
τ̂ f ri,2 =
Note that the LMI toolbox of MATLAB can be used to solve this
LMI simultaneously for Y and Γ when Γ is not known.
−0.0077θ̇2 + 0.0527
−0.0065θ̇2 − 0.0776
θ̇2 > 0
θ̇2 < 0
(11)
Simplified versions of the friction models were also developed, mainly to eliminate detrimental effects of the discontinuities at zero velocity on friction torque estimations. The simplified models provide a more gradual change around zero velocity
to avoid high frequency noise in estimations due to high sensitivity of original friction models at zero velocity. The simplified
3.3
Friction Modeling
Friction modeling was completed separately for each joint.
The robot was oriented in a configuration that eliminated effects
of gravity for the joint of interest, and the joint was driven under
5
Copyright © 2015 by ASME
Joint 2 torque (N.m)
Joint 1 torque (N.m)
Dynamic model+Friction model
Motor torque
0.4
0.2
0
−0.2
−0.4
0
2
4
6
8
10
12
14
16
18
Dynamic model+Simplified friction model
Motor torque
0.4
0.2
0
−0.2
−0.4
20
0
2
4
6
8
0.05
0
−0.05
−0.1
−0.15
−0.2
0
2
4
6
8
10
10
12
14
16
18
20
12
14
16
18
20
time (sec)
Joint 2 torque (N.m)
Joint 2 torque (N.m)
time (sec)
12
14
16
18
0.05
0
−0.05
−0.1
−0.15
−0.2
20
time (sec)
0
2
4
6
8
10
time (sec)
(a)
(b)
FIGURE 5: VALIDATION OF ROBOT’S DYNAMIC AND FRICTION MODELS. DURING SINUSOIDAL TRAJECTORY TRACK-
ING UNDER PD CONTROL. RECORDED MOTOR TORQUES ARE COMPARED WITH MOTOR TORQUES ESTIMATED BY
USING ROBOT’S DYNAMIC AND FRICTION MODEL. UPPER PLOTS ARE FOR JOINT 1, LOWER PLOTS ARE FOR JOINT 2.
ORIGINAL FRICTION MODELS ARE USED IN (a), SIMPLIFIED FRICTION MODELS ARE USED IN (b).
models were parametrized as follows:
(Figure 5(a)). These errors are effectively removed by use of the
simplified friction models (Figure 5(b)).

−0.0036θ̇1 + 0.1382




 0.0523θ̇1 + 0.1100
τ̂ f ri,1 = 1.210θ̇1


0.0475θ̇1 − 0.1010



−0.0042θ̇1 − 0.1271
Figure 6 presents the results for validation of the implemented nonlinear disturbance observer (NDO) algorithm. The
data presented is from the same experimental recording presented in Figure 5. The NDO estimates the total lumped disturbance applied on the robot and takes into account only the robot
dynamics. When a user does not apply any interaction forces
to the robot’s handle, the only disturbance source is the friction,
therefore the two completely different methods of estimating the
frictional torque should agree for validation (the friction model
estimations are highly reliable and replicable). Figure 6 compares the friction torques estimated from the identified friction
models with the estimated disturbance torques. The two signals
agree well, especially for the second joint of the robot (lower
plots). Disagreement between the torque signals on the order of
one second rise time and 20% amplitude is observed for the first
joint. We believe that it would be possible to minimize these
errors through (1) more accurate, experimental identification of
robot model parameters and (2) further tuning of NDO gains.
High frequency erroneous behavior is effectively removed by using simplified friction models over original friction models. It
should be noted that using a simplified friction model just removes the high frequency noise in the estimated torque, and has
a negligible effect on the RMS of the error.

 −0.0077θ̇2 + 0.0527
τ̂ f ri,2 = 0.645θ̇2

−0.0065θ̇2 − 0.0776
θ̇1 > 0.5
0.1 < θ̇1 < 0.5
−0.1 < θ̇1 < 0.1
−0.5 < θ̇1 < −0.1
θ̇1 < −0.5
(12)
θ̇2 > 0.1
−0.1 < θ̇2 < 0.1
θ̇2 < −0.1
(13)
4
Results and Discussion
Figure 5 presents the experimental results for validation
of the derived robot dynamic model and the friction models.
Torque applied by the motors (calculated from motor currents)
to the joints are recorded during a sinusoidal trajectory tracking scenario under PD control, simultaneously for joints 1 and 2.
These torques are compared to the torques estimated at the joints
through feeding the position and velocity data to the robot inverse dynamic model and the friction models. Torques estimated
by the models track closely the recorded torques, establishing the
validity of the models. An important detail to note is the high frequency, large amplitude errors in torque estimations when original friction models with high sensitivity at zero velocity are used
Overall, the results obtained validated accuracy and correctness of the robot dynamic model and the NDO algorithm for
6
Copyright © 2015 by ASME
Joint 1 torque (N.m)
Joint 1 torque (N.m)
Estimated torque
Friction model
0.2
0.1
0
−0.1
−0.2
0
2
4
6
8
10
12
14
16
18
0.1
0
−0.1
−0.2
20
Estimated torque
Simplified friction model
0.2
0
2
4
6
8
Joint 2 torque (N.m)
Joint 2 torque (N.m)
0.1
0.05
0
−0.05
−0.1
0
2
4
6
8
10
10
12
14
16
18
20
12
14
16
18
20
time (sec)
time (sec)
12
14
16
18
0.1
0.05
0
−0.05
−0.1
20
0
2
4
6
8
10
time (sec)
time (sec)
(a)
(b)
FIGURE 6: VALIDATION OF TORQUE ESTIMATIONS WITH THE NONLINEAR DISTURBANCE OBSERVER. DURING SI-
NUSOIDAL TRAJECTORY TRACKING UNDER PD CONTROL WHERE FRICTION IS THE ONLY DISTURBANCE, FRICTION
ESTIMATIONS ARE COMPARED WITH ESTIMATED DISTURBANCES. UPPER PLOTS ARE FOR JOINT 1, LOWER PLOTS
ARE FOR JOINT 2. ORIGINAL FRICTION MODELS ARE USED IN (a), SIMPLIFIED FRICTION MODELS ARE USED IN (b).
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5
Conclusion
We presented successful results from implementation of a
nonlinear disturbance observer (NDO) algorithm on Wrist Gimbal 2 DOF exoskeleton, for torque estimation purposes. The
robot dynamic model and two types of friction models (original
and simplified) were developed. Accuracy of these models were
verified through sinusoidal trajectory tracking experiments under
PD control. Correct operation of the NDO was verified through
the same experiments. High agreement was observed between
friction torques estimated at robot joints via the friction models,
and via the disturbance observer estimations. The pursued NDO
structure is found to have potential for enabling accurate interaction force/torque estimations for force feedback robots. Future
work will involve validation of the proposed method for estimation of interaction force/torque with measurements obtained
using dynamic payloads and a force sensor.
7
Copyright © 2015 by ASME
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