contact and tracking hybrid control with impulse …

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


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


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


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


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


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



of 20 kg. The figure shows (top) the ground reaction force and

(bottom) the control input.


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.


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


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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