Proceedings of Proceedings of DSCC 2018ASME 2018 Dynamic Systems and Control Conference
October 1-3, 2018, Atlanta, Georgia, USA
DSCC2018-8945
CONTACT AND TRACKING HYBRID CONTROL WITH IMPULSE-MOMENTUMSLIDING SURFACE AND TERMINAL SLIDING MODE
Hanz Richter
ProfessorDept. of Mechanical Engineering
Cleveland State University
Cleveland, Ohio 44115Email: [email protected]
Saleh Mobayen
Assistant ProfessorDept. of Electrical Engineering
University of Zanjan, Iran
Dan Simon
ProfessorDept. of Electrical Engineering
and Computer Science
Cleveland State UniversityCleveland, Ohio 44115
ABSTRACT
The control system proposed in the paper is motivated by
robotic testing of prosthetic legs, where a test robot is used to
emulate the mechanics of walking. Previously, robust trajectory
tracking of walking profiles was used on the test robot’s hip axes,
both for swing and stance phases. For the stance phase, a track-
ing controller does not reproduce the dynamics of weight transfer
and release during the swing-stance transitions of human walk-
ing. To address this problem, a novel contact mode controller is
proposed which allows the test robot to emulate weight transfer
and momentum exchange in the stance phase. The swing phase
controller is still designed for tracking, but introduces a fast ter-
minal sliding mode controller for rapid convergence to reference
swing trajectories with small chattering. This paper introduces
the concept of an impulse-momentum sliding surface and devel-
ops the control system for a one-degree-of-freedom electrome-
chanical system moving in a vertical axis. A simulation study
and a successful real-time implementation are described that il-
lustrate the practical validity of the concept, which can be used
in conjunction with more realistic walking models.
1 Introduction
Research in the field of prosthetic legs and other lower-
limb assistive devices has shown remarkable activity in recent
years. Specifically, the area of novel designs for both powered
and passive controllable devices continues to grow, as evidenced
by funding and publication trends. A recent review [1] accounts
for more than 20 active (powered) designs for transfemoral knee
prostheses and compares their capabilities to the daily needs of
a group of patients. The review included commercially-available
and human-tested devices, and excluded low-cost, passive de-
signs and promising, innovative concepts which are still at a low
readiness level.
Design innovations often begin with high-risk, debatable
concepts that must be proven by simulations first. Human tri-
als are reserved for sufficiently mature designs whose safety and
functionality have been assured by an intermediate phase of test-
ing. Such intermediate stage must be a reasonable approximation
of human operation. Prosthetic leg prototype evaluation can thus
be divided in the following stages:
1. Simulation studies involving a model of the prosthetic leg
and various levels of representation of human motion [2,3,4]
2. Using a test robot or other machine to represent walking [5,
6, 7, 8]
3. Using a leg bypass device such as a bent-knee adaptor and
an abled-bodied subject [9, 10]
4. Recruiting amputees to wear the device [11, 12, 13]
Machine-based testing offers the benefits of safety and high re-
peatability [8], and is not subject to the constraints of human
fatigue and test subject time. Additionally, the mechanical in-
tegrity and safety of the prototype can be reliably tested prior to
any human involvement.
A one-legged test robot was developed and extensively used
for intermediate-stage testing of a powered transfemoral prosthe-
1 Copyright c© 2018 by ASME
BallscrewLinear Guides
Vertical CarriageRotary Plate
Rotary Attachment
Rotary Servomotor
Thigh
Vertical Servomotor
Prosthesis Body
Knee Damper
Knee Joint
Ankle Attachment
Prosthetic Foot/Shoe
Force Sensor
FIGURE 1: Prosthesis test robot at the Center for Human-
Machine Systems, Cleveland State University
sis [8,14]. A prosthesis prototype is attached to the machine and
cyclic hip and thigh reference trajectories are tracked by a con-
trol system, as the prosthesis walks on a built-in treadmill under
the action of its own control system, as shown in Fig. 1. Since
the treadmill belt is a compliant (elastic) environmental bound-
ary, a properly-designed controller can maintain stability, reject
ground reaction forces and deflect the belt while closely main-
taining tracking [8, 14]. The robot is operated without contact
with the treadmill, that is, walking “on air”. The researcher then
biases the hip reference trajectory downward until a stance phase
appears. Bias is increased while monitoring the ground reaction
force, until it reaches a prescribed level matching human walking
data.
In the past decades, several robust control techniques have
been established for the tracking control of uncertain nonlin-
ear systems. Sliding mode control (SMC) is a robust technique
which has been widely employed for tracking in nonlinear sys-
tems ranging from chemical reactors to magnetic levitation sys-
tems, robot manipulators, and spacecraft propulsion and guid-
ance [15, 16, 17, 18].
SMC uses a discontinuous function to drive state responses
to a pre-specified switching surface (also termed sliding surface)
and holds states on the surface afterwards. Moreover, SMC con-
trollers are designed to reach the sliding surface in finite time and
robustly against certain classes of model errors and disturbances
termed matched disturbances. When system states reach the slid-
ing surface, the system adopts reduced-order dynamics specified
by the designer and becomes totally insensitive to matched dis-
turbances and model errors [19, 20].
The sliding surface is typically specified as a linear combi-
nation of the tracking error and its derivatives [21]. Thus, when
states lie on the sliding surface, the tracking error behaves like the
solution of a homogeneous linear differential equation, and the
error converges to zero asymptotically (i.e., not in finite time).
Another disadvantage of such conventional approach is the re-
liance on high-frequency control switching to reach the sliding
surface. Such chattering phenomenon is impractical or detrimen-
tal to actuation systems.
In comparison with conventional SMC, the terminal sliding
mode control (TSMC) approach provides finite-time error con-
vergence [22, 23] Unlike SMC, TSMC is based on a category
of recursive nonlinear non-smooth differential equations. TSMC
may not supply the same convergence rate as SMC when the
states are far away from the equilibrium. To address this issue, a
nonlinear term is added to the switching surface to improve the
convergence rate. This approach offers all the benefits of SMC
and it improves convergence speed. In the last decade, the tech-
nique has advanced both in theory and applications [24, 25].
While pure tracking can be used to reproduce the kinemat-
ics of walking in the test robot joints and to match the ground
reaction force peak, it does not mimic the dynamics of weight
transfer and support that would be observed in a human wear-
ing the prosthesis. More precisely, the vertical hip displacement
SMC guarantees trajectory tracking at any ground disturbance
load, provided it remains within the disturbance bound assumed
for the controller design and the motor has the necessary torque
rating. It follows that the same trajectories will be observed at
any level of bias and corresponding peak ground reaction force
and peak belt deflection. As the belt returns to its undeflected
position, the hip may be lifted by the motor rather than being
fully supported by the prosthetic knee against the simulated in-
ertia and weight. As a result, a prosthesis controller is tested
in unrealistic conditions. One approach to address this problem
was to modify the vertical reference trajectories online to match
a ground reaction force profile obtained from clinical walking
data [26]. However, this approach is indirect, as it does not allow
the specification of patient weight in the control system.
To address these shortcomings, a controller that considers
weight and mass is proposed for the stance phase to reproduce
the transition between swing and stance phases and the stance
phase itself. A novel reset integral sliding function based on
the impulse-momentum principle is introduced and used for the
stance phase. Since the machine is one-legged, the robot must
be supported during the swing phase. As before, this is accom-
plished using a tracking controller. A TSMC is used to guarantee
quick convergence to the swing phase reference trajectory with
small control chattering. This paper only considers a one-degree-
of-freedom mechanism to introduce the concept, but a similar
idea can be readily applied to a more general walking model.
The remainder of the paper is organized as follows: Sec-
tion 2 presents a one-degree-of-freedom model for an electrome-
chanical system moving in a vertical axis, Section 3 introduces
the proposed impulse-momentum sliding surface and derives
the stance phase controller, Section 4 summarizes the concept
2 Copyright c© 2018 by ASME
Gearmotor
Torque-mode
servo drive
x
+
Fe
g
m
u
FIGURE 2: Motor-driven vertical mechanism with environmen-
tal interaction force. The control input u is proportional to the
force applied to the moving carriage.
of non-singular terminal SMC and derives a swing phase con-
troller, Section 4.1 discusses possible switching rules between
controllers, Section 5 presents a simulation study and Section 6
describes real-time control experiments using the vertical axis
of a test robot. Finally, Section 7 offers conclusions and future
work.
2 One-axis electromechanical modelConsider a motor-driven vertical mechanism and environ-
mental boundary as illustrated in Fig. 2. A simple derivation
results in the following differential equation of motion:
mx+ δ (x, x) = ku−Fe (1)
where x is the position (with a positive down convention), m is
the mechanism mass, k is a constant associated with the motor
and servo amplifier, Fe is the environmental force and δ (x, x)is an uncertain term capturing possibly nonlinear friction in the
mechanism. The control input u is assumed to already contain
a gravity compensation term, that is, u = u′+mg/k, where u′ is
the control input to be applied to the motor servo amplifier. The
uncertain term is assumed to be bounded by a known constant ∆:
|δ (x, x)| ≤ ∆ (2)
The environmental force is strictly speaking a function of x, and
also of x and x if environment mass and damping are considered.
TABLE 1: System Parameters. All values are reflected to the
linear coordinate and include gearing effects.
Parameter Symbol Value Units
Total inertia m 146 kg
Damping constant b 2570 N-s/m
Coulomb friction Ff 83 N
Motor+servo constant k 600 N/V
In prosthesis testing, the environment is either rigid (floor) or
compliant with a very high stiffness (treadmill belt). Provided the
controller is designed so that the closed-loop system is passive
with respect to Fe, coupled stability against the passive environ-
ment is guaranteed [27] and one may regard Fe as an exogenous
input. This approach is followed here.
The test robot considered in the simulation example and
used for the real-time experiment follows the model of Eq. 1 for
its vertical axis. The parameters are listed in Table 1. Uncertainty
may be reduced by modeling Coulomb and viscous friction when
the respective parameters are known:
δ (x, x) = Ff sign (x)+ bx+ δ ′(x, x) (3)
where δ (x, x)′ is a new uncertainty.
3 Impulse-Momentum Sliding Mode Controller: Con-
tact PhaseThe control objective during the contact phase is to replace
the mechanism mass and weight by target values reflecting pa-
tient properties. Also, the electromechanical system should have
a prescribed response to environmental force rather than follow
a preset trajectory. This objective falls in the general category of
impedance control, although a novel hybrid integral approach is
presented here.
During contact, the following target dynamics are to be sat-
isfied:
Mx =−Fe +W (4)
where M is the new inertia and W is the new weight, taken as
W = Mg. As shown in Fig. 3, the target dynamics correspond to
a synthetic inertia and weight to be applied to the environment.
In conventional (unity relative degree) sliding mode control, one
differentiation of the sliding function is used to access the control
input and define a feedback law. The above dynamics have zero
3 Copyright c© 2018 by ASME
x
+
Fe
M
W
FIGURE 3: Target dynamic behavior for the contact phase and
the sliding regime.
relative degree with respect to u and cannot be directly used, thus
an integral form will be introduced. Let tI denote the impact time
(initiation of contact) and integrate Eq. 4:
Mx(t)−Mx(tI) =
∫ t
tI
(W −Fe(τ))dτ (5)
The left side of the equation is the change of momentum of the
virtual mass and the right side is the net impulse applied by the
virtual weight and the environmental force. This can be used to
define a sliding surface:
s1 = Mx+
∫ t
tI
(Fe(τ)−W)dτ −Mx(tI) (6)
If s1 is driven to and maintained at zero, the machine will react
to the environmental force exactly as a virtual object with mass
M and weight W = Mg. With an elastic boundary, energy will be
transferred from the machine to the environment and stored as
elastic potential energy, and subsequently released without loss.
Bouncing will occur, but only until the trajectory controller be-
comes active and trajectory tracking is established.
Following the standard techniques of sliding mode con-
trol [21], a control law is derived to drive s1 to zero in finite
time, maintaining s1 = 0 invariant thereafter despite the presence
of the uncertain term δ (x, x). The commonly-used reaching law
s1 =−η2 sign (s1) is imposed, resulting in the feedback law:
u = u1 =−1
k[r(η1 sign (s1)−W)+ (r− 1)Fe] (7)
where η1 > 0 must be chosen large enough to overcome the max-
imum uncertainty, i.e., η1 ≥ ∆. This can be directly shown by
using the Lyapunov function V1 = s21 and checking that its deriva-
tive is negative for nonzero values of s1. Moreover, the reaching
law can be integrated to show that s1 = 0 is reached at some finite
time tR > tI .
3.1 Including damping
If viscous damping is present in the plant (information about
δ is available) and damping is also desired for the target dynam-
ics during contact, the following plant equation applies:
mx+ δ ∗(x, x) = ku−Fe− bx (8)
where b is the damping coefficient of the plant and δ ∗(x, x) is an
uncertainty bounded in absolute value by a known constant ∆∗.
The target dynamic behavior for the contact phase is now:
Mx =−Fe+W −Bx (9)
where B is the desired virtual damping.
Following a similar idea as for the undamped case, the slid-
ing surface is defined as
s1 = Mx+Bx+
∫ t
tI
(Fe(τ)−W)dτ −Mx(tI)−Bx(tI) (10)
and again, the integral is reset to zero and x(tI) and x(tI) are reset
to the sensed values at each impact time. It can be easily shown
that the feedback law is now
u1 =−1
k[r(η1 sign (s1)−W)+ (r− 1)Fe+(Br− b)x] (11)
The approach offers a key property: the net impulse-
momentum exchange virtualized by the controller holds even
during the reaching phase, before s1 has become zero. To see
this, consider first the undamped case and calculate the value of
s1 at t = tR:
s1(tR) =
∫ tR
tI
Fe(τ)−Wdτ +Mx(tR)−Mx(tI) = 0 (12)
Rearranging terms:
∫ tR
tI
W −Fe(τ)dτ = M(x(tI)− x(tR)) (13)
The term on the left is the net impulse of the virtual weight and
environmental force and the term on the right is the change of
momentum during the reaching phase.
Another key observation is that the inclusion of a damping
term in the target dynamics is equivalent to adding a correspond-
ing virtual damping to the environment and using the undamped
4 Copyright c© 2018 by ASME
contact dynamics. To see this, simply define a new external force
F ′e = Fe+Bx and move the term Bx−Bx(tI) inside the integral in
Eq. 10 to recover the impulse-momentum properties of the un-
damped case. Therefore, the impulse-momentum property of the
reaching phase is still effective when damping is used in the tar-
get dynamics, relative to a virtual environment which includes
the same damping.
Finally, we observe that the contact-mode controller requires
position, velocity and force feedback from sensors or estimators,
but acceleration measurements are not required.
4 Terminal Sliding Mode Control: Off-Contact Track-
ing
Define state variables for the plant model of Eq. 1 as q1 , x
and q2 , x. With these definitions and taking Fe = 0 (non-contact
mode), the plant can be represented by
q1 = q2 (14)
q2 = −b
mq2 −
δ ∗
m+
k
mu (15)
The tracking sub-problem is to track a reference trajectory
qd1 , q
d1, q
d1 with finite-time convergence of the tracking error. The
following result concerning finite-time stability [28] is the basis
of TSMC:
Lemma 1. Let V : R 7→ R be a positive-definite function satis-
fying the inequality
V (t)≤−αV (t)− βV η (t) (16)
for t ≥ t0 and some α > 0 and β > 0, with 0 < η < 1 equal to the
ratio of two odd positive integers. Then V (t) converges to zero in
a finite time ts upper bounded as follows:
ts ≤ t0 +1
α(1−η)ln
αV 1−η(t0)+ β
β(17)
Now define a sliding function for the tracking regime by
s0 = e+ϕe+ µeη (18)
where e = q1 − qd1 is the tracking error and µ and ϕ are positive
tuning parameters. As above, η is assumed to be the ratio of
two positive integers such that η < 1. The following is the main
result concerning the tracking controller:
Theorem 1. Suppose the control input
u0 = −m
k(κ |s0|
η sgn(s0)+ γs0+ (19)
χ sgn(s0)+ (ϕ + µηeη−1)(q2 − qd1)− qd
1)
is applied to system 14, 15, where γ and κ are positive tuning
parameters and χ is chosen to overcome the disturbance bound,
that is, χ > ∆∗/m. Then the sliding function s0 and the tracking
error e converge to zero in finite time.
Proof. Consider the candidate positive-definite Lyapunov func-
tion V (s0) =12s2
0. The derivative of V along the closed-loop tra-
jectories can be derived as
V = s0s0 =−δ (q1,q2)
ms0 −κ |s0|
η+1 − γs20 − χ |s0| (20)
Using the disturbance bound ∆∗ and the condition χ > ∆∗/m
yields the inequality
V ≤−γs20 −κ |s0|
η+1 ≤−αV − βV η (21)
where α = 2γ > 0, β = 2ηκ > 0, 0< η = η+12
< 1. By Lemma 1,
V converges to zero in finite time. Furthermore, since V is
positive-definite, V (s0) = 0 implies s0 = 0, and from Eq. 18, the
tracking error has dynamics
e+ϕe+ µeη = 0 (22)
It can be shown that under the assumptions made for ϕ , µ and η ,
e must also converge to zero in finite time. [29, 30]
4.1 Switching rules and stability considerations
Having established the controllers for the off- and on-contact
modes, it remains to specify a rule to switch controllers. Intro-
duce a binary state q, where q = 0 corresponds to the off-contact
mode and q = 1 to the contact mode. When a feedback rule
for the next state q+ is included, a closed-loop system with hy-
brid dynamics is created. Thus, we consider rules of the form
q+ = q+(q,x,Fe), where x is the continuous state of the system.
For the remainder of the paper, we assume that the initial
value of q is zero, as dictated by the application. Clearly, Fe = 0
holds whenever q = 0, and Fe > 0 is true for q = 1. Therefore, a
force threshold (termed inbound threshold) can be used to trigger
a transition from q = 0 to q = 1. The opposite transition is not so
straightforward. In numerical simulations, a force-based rule can
also be used for the 1 → 0 transition provided a hysteresis band
5 Copyright c© 2018 by ASME
between inbound and outbound thresholds is included. This must
be done to prevent an infinite sequence of transitions or high-
frequency switching between modes. This rule is implemented
with a hysteresis relay in the simulation of Section 5.
As explained earlier, a force-based 1 → 0 transition relies
on bouncing, which allows the force to become small enough
to reach the prescribed threshold while in the contact mode.
High bouncing is not compatible with the application, thus vir-
tual damping must be included. Damping reduces or eliminates
bouncing, but unfortunately it also prevents the force from be-
coming small enough in the contact mode to cross the outbound
threshold and trigger a transition. If the outbound threshold is
raised, it approaches the inbound threshold and the hysteresis
deadband is reduced, increasing the posibility of high-frequency
mode switching.
In this study, a dwell time strategy is included to pre-
clude excessive switching and ensure the desired cyclic opera-
tion. Specifically, the 0 → 1 transition will be established when-
ever the force crosses the inbound threshold, provided the time
elapsed since the previous 1 → 0 transition is larger than a spec-
ified tracking dwell time. Since the TSMC guarantees a finite
error convergence time, the tracking dwell time can be suitably
selected. The 1 → 0 transition is triggered solely on the basis of
a contact dwell time. The switching rules are thus summarized
as follows:
q+ =
{
1, q = 0 and Fe ≥ Fth and ttrack ≥ T 0D
0, q = 1 and tcont ≥ T 1D
(23)
where Fth > 0 is the force threshold, ttrack and tcont are resettable
time counters and T 0D and T 1
D are the corresponding dwell times
for the tracking and contact modes. Each time counter is reset at
the start of the transition to their corresponding modes. Figure 4
shows the timing pattern arising from the above logic.
Stability considerations: The control objective can be
stated as follows:
1. The state of the closed-loop system must enter a periodic
orbit containing exactly one 0 → 1 and one 1 → 0 transition
per cycle, with a specified lower bound for the time intervals
between successive transitions.
2. The error between the closed-loop periodic orbit and the ref-
erence trajectory must be bounded.
Straightforward logical reasoning is used to show that the two
controllers and switching rules meet the desired objective. In-
deed, whenever either controller becomes active, the tracking er-
rors are bounded and are guaranteed to converge to zero. For
the TSMC, the error converges to zero in finite time. For the
IM-SMC, it is sufficient to show that the error remains bounded.
Hence the second objective is implied by the first one.
q
1
0t
ttrack
T 0
D
undefined
undefined
tcont
T 1
D
undefined
undefined
undefined
t
t
FIGURE 4: Timing pattern for q and time counters arising from
the proposed switching rules. The 1 7→ 0 transitions happen ex-
actly after T 1D seconds, while T 0
D only specifies a minimum dwell
time.
For the first objective, assume that q = 0 at the initial
time and that the reference trajectory crosses the environmen-
tal boundary. Assume also that the initial position and velocity
are such that the tracking error will become zero prior to con-
tact. Then a 0 → 1 transition will occur, and a contact dwell
time counter will start. The rules guarantees a 1 → 0 transition,
and since the TSMC is globally stable, the tracking error will be-
come zero after a finite time T0. The tracking dwell time counter
is started and the position will follow the reference trajectory un-
til the environmental boundary is again encountered. Since the
tracking dwell time has been chosen to be larger than T0, the rule
dictates a 0 → 1 transition and the cycle is repeated.
5 Simulation
Two simulations were run to illustrate the operation of the
control system and show the limitations of force-only switching
rules. A virtual mass of M = 40 kg was used, along with inbound
and outbound force thresholds of 1 and 0 N respectively. A hys-
teresis relay with those switch-on and switch-off points was used.
In the first simulation, virtual damping was set to zero.
As shown in Fig. 5, there is significant bouncing after im-
pact, which reduces the force enough to cross the outbound
threshold and trigger the transition to the tracking mode. This
can be further explained by forming a characteristic polynomial
with the virtual mass, the virtual damping and the actual envi-
ronmental stiffness. In the zero-damping case, the presence of
6 Copyright c© 2018 by ASME
FIGURE 5: Simulation results with a virtual mass of 40 kg
and zero virtual damping. Significant bouncing allows the force
threshold to be crossed to trigger a return to the tracking mode.
oscillation is expected.
The TSMC achieves tracking after a finite time while q =0. Verification of the target contact dynamics is deferred to the
real-time experiment. The figure also shows how the two sliding
functions alternately converge to zero whenever their controllers
are active. In the second simulation, a virtual damping of B =2500 Ns/m was introduced. The characteristic polynomial with
M = 40, K = 37000 and B = 2500 has overdamped roots and no
bouncing is expected. This is confirmed by the simulation, which
shows that the system “gets stuck” to the environment boundary,
since the force does not cross the outbound threshold.
6 Real-Time experiment
The dwell time switching approach was used in a real-time
experiment using the prosthesis test robot. Two virtual masses
were tried, namely M =20 kg and M =60 kg. Due to the large
stiffness of the treadmill belt (estimated at 37000 N/m), a high
bounce would be observed with a pure impulse-momentum tar-
get behavior. Therefore, an appropriately high value of damping
was specified for both cases, namely B = 2500 Ns/m.
The inclusion of damping in the target dynamics is equiva-
lent to the creation of virtual damping in the environment. With
a virtual mass of 20 kg, impact dynamics has two real poles at
-108 and -17 rad/s. With 60 kg, the poles are complex with a fre-
quency of about 25 rad/s and a damping ratio of 0.84. As shown
by the experiment, these characteristics are enough to prevent
excessive bouncing.
The TSMC was tuned in simulation and then adjusted in
real-time for satisfactory tracking performance. Figures 7 and 8
show the experimental results for 20 kg, and figures 9 and 10
FIGURE 6: Simulation results with a virtual mass of 40 kg and a
virtual damping of 2500 Ns/m. The absence of bouncing results
in the system “getting stuck” in the contact mode when switching
rules based on force only are used.
FIGURE 7: Real-time experimental results with a virtual mass
of 20 kg. The impact dynamics have overdamped poles, which
is confirmed by the lack of bouncing after the contact phase has
been initiated. The figure shows (top) the actual and reference
positions and (bottom) the sliding functions.
show the 60 kg case.
Since the impulse-momentum sliding function remains close
to zero whenever its corresponding controller is active, it can
be anticipated that the target dynamics have been precisely en-
forced. To verify this, the experimental data is tested for confor-
mity to Eq. 9 by a separate simulation. Specifically, Eq.9 was
numerically integrated using the experimentally-recorded force
as an input and impact-time initial conditions obtained from the
7 Copyright c© 2018 by ASME
FIGURE 8: Real-time experimental results with a virtual mass
of 20 kg. The figure shows (top) the ground reaction force and
(bottom) the control input.
FIGURE 9: Real-time experimental results with a virtual mass
of 60 kg. The impact dynamics have underdamped poles, which
is confirmed by the presence of some bouncing after the contact
phase has been initiated. The figure shows (top) the actual and
reference positions, the discrete state and (bottom) the sliding
functions and the discrete state
experiment.
Figure 11 shows the predicted and experimental position and
velocity for the 60 kg case. It can be seen that the contact behav-
ior has been controlled to closely match the specified target. The
20 kg case is not shown but the results were also accurate. Ta-
ble 2 summarizes the errors as root mean squares measured over
one contact cycle for both cases.
FIGURE 10: Real-time experimental results with a virtual mass
of 60 kg. The figure shows (top) the ground reaction force and
(bottom) the control input and the discrete state.
FIGURE 11: Real-time experimental verification of the target
contact dynamics with a virtual mass of 60 kg. The figure shows
(top) the predicted and actual positions and (bottom) the pre-
dicted and actual velocities.
TABLE 2: RMS errors during the tracking and contact phases
Case RMSE Units
Contact (position), 20 kg 3.93 ×10−5 m
Contact (velocity), 20 kg 4.90 ×10−3 m/s
Contact (position), 60 kg 2.17 ×10−4 m
Contact (velocity), 60 kg 7.70 ×10−3 m/s
8 Copyright c© 2018 by ASME
7 Conclusions and Future Work
Integral sliding mode with resetting provides a simple and
highly effective way to specify and achieve desired dynamics
for robot-environment interaction. For both virtual masses tested
experimentally, achievement of the target contact dynamics was
measured by predicting position and velocity by numerical inte-
gration of the target dynamics driven by the experimental force.
The maximum velocity errors for the 20 kg and 60 kg case were
7% and 3.8 % respectively, measured as root-mean-square (rms)
errors normalized to the velocities at the initiation of the con-
tact phase. These figures are conservative, since velocity sensor
errors visible in Fig. 11 increase the rms value.
The paper has focused on the one degree of freedom, linear
motion case, but the core idea can be readily generalized to other
motions. Our simulation results used contact force thresholds as
a criterion to switch between the tracking and contact controllers.
In an experimental situation, the use of force to trigger the off-
contact to on-contact transition is adequate. However a force
threshold cannot be reliably used for the opposite transition, as
the force reduction necessary to cross an exit threshold can only
be obtained by letting the system bounce off the environmental
boundary. In this paper, a dwell time strategy was used to trigger
the transition to the tracking state.
Future work includes the application of the impulse-
momentum SMC concept to emulate walking in one plane. Here
force thresholds may be allowable for transitions in both direc-
tions, since the effect of thigh and knee flexions may provide the
necessary force reductions without relying on bouncing. Alter-
natively, thigh or knee angles may be used instead of force to
trigger the transitions to the tracking state. Finally, the ability to
introduce virtual damping in the environment can be exploited to
emulate walking on terrains with dynamic characteristics which
are different from those of the physical surface.
8 Acknowledgement
This work was supported by NSF grants #1344954 and
#1544702.
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