Robust Tracking Control Hanz Richter of a Prosthesis Test...

Hanz RichterAssociate Professor

Mechanical Engineering Department,

Cleveland State University,

Cleveland, OH 44115

e-mail: [email protected]

Dan SimonProfessor

Electrical and Computer Engineering Department,

Cleveland State University,

Cleveland, OH 44115

e-mail: [email protected]

Robust Tracking Controlof a Prosthesis Test RobotThis paper develops a passivity-based robust motion controller for a robot used in pros-thetic leg performance studies. The mathematical model of the robot and passive prosthe-sis corresponds to a three degree-of-freedom, underactuated rigid manipulator. A form ofrobotic testing of prostheses involves tracking reference trajectories obtained fromhuman gait studies. The robot presented in this paper emulates hip vertical displacementand thigh swing, and we consider a prosthesis with a passive knee for control develop-ment. The control objectives are to track commanded hip displacements and thigh anglesaccurately, even in the presence of parametric uncertainties and large disturbance forcesarising from ground contact during the stance phase. We develop a passivity-based con-troller suitable for an underactuated system and compare it with a simple independent-joint sliding mode controller (IJ-SMC). This paper describes the mathematical model andnominal parameters, derives the passivity-based controller using Lyapunov techniquesand reports success in real-time implementation of both controllers, whose advantagesand drawbacks are compared. [DOI: 10.1115/1.4026342]

1 Introduction

The development of novel prosthetic devices has progressed ata rapid pace in the last decades. Transfemoral leg prostheses, inparticular, have become increasingly sophisticated in terms of thetechnologies used to modulate knee behavior or even producepush-off forces with battery power. Knee torque may be varied bypneumatic, hydraulic, or fluid-rheological means, and embeddedmicroprocessors are typically used for control [1]. Much researcheffort continues to be focused on the design and control of pros-theses capable of fully restoring normal gait [2–6]. Robotic testingis a viable method to evaluate and compare prosthesis prototypeswithout clearance burdens and with improved repeatability incomparison with human trials. Moreover, a machine may be fittedwith sensors which would be inadequate or difficult to install in apatient. Robotic testing has the potential to accelerate the develop-ment of new prosthesis concepts, as new hardware and controlideas can be examined in safe and controlled conditions. Someuse of commercial manufacturing robots in prosthesis testing hasbeen reported by the Fraunhofer Institute [7] and the ClevelandClinic [8]. This paper is concerned with a recently developedrobot [9,10] designed for transfemoral prosthesis tests using hipvertical displacement and thigh angle as input trajectories. Thesetwo degrees of freedom are the minimum necessary to emulatehuman walking in the sagittal plane. The overall robot configura-tion depends on the type of prosthesis attached to the machineinterface plate. When a passive prosthesis for transfemoral ampu-tees is installed, the overall system can be modeled as a three-linkrigid robot with one underactuated joint (i.e., the knee joint) withdamping. As shown in Fig. 1, the machine’s vertical axis is fixed,and walking is emulated using a treadmill. When a test is con-ducted, hip displacement and thigh angle histories measured fromnormal human gait are used as reference commands to the robot’sactuated degrees of freedom. Variables, such as knee angle,ground reaction force, and hip torque are either measured directlyor calculated from the measurements using models of the prosthe-sis and the machine. It is thus imperative for the controller toachieve accurate trajectory tracking in the presence of parametricuncertainty and modeling error associated with machine and pros-thesis. Moreover, this paper considers the control objective to bepure trajectory control, even during periods of contact with the

treadmill, where large forces applied to the foot are generated.Thus, ground reaction is regarded as an external disturbance to beaccommodated by the controller. Since the prosthetic foot and thetreadmill belt are compliant, the user may adjust the peak value ofground reaction force by lowering the center of oscillation of hipdisplacement. Note that hybrid force-position control [11,12] andimpedance control [13] are also possible with this equipment.

A robust control approach is mandatory, given that keydynamic parameters, such as some masses, moments of inertia,center-of-mass locations, and friction are subject to significantmeasurement errors. The robot controller should also require littleor no modification when different (passive) prostheses are in-stalled or when the reference trajectories are changed. This paperdevelops a coupled controller following the passivity-basedrobotic control technique, suitably extended to account for under-actuation and external disturbance. This design is presented incontrast with an independent-joint (decoupled) sliding mode con-troller which is based on actuator models only.

The paper is organized as follows: Section 2 describes machinefunctionality and components and presents the mathematicalmodel and its structural properties; Sec. 3 derives the passivity-based controller, proving uniform ultimate boundedness of thetracking error; Sec. 4 presents the independent-joint sliding modecontroller; Sec. 5 reports the real-time control results, and Sec. 6offers conclusions and recommendations for further development.

2 Test Robot Description

A schematic diagram of the robot and its components and aphotograph of the robot with a prosthesis and flexible foot areshown in Fig. 1. Vertical motion is generated by a linear slidedriven by a ballscrew with a pitch of 12.7 mm/rev. A brushless dcmotor (Mitsubishi HF-KP73) is directly coupled to the ballscrew.The motor is powered by a torque-mode servo amplifier (Mitsu-bishi MR-J3-70A). An analog input voltage applied to the servoamplifier results in a proportional torque on the motor shaft andballscrew. Vertical position (hip displacement) is measured withan incremental encoder rotating synchronously with the motor.An absolute position reference is established with a limit switchmounted on the vertical guides. The total useful travel of the slideis 12 in. The rotary motion stage, including motor, is carried bythe vertical slide. The rotary servo system is composed of a brush-less dc motor (ElectroCraft RapidPower RP34) coupled to a rotaryplate through an inchworm-gear reducer with ratio 80:1. Angularposition (thigh angle) is measured with an incremental encoder

Contributed by the Dynamic Systems Division of ASME for publication in theJOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedFebruary 1, 2013; final manuscript received December 1, 2013; published onlineFebruary 19, 2014. Assoc. Editor: Won-jong Kim.

Journal of Dynamic Systems, Measurement, and Control MAY 2014, Vol. 136 / 031011-1Copyright VC 2014 by ASME

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rotating synchronously with the motor. An absolute position refer-ence is established with a limit switch mounted on the rotaryplate. Prostheses are attached to the rotary plate by means of anadjustable threaded rod, which is secured to a bracket on the platewith two 2.75-in. nuts. The threaded rod, in combination with theadjustable center of oscillation of the vertical stage, offers flexibil-ity for standoff adjustments. Although thigh angular excursion inthe normal gait data does not exceed 50 deg, the rotary actuatorhas an unlimited angular range. The linear slide is attached to arigid steel frame. The prosthesis used for proof-of-concept testswith the robot is the Mauch MicroLite S [14], which consists oftwo rigid links connected by a rotary joint. A damper is connectedbetween the links, as shown in Fig. 1, to stabilize the knee duringthe stance phase and limit knee angle during the swing phase. Theflexible prosthetic foot is fitted with a strain gage calibrated tomeasure the vertical component of ground reaction force. A tread-mill is secured to the frame, and anchor bolts are used to securethe entire assembly to the lab floor. Normal gait profiles used as aguideline for design are a subset of the data collected by van denBogert [15], which includes walking and running in healthysubjects. The machine is designed for hip displacement ampli-tudes of up to 50 mm with a maximum velocity of 1 m/s. Verticalforce capacity is specified at 1200 N, which exceeds the groundforce generated by a 78 kg normal subject during fast walking orslow running. Real-time instrumentation and control is handled bya dSPACE DS-1102 system and associated software.

2.1 Mathematical Model. The overall machine-and-prosthe-sis system has been modeled in the standard framework ofrobotics [16]. When a passive leg prosthesis is attached, thesystem is underactuated, since the torque of the knee joint cannotbe externally controlled. Knee damping for the Mauch MicroLiteS knee may be manually set to two separate constant values forflexion and extension using the prosthesis’ adjustment rings, how-ever. When advanced prototypes featuring actively controlleddamping or powered knee actuation are attached, the system canbe regarded as semi-actuated or fully actuated. A forward kine-matics model is available [9,10], which allows the computation ofthe position and velocity of any point given the positions andvelocities of the joints. For the purposes of this paper, the point ofapplication of the vertical ground reaction and its corresponding

Jacobian are of interest. These quantities have been listed in theAppendix. Figure 2 shows the coordinate frames used in thekinematic model, where l2 is the length of link 2 (thigh), d0 is the

Fig. 1 Machine schematic and overall robot installation

Fig. 2 Denavit–Hartenberg coordinate frame assignments

031011-2 / Vol. 136, MAY 2014 Transactions of the ASME


offset of link 2, and c3 is the distance between the knee joint andthe center of mass of link 3. Frame assignments follow theDenavit–Hartenberg convention [17]. The world-frame coordi-nates of points of interest can be readily computed assuming thejoint coordinate vector q is known. In frame 3 coordinates, groundreaction is applied at ½lcx � lcy0�T . The vertical coordinate ZLC ofthe point of application can be found using the composite transfor-mation from frame 3 to the world frame (see Appendix). Thedynamic model for the robot and passive prosthesis is given injoint coordinates as

MðqÞ€qþ Cðq; _qÞ _qþ B _qþ Nðq; _qÞ þ gðqÞ þ Ff þ JTe Fe ¼

Kaua

0

� �(1)

where qT ¼ ½q1 q2 q3� is the vector of joint displacements (in ourcase q1 is the hip vertical displacement, q2 is the thigh angle, andq3 is the knee angle), ua is the vector of control inputs, M(q) is themass matrix, Cðq; _qÞ is a matrix accounting for centripetal andCoriolis effects, Nðq; _qÞ is a nonlinear damping matrix (in ourcase due to the knee damper), B is a viscous damping matrix, Ff isa vector of Coulomb friction terms, Je is the kinematic Jacobianrelative to the point of application of external forces Fe, g(q) is thegravity vector, and Ka is a diagonal matrix of actuator gains. Theexplicit entries of the above matrices are found in Refs. [9,10],and their regressor form is listed in the Appendix. Note that actua-tor dynamics can be considered in this model by adding reflectedactuator inertias (i.e., weighted by the square of gear ratio) in theentries of M. Viscous damping is significant only in the rotarymechanism, with coefficient b. Coulomb friction of magnitude f isconsidered only for the vertical slide, where it is significant. Thenonlinear damping term Nðq; _qÞ corresponds to a Rayleighdissipation function for the knee damper [9,10]. We decomposethe vector of joint coordinates into actuated and unactuated

components as qT ¼ ½qTa qT

u �. The friction terms are decomposed

along actuated and unactuated coordinates as FTf ¼ ½FT

fa 0�; ðB _qÞT

¼ ½ðBa _qaÞT0� and NT ¼ ½0 Rðq3; _q3Þ�, with Ba¼ diag([0 b]) and

FTfa ¼ ½ f signð _q1Þ0�.The prosthesis is regarded as links 2 and 3 of the overall robot.

Joint 3 is subject to an internal torque due to damper action. Thisnonlinear damping torque can be calculated as @R=@ _xd, where Ris the Rayleigh dissipation function

R ¼ 1

2bk _x2

d (2)

where bk is a direction-dependent damping coefficient and _xd isthe expansion rate of the damper. The expansion rate is readilyfound by considering the geometry of the damper attachment andusing differentiation of the cosine law. The nonlinear dampingterm of Eq. (1) is thus found to be

Rðq3; _q3Þ ¼ �bko2

dr2d cos2ðq3Þ _q3

l2d

(3)

where od, rd, and ld are the damper offset, swing radius, and in-stantaneous damper length, as shown in Fig. 3. It is important tonote that the external force Jacobian Je contains only link lengthsas parameters. Since lengths can be measured with very smallerrors and the vertical component of ground reaction Fe is meas-ured in real-time, the term JT

e Fe can be regarded as a measurabledisturbance that can be compensated directly by the real-timecontroller. Partition this term along actuated and unactuated coor-dinates as JT

e Fe ¼ ½FTa jFu�T and define a control input wa as

wa ¼ Kaua � Fa (4)

The horizontal component of ground reaction is not measuredin the real-time system and is difficult to describe without an

ankle/foot model. However, it is anticipated to have a small effecton thigh rotation due to the high gear ratio used in the mechanism.It is not considered for the theoretical derivation. The model ofEq. (1) is now partitioned into actuated and unactuated coordi-nates as

M11 €qa þM12 €qu þ C11 _qa þ C12 _qu þ Ba _qa þ ga þ Ffa ¼ wa (5)

M21 €qa þM22 €qu þ C21 _qa þ C22 _qu þ Rð _qu; quÞ þ gu þ Fu ¼ 0 (6)

where the dependence of M, C, Ffa, ga, and gu on the jointcoordinates and their velocities has been omitted from thenotation.

2.2 Structural Properties. We note that the well-knownproperties of skew symmetry, linearity in the parameters, andinertia matrix boundedness apply to the above partitioned modelas shown next. The model of Eq. (1) satisfies the property that_MðqÞ � 2Cðq; _qÞ is a skew-symmetric matrix. Clearly, the prop-

erty also applies to the block-diagonal elements of any partitionof M(q) and Cðq; _qÞ. In particular, _M11 � 2C11 must be skewsymmetric.

The linearity-in-parameters property states that MðqÞ€qþCð _q; qÞ _qþ gðqÞ can always be factored as Yð€q; _q; qÞH, wherethe regressor matrix Y does not contain any physical or dimen-sional constants [16]. All such constants appear in combination asa set of parameters forming the entries of a parameter vector H. Aminimal set of parameters exists for each kinematic configuration,and their determination must be done by algebraic manipulationon a case-by-case basis. We note that the linear parameterizationproperty is applicable to a partitioned model. Furthermore, whenCoulomb friction and linear viscous damping terms are present, itis still possible to obtain a linear parameterization. For the robotand passive prosthesis, the minimal parameterization leads to aregressor Y with three rows and nine columns (parameters). Aselaborated below, Eq. (5) is partitioned again along each actuatedcoordinate to facilitate tuning of certain control gains. This opera-tion yields the set of regressors and corresponding parametersshown in the Appendix.

The inertia matrix boundedness property guarantees that thereexist constants �qM and q

Msuch that q

M�k M k� �qM, where any

suitable norm may be used. Evidently, the block entries of a parti-tion of M must also possess absolute upper and lower bounds.This property will be used below to guarantee that the knee accel-eration estimation error term will not cause a Lyapunov derivativeto be positive.

Fig. 3 Damping cylinder and knee geometry. The indicateddirection for q3 is positive (flexion), the opposite is extension.The cylinder has separate adjustment rings for the dampingcoefficient during flexion and extension.

Journal of Dynamic Systems, Measurement, and Control MAY 2014, Vol. 136 / 031011-3


3 Robust Passivity-Based Controller

The following development follows the framework ofpassivity-based control [16]. Significant extensions to the standardcase are required to handle underactuation, to compensate for fric-tion and disturbance and to facilitate tuning. Suppose that theactuated joints must follow reference trajectories given by func-tions of time qd

aðtÞ, while the unactuated joints must remainbounded. For this, define a function sa ¼ _qa � _qd

a þ Kðqa � qdaÞ,

where K is a positive-definite diagonal matrix. Note that thisdefinition coincides with the notion of sliding function used inthe sliding mode control of second-order systems. Thus, if sa

and its derivatives become zero, first-order decay dynamics willbe obtained for the tracking error ~qa ¼ qa � qd

a . Definingva ¼ _qd

a � Ka ~qa, we have

sa ¼ _qa � va (7)

_sa ¼ €qa � aa (8)

where aa ¼ _va. Define the Lyapunov function candidate

Vr ¼1

2sT

a M11sa þ ~qTa KðBa þPÞ~qa (9)

where P is a positive-definite diagonal matrix. The time deriva-tive of Vr is given by

_Vr ¼ sTa M11 _sa þ

1

2sT

a_M11sa þ 2~qT

a KðBe þPÞ _~qTa

The skew-symmetry property applied to M11 and C11 implies that

_Vr ¼ sTa ðM11 _sa þ C11saÞ þ 2~qT

a KðBa þPÞ _~qTa

Using Eqs. (7) and (8) yields

_Vr ¼ sTa ð�M11aa � C11va �M12 €qu � C12 _qu

� Basa � Bava � ga � Ffa þ waÞ þ 2~qTa KðBa þPÞ _~qT

a

To construct a control law, we assume that only the structure ofthe model matrices is known, but that the parameters are uncer-tain. However, we assume that parametric uncertainties arebounded. Also, we assume that joint positions and velocities aredirectly available, and that the acceleration of the unactuated jointis estimated (for instance through numerical differentiation), witha known estimation error bound. Therefore we postulate a controlinput of the form

wa ¼ M11aa þ ðC11 þ BaÞva þ M12au

þ C12 _qu þ ga þ Ffa �Psa þ w2 (10)

where the circumflex notation has been used to indicate estimatedquantities. In particular, au is the estimated acceleration of theunactuated joints. Also, w2 is an additional control term used tocompensate for the acceleration estimation error, as shown below.The linear parameterization property establishes that there exist aregressor Yðaa; va; aa; _qu; _qu; qa; quÞ and corresponding parametervector H such that

M11aa þ ðC11 þ BaÞva þ M12au þ C12 _qu þ ga þ Ffa

¼ Yðaa; va; aa; _qu; _qu; qa; quÞH

To simplify control gain tuning, the above equation is partitionedagain along the individual actuated joints. In the case of the legprosthesis test robot, there are two actuated joints: hip verticaldisplacement and thigh angle. For this case, we thus consider thedecomposition

wa ¼Y1H1

Y2H2

" #�Psa þ w2 (11)

where the arguments of the regressors have been omitted. Substi-tuting Eq. (11) into the Lyapunov derivative yields

_Vr ¼ s1Y1ðH1 �H1Þ þ s2Y2ðH2 �H2Þ � sTa ðBa þPÞsa

� sTa M12 ~au þ sT

a w2 þ 2~qTa KðBa þPÞ~qa

where ~au ¼ €qu � au is the acceleration estimation error for theunactuated joint. In this robust control approach, we assume thatonly nominal values for the parameters are available, namely, H10

and H20, where known bounds exist for their errors relative to thetrue parameters

kH10 �H1k � q11 (12)

kH20 �H2k � q12 (13)

The inertia matrix boundedness property implies that kM12ðqÞk isupper-bounded by a constant (see Appendix). Assuming that theunactuated acceleration is computed with some known maximumerror, a norm bound can be established for the term M12 ~au

kM12 ~auk � q2 (14)

Now choose Hi ¼ Hi0 þ DHi for i¼ 1,2, where DHi are new con-trols to be determined below. Substituting the definition of sa intothe term sT

a ðBa þPÞsa and combining terms leaves the Lyapunovderivative in the form

_Vr ¼ s1Y1ð ~H1 þ DH1Þ þ s2Y2ð ~H2 þ DH2Þþ sT

a ðw2 �M12 ~auÞ � eTQe (15)

where the error state has been defined as eT ¼ ½~qTa j _~qT

a � and Q is ablock-diagonal matrix with blocks KðBa þPÞK and ðBa þPÞ.

Since ~Hi are norm-bounded, the terms DHi can be adjustedaccording to the switching feedback laws DHi ¼ �q1iðsiYi=ksiYikÞ whenever ksiYik 6¼ 0, with DHi ¼ 0 otherwise. Indeed,the following inequality holds:

siYið ~Hi þ DHiÞ � ksiYik k ~Hik � q1iksiYik � 0

Similarly, since the M12 ~au is norm-bounded, w2 can be chosen as�q2ðsa=ksakÞ whenever ksak 6¼ 0 and w2¼ 0 otherwise. The fol-lowing inequality holds:

� sTa M12 ~au � sT

a q2

sa

ksak� ksak k M12 ~auk � q2ksak � 0

Since the term �eTQe is negative-definite, the above choicesrender the Lyapunov derivative negative-definite.

These laws are of the variable-structure type and their switch-ing action requires unlimited actuator bandwidth [18]. A continu-ous approximation of these laws is therefore adopted

DHi ¼�q1i

siYTi

ksiYik; if k siYik > e1i

�q1i

siYTi

e1i; if k siYik � e1i

8>>><>>>:

(16)



where e1i > 0 are small positive constants. Similarly, w2 is chosenas

w2 ¼�q2

sa

ksak; if k sak > e2

�q2

sa

e2

; if k sak � e2

8>><>>: (17)

where e2 is a small positive constant. With this approximation, theclosed-loop control system is guaranteed to possess the propertyof uniform ultimate boundedness [19] (uub) of the error trajecto-ries. The guaranteed uub radius is estimated as follows: Assumingthe “worst-case” where ksiYik � e1i and ksak � e2 at the sametime, the following inequality holds:

_Vr ¼X2

i¼1

siYi~Hi � q1iY

Ti

si

ksiYikHi

� �

þ sa �M12 ~au � q2

sa

ksak

� �� eTQe

�X2

i¼1

ksiYikq1i 1� ksiYike1i

� �þ ksakq2 1� ksak

e2

� �� eTQe

which reduces to

_Vr �X2

i¼1

e1iq1i þ q2e2 � eTQe (18)

For _Vr < 0, it is then sufficient that the quadratic term on e is largeenough to overcome the positive contributions of the other terms.Since Q is positive-definite, this can be guaranteed by the follow-ing equation:

ke k>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX2

i¼1

e1iq1i þ q2e2

kðQÞ

vuuuut ¼D ru (19)

where ru represents the uub radius and kðQÞ is the smallest eigen-value of Q. For the actuated joints, closed-loop trajectories willultimately enter and remain on the smallest level set of Vr whichcontains a ball of radius ru. The designer must adjustq1i; q2; e1i; e2, K, and P to achieve an acceptable tradeoff betweengood tracking accuracy (decreasing ru), while avoiding excessivecontrol chattering (increasing ru). Remarks:

(1) Although damping was considered for the actuated jointsthrough matrix Ba, the stability proof does not require it. Itspresence can help reduce control effort, however.

(2) The regressors are functions of actuated joint positions andvelocities only, assumed to be available as real-time meas-urements. An estimate of the unactuated acceleration can beobtained by online numerical differentiation, and a bound forthe acceleration estimation error can be obtained accordingto the differentiation method. A simpler approach is to use azero estimate for the acceleration and set the error bound tothe maximum knee acceleration that can be reasonablyexpected during prosthesis operation. This eliminates theneed for acceleration estimation altogether.

(3) Splitting the regressor along individual actuated coordi-nates allows us to use separate gains q1i whose magnitudesare tailored to those of the corresponding parameters andregressors. If this is not done, a conflict may arise in prac-tice, when tuning the value of a single q. Indeed, q mayneed to be increased so that sufficiently accurate tracking isobtained in some coordinates. At the same time, q may

need to be reduced to avoid excessive chattering in controlchannels which are already achieving good tracking.

(4) The regressors and corresponding parameters for the pas-sive leg prosthesis have been listed in the Appendix, alongwith the bound for kM12k.

The designer must tune P;Ki;qi1; q2, and the e constants. Thefirst matrix increases the damping effect on the actuated joints, thesecond accelerates the convergence of the tracking error andthe remaining gains are used to guarantee robustness against para-metric uncertainty. Note that boundedness of the unactuated coor-dinate follows from rearranging Eq. (6) as follows:

M22 €qu þ C22 _qu þ Rð _qu; quÞ þ gu ¼ �ðM21 €qa þ C21 _qa þ FuÞ(20)

Since tracking of qdaðtÞ is achieved with bounded errors, qa and its

derivatives must be bounded. Matrices M21 and C21 are them-selves bounded, and so is Fu. Therefore, the term on the right-hand side of Eq. (20) acts as a bounded input to the unactuatedcoordinate. Equation (20) represents the dynamics of a passivemechanical system driven by a bounded input. Its position and ve-locity states must then remain bounded.

3.1 Simulation. A simulation was conducted for the robustpassivity-based controller (RPBC) under motion profile tracking.A set of reference profiles qd

1ðtÞ and qd2ðtÞ were available from

human motion studies. Their first two derivatives were computedoffline by numerical differentiation and spline smoothing. Thecontrol tuning parameters used in this simulation were set to thebest values found in subsequent real-time trials (Sec. 5). An inten-tional parametric mismatch in H between plant and controller wasintroduced to evaluate robustness. In particular, the controllerused the nominal H0, while the plant used H0 þ DH, where thecomponents of DH were selected randomly from the interval½�0:5H0; 0:5H0�. Figure 4 shows the results, indicating that thecontroller is capable of achieving accurate tracking withoutexceeding amplifier saturation limits. Convergence of the Lyapu-nov function of Eq. (9) is also shown, although it increases itsvalue during brief instants due to the nonideality of the deadzoneimplementation.

4 Independent-Joint Sliding Mode Control

A simple alternative to the coupled controller is to take advant-age of the high transmission ratios between the motors and theirdriven links and treat the robot as a decoupled mechanical system.As shown in [9,10], the actuator-centric models have the form

J €hþ b _h ¼ kuþ sd (21)

where hðtÞ is the controlled position variable, u(t) is the controlvoltage (assumed proportional to motor torque), k is a constantreflecting a combination of servo amplifier gain, motor torqueconstant and rotary/linear motion conversion, J is the inertia ofthe load and motor, and b is a viscous damping coefficient. Vari-able sd represents an uncertain torque input consisting of externaltorque disturbances, unmodeled dynamics, parametric uncertain-ties, and unmodeled static effects, such as friction torque. Unmod-eled dynamics include gravity torque and inertial coupling. SMCwas chosen due to its good robustness properties and straightfor-ward implementation [18,20,21]. In terms of the tracking errore ¼ hd � h, the system of Eq. (21) admits a sliding function of theform s ¼ _eþ ke, with k > 0 a tunable constant. Note that ifs(t)¼ 0 for all t after some reaching time tr, ideal first-order decayis achieved for the tracking error. An SMC law capable of achiev-ing and maintaining s¼ 0 in finite time despite the presence of theuncertain term sd has the form



u ¼ J

kð€hd þ k _hdÞ þ b

J� k

� �_hþ g sign ðsÞ

� �(22)

where g > 0 is chosen according to an assumed bound for sd.Note that motion profiles to be tracked and their derivatives enterthe control law through the feedforward term €hd þ k _hd. To limitchattering, a continuous approximation to the signum function isused, namely, a saturation function. Implementation of this con-trol law is simpler than that of a PID controller, since no onlineintegration or differentiation are needed, and only position andvelocity measurements are required, which are available fromoptical encoders. Figure 5 shows a simulation of the IJSMC to therobot and prosthesis model. In this simulation, point contactbetween the prosthesis endpoint and the treadmill is assumed. Atreadmill belt compliance for vertical deflections is taken fromexperimental measurements, and vertical ground reaction is calcu-lated from ZLC shown in the Appendix. It can be seen that accu-rate tracking is achieved and that the sliding functions converge toand remain within their boundary layers.

5 Real-Time Control Tests

The purpose of the tests is twofold: to demonstrate the validityof the passivity-based controller developed in this paper and tocompare it with the IJ-SMC. Tests were conducted under puremotion tracking mode, and were initiated without ground contact.Then, manual biasing of the vertical stage was used to gradually“land” the foot on the treadmill, monitoring the force sensed bythe foot-mounted strain gage. The speed of the treadmill was setto 2.6 miles per hour, the value at which the normal walk gait datawere taken. The center of oscillation of the thigh was also biasedmanually once the foot was engaged in a swing-stance cycle. Thiswas done so that overall motion resembled natural walkingas much as possible, given the limitations associated with apassive prosthesis. Achievement of natural motion was judged

quantitatively by achieving a ground reaction magnitude and aknee angle variation which matched human data for the hip andthigh motion profiles being tracked. Natural gait was judged quali-tatively by ensuring that the stance phase was initiated with heelcontact, transitioning to forefoot contact, and without using exces-sive thigh angle biasing.

The Mauch MicroLite S knee was adjusted to a flexion settingof 7 and an extension setting of 2. Although manufacturer data areunavailable regarding the damping constants associated with thesesettings, custom measurements carried out by the authors provideestimated values of b¼ 1.57 N s/mm for flexion and b¼ 0.533 Ns/mm for extension.

Controllers are compared on the basis of their tracking abilitiesunder full loading conditions, considering the amount of chatter-ing required to achieve a given tracking accuracy. Test conditionswere maintained as similar as possible for both controllers,although some differences to be elaborated below were unavoid-able due to manual biasing during the tests. Note that the thighangle coordinate q2 is measured relative to a horizontal referencein the robotic model; however, results are presented in terms of itscomplement q2v, which is more commonly used in biomechanicsstudies.

5.1 Robust Passivity-Based Controller. Two tests were con-ducted: one with nominal H parameters arising from modelingand parameter estimation, and another one with an intentional ran-dom perturbation in each component of H with a deviation of upto 10%. The RPBC was initially tuned in simulation with satisfac-tory results. Although the initial simulation values led to stablemotion, chattering was large and it was evident that trackingaccuracy could be improved with online tuning. Final values ofthe gains were K ¼ diagð80; 7:5Þ; P ¼ diagð10; 0:1Þ; q11 ¼ 140;q12 ¼ 5; e11 ¼ 2; e12 ¼ 7; q2 ¼ 1; and e2 ¼ 1. Figure 6 shows thehip displacement and thigh angle tracking performance achievedby the RPBC, as well as the control voltage histories for thenominal case. Figure 7 shows the knee angle and reaction force

Fig. 4 Simulation of the RPBC in motion reference tracking with 50% parameter perturbation



histories. During the real-time tests, manual biasing was applieduntil a force peak of 1000 N was observed. As expected, theground forces act as disturbances preventing exact tracking of ver-tical hip displacement and thigh angle during ground contact.

Although knee angle is not being controlled, the prosthesis mech-anism maintains it between 0 and 70 deg, a range compatible withnormal human walking at the test speed. Note also that the controlvoltages reach the amplifier saturation limits of 68 V for the

Fig. 5 Simulation of IJ-SMC

Fig. 6 RPBC: hip displacement and thigh angle tracking performance and control voltages(nominal)



vertical stage and 610 V for the rotary stage. The tracking resultswith RPBC and off-nominal plant parameters are shown in Fig. 8.Chattering in the respective control channels can be measured bythe spectral energy above an arbitrary frequency. The fundamental

frequency of the control inputs (i.e., the one correlated with theperiodicity of the gait data) is 0.87 Hz. Fast Fourier transform(FFT) analysis reveals that the motion references do not containsignificant frequency components beyond 10 Hz. Spectral

Fig. 7 RPBC: knee angle and vertical ground reaction force (nominal)

Fig. 8 RPBC: hip displacement and thigh angle tracking performance and control voltages(off-nominal)



energies found in the control signals above this frequency can beattributed to the control algorithm and electronics and can be rea-sonably used to compare control activity. The length of the datarecords and a sampling period of 5� 10�4 s implies a frequencyresolution of 0.2 Hz and an FFT frequency range of 1 kHz. Ameasure of chattering is obtained by an rms average of FFT mag-nitudes between 10 Hz and 1 kHz normalized by the length of thedata record. Chattering levels according to this measure for thevarious tests are shown in Table 1, along with tracking perform-ance data.

5.2 Independent-Joint Sliding Mode Controller. Decoupledcontrol laws of the form of Eq. (22) were used for the vertical androtary servos. Servo system constants were obtained by systemidentification and custom measurements [9,10] as J1¼ 1 kg m2,b1¼ 44.4 N m s, and k1¼ 1610 N m/V for the vertical system asJ2¼ 1 kg m2, b2¼ 6.5 N m s, and k2¼ 10 N m/V for the rotarysystem. Sliding mode control parameters for the vertical servosystem were chosen as g1 ¼ 104; k1 ¼ 90; and /1 ¼ 166. The cor-responding parameters for the rotary servo system wereg2 ¼ 50; k2 ¼ 10; and /2 ¼ 0:5.

Figure 9 shows the tracking performance and control input his-tories for the same motion references as used with the RPBC. Thetracking accuracy is given by the rms errors of 2.28 mm for q1 and2.15 deg for q2v. The chattering levels for u1 and u2 are 8.52 V and

6.70 V, respectively, calculated as described for the RPBC. It isobserved that the control inputs do not reach amplifier saturationlevels. Figure 10 shows the knee angle and vertical ground reac-tion force. Vertical biasing was used to achieve a force peak of1000 N, and the thigh angle was manually biased during operationto obtain the initial heel strike indicative of a normal gait pattern.As with RPBC, a test was conducted using an intentional 10%perturbation in plant parameters used by the controller (b, J, andk for each channel). Tracking results for this case are shown inFig. 11.

5.3 Discussion. Both controllers achieved rms tracking errorsof 1.75 mm or less for vertical hip displacement (4.6% of themotion reference range) and 2.6 deg or less for thigh swing (5.5%of the motion reference range). Chattering measures were signifi-cant larger for u2 when using RPBC. Table 1 summarizes theresults. Two observations are in order in regard to Table 1. First,tracking performance and chattering levels are essentially main-tained when plant parameters are intentionally perturbed. Second,chattering levels in u2 are significantly higher for RPBC regard-less of nominal or off-nominal conditions, but this does not occurin simulation studies, even when large parameter perturbations areintroduced. Therefore, excessive chattering is attributed to effectspresent in the real system that were not modeled. We were able totrace the source of chattering to backlash in the thigh rotation gearmechanism. The vertical positioning mechanism uses a ballscrewwhich is essentially backlash-free, which explains the absence ofchattering in u1. Although backlash is present in the systemregardless of control strategy, significantly smaller chattering lev-els were observed in u2 when using IJ-SMC. This is becauseRPBC implements an approximate model inversion throughEq. (10), which includes the actuator models and the coupleddynamics of the robotic links. Since the actuator models do notinclude backlash, a significant mismatch occurs between actualand modeled plant behavior, particularly at the times when the

Table 1 Summary of tracking performance and chatteringmeasures

~q1 rms ~q2 rms u1 chatter u2 chatter

RPBC nominal 1.64 2.61 7.39 22.9SMC nominal 1.15 2.08 1.69 2.18RPBC off-nominal 1.76 2.44 8.67 21.9SMC off-nominal 1.03 1.93 1.31 2.16

Fig. 9 IJ-SMC: hip displacement and thigh angle tracking performance and control voltages(nominal)



gears become disengaged. Unlike errors due to parametric uncer-tainty, backlash (or other nonlinearity) are not explicitly addressedby RPBC. In contrast, the approximate model inversion used inthe IJ-SMC law of Eq. (22) contains only actuator dynamics up to

the driving gear, treating inertial coupling among links, frictionand other effects as a disturbance. Model inversion is moreaccurate in this case, since actuator dynamics were determinedaccurately by system identification. This explanation can be

Fig. 10 IJ-SMC: knee angle and vertical ground reaction force (nominal)

Fig. 11 IJ-SMC hip displacement and thigh angle tracking performance and control voltages(off-nominal)



confirmed in simulation by introducing a backlash element in thethigh rotation drivetrain and testing each controller. We were ableto reproduce the same effect in simulation: chattering due to back-lash is higher when using RPBC. Figure 12 shows the simulatedu2 with each controller. Note that chattering has been identified asproblematic in similar passivity-based implementations [22].Another reason for the higher chattering levels found with RPBCis that the “nominal” values of H come from measurements whichare subject to errors and should not be regarded as “true” parame-ters. The difference between nominal and actual parameters mustbe overcome with high values of the switching gains q duringonline tuning, giving rise to chattering. IJ-SMC relies on only sixactuator parameters which exclude highly uncertain quantities,such as friction coefficient, link inertias, and center-of-gravitylocations. RPBC, in contrast, requires a complete dynamic modeland nine independent parameter estimates. Note that althoughextensive online tuning was conducted with each controller toachieve good performance, the numerical values of rms trackingerror and chattering measure are tailored to this particular set ofgains.

The observed ability of the IJ-SMC to offer good performanceis due to the high transmission ratios used in each servo. Indeed,inertial coupling among links and gravity forces appear at themotor shaft after division by the square of the gear ratio. Thus, theservos operate practically in a decoupled mode. In this situation,the simplicity of the IJ-SMC (only three adjustable control param-eters per channel) represents an advantage over the RPBC, whichhas five adjustable control parameters per channel, making tuningmore difficult.

Although the peak magnitude of the ground reaction matchedthe RPBC case, its shape is different. This is due to the fact thatthe prosthesis is not actuated at the knee. This means that the tra-jectories exhibited by uncontrolled variables, such as knee angleor ground reaction, are highly influenced by the orientation andvelocity of the foot at the strike instant. In turn, these two varia-bles depend on how manual biasing is used. This difference does

not prevent a meaningful comparison of controllers in terms oftheir tracking accuracy against the amount of chattering required.

The RPBC has the advantage of having guaranteed robust sta-bility, backed by theoretical Lyapunov analysis. To the extent ofthe authors’ knowledge, there is no proof available that anindependent-joint sliding mode controller (without gravity com-pensation) achieves asymptotic tracking of the actuated coordi-nates while maintaining the unactuated coordinates bounded.

6 Conclusions

A robust controller for motion tracking was developed by suit-ably extending the passivity-based robot control technique toaccount for the lack of actuation at the knee joint and to facilitatetuning. The developed controller was compared with a simple,decoupled sliding mode controller. Good tracking of hip displace-ment and thigh angle normal walk motion profiles was achievedwith both controllers, even during ground contact, where largedisturbance forces were applied. In our tests, the passivity-basedcontroller exhibited higher chattering in the thigh rotation controlchannel for the same tracking accuracy. We were able to trace thiseffect to backlash in the thigh rotation drive train. Modeling, con-trol development, and real-time operation were conducted with apassive leg prosthesis. For prostheses which produce knee torqueby means other than passive damping, an appropriate model mustbe used and the controller redesigned. Also, when activelydamped or powered knee prosthesis is tested using this robot, thepassivity-based approach may prove more useful than decoupledcontrol due to the availability of a systematic theoretical frame-work from where to derive control laws with guaranteed stabilityand robustness. The higher sensitivity of RPBC to backlash moti-vates further research into the subject.

Although a motion control approach was adopted in this work,other test modalities are possible. For instance, a mixed motion/force approach may be used, where motion control is still used forboth degrees of freedom in the swing phase and for thigh angle in

Fig. 12 Simulation of backlash effects: RPBC versus IJ-SMC



the stance phase, but hip displacement is controlled in the stancephase so that a desired ground force profile is tracked. Also, withactively damped or powered knees, an impedance synthesisapproach may be undertaken, where control is used to track hipand thigh motions while controlling the knee actuator to replicatethe moment/velocity impedance observed in human knees.

Acknowledgment

This work was supported by the State of Ohio, Department ofDevelopment and Third Frontier Commission, which providedfunding through the Cleveland Clinic in support of the projectRapid Rehabilitation and Return to Function for AmputeeSoldiers.

Appendix

Vertical coordinate of the point of application of the groundreaction (point-foot model)

ZLC ¼ q1 � lcy cosðq2 þ q3Þ þ ðc3 þ lcxÞ sinðq2 þ q3Þ þ l2 sinðq2Þ(A1)

External force Jacobian at point of application

Jeð1; 1Þ ¼ 0

Jeð1; 2Þ ¼ �ðc3 þ lcxÞ sinðq2 þ q3Þ þ lcy cosðq2 þ q3Þ � l2 sinðq2ÞJeð1; 3Þ ¼ �ðc3 þ lcxÞ sinðq2 þ q3Þ þ lcy cosðq2 þ q3ÞJeð2; 1Þ ¼ Jeð2; 2Þ ¼ Jeð2; 3Þ ¼ 0

Jeð3; 1Þ ¼ 1

Jeð3; 2Þ ¼ ðc3 þ lcxÞ cosðq2 þ q3Þ þ lcy sinðq2 þ q3Þ þ l2 cosðq2ÞJeð3; 3Þ ¼ ðc3 þ lcxÞ cosðq2 þ q3Þ þ lcy sinðq2 þ q3Þ

Regressors and parameter vectors

Y1ð1Þ ¼ a1

Y1ð2Þ ¼ a1 � g

Y1ð3Þ ¼ a2 cosðq2Þ � _q2v2 sinðq2ÞY1ð4Þ ¼ a2 cosðq2 þ q3Þ � v3ð _q2 sinðq2 þ q3Þ þ _q3 sinðq2 þ q3ÞÞ

�v2ð _q2 sinðq2 þ q3Þ þ _q3 sinðq2 þ q3ÞÞ þ a3 cosðq2 þ q3ÞY1ð5Þ ¼ signðq1ÞY2ð1Þ ¼ a1 cosðq2Þ � g cosðq2ÞY2ð2Þ ¼ a1 cosðq2 þ q3Þ � g cosðq2 þ q3ÞY2ð3Þ ¼ a2

Y2ð4Þ ¼ 2a2 cosðq3Þ þ a3 cosðq3Þ � v3ð _q2 sinðq3Þ þ _q3 sinðq3ÞÞþ _q3v2 sinðq3Þ

Y2ð5Þ ¼ a3

Y2ð6Þ ¼ _q2

where ai and vi are the entries of vectors a and v.Parameter vectors

H1 ¼

m0

m1 þ m2 þ m3

m2ðc2 þ l2Þ þ m3l2

m3c3

f

26666664

37777775

H2 ¼

m2ðc2 þ l2Þ þm3l2

m3c3

I2z þ I3z þ Jmr2 þ c22m2 þ c2

3m3 þ l22m2 þ l22m3 þ 2c2l2m2

l2c3m3

I3z þm3c23

b

2666666664

3777777775

See Refs. [9,10] for the definition of inertial parameters.Bound on kM12k

M12 ¼m3c3 cosðq2 þ q3Þ

m3c23 þ l2m3c3 cosðq3Þ þ I3z

� �

¼H2ð2Þ cosðq2 þ q3Þ

H2ð5Þ þH2ð4Þ cosðq3Þ

� �

Using the Euclidean norm kM12k2 � H22ð2Þ þ ðH2ð5Þ þH2ð4ÞÞ2.

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