Intelligent control of a prosthetic ankle joint using gait recognition

Control Engineering Practice 49 (2016) 1–13

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Intelligent control of a prosthetic ankle joint using gait recognition Anh Mai a, Sesh Commuri b,n,1 a b

School of Electrical and Computer Engineering, University of Oklahoma, Devon Energy Hall, Room 150 110 W. Boyd St., Norman, OK 73019, USA School of Electrical and Computer Engineering, University of Oklahoma, Devon Engineering Hall, Room 432 110 W. Boyd St., Norman, OK 73019, USA

art ic l e i nf o

a b s t r a c t

Article history: Received 3 February 2015 Received in revised form 15 January 2016 Accepted 19 January 2016

Desire for better prosthetic feet for below-knee amputees has motivated the development of several active and highly functional devices. These devices are equipped with controlled actuators in order to replicate biomechanical characteristics of the human ankle, improve the amputee gait, and reduce the amount of metabolic energy consumed during locomotion. However, the functioning of such devices on human subjects is difficult to test due to changing gait, unknown ankle dynamics, complicated interaction between the foot and the ground, as well as between the residual limb and the prosthesis. Commonly used approaches in control of prosthetic feet treat these effects as disturbances and ignore them, thereby degrading the performance and efficiency of the devices. In this paper, an artificial neural network-based hierarchical controller is proposed that first recognizes the amputees' intent from the actual measured gait data, then selects a displacement profile for the prosthetic joint based on the amputees' intent, and then adaptively compensates for the unmodeled dynamics and disturbances for closed loop stability with guaranteed tracking performance. Detailed theoretical analysis is carried out to establish the stability and robustness of the proposed approach. The performance of the controller presented in this paper is demonstrated using actual gait data collected from human subjects. Numerical simulations are used to demonstrate the advantages of the proposed strategy over conventional approaches to the control of the prosthetic ankle, especially when the presence of noise, uncertainty in terrain interaction, disturbance torques, variations in gait parameters, and changes in gait are considered. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Prosthetic foot Gait analysis Intelligent control Biomechanics Amputation

1. Introduction Loss of limbs has far reaching impact on the mobility, emotional and physical health of an individual. According to the recent estimates (Ziegler-Graham, MacKenzie, Ephraim, Travison, & Brookmeyer, 2008), over 1.6 million Americans, or roughly 1 in 190 persons, are currently living with a loss of a limb. Amongst them, 62% underwent an amputation procedure to the lower extremity. It is anticipated that the number of people living with limb loss would increase to 3.6 million by the year 2050, with over 1.4 million of them being under 65 years of age. Clinical studies indicate that 68–88% of amputees wear prostheses at least 7 h a day (Pohjolainen, Alaranta, & Karkkainen, 1990; Walker, Ingram, Hullin, & McCreath, 1994) and lead an active lifestyle (Brodzka, Thornhill, Zarapkar, Malloy, & Weiss, 1990). These people need highly functional prostheses in order to maintain healthy and good quality of life (Ziegler-Graham et al., 2008). In the short term, the n

Corresponding author. E-mail addresses: [email protected] (A. Mai), [email protected] (S. Commuri). URL: http://hotnsour.ou.edu/commuri/scommuri.html (S. Commuri). 1 http://www.springer.com/sgw/cda/frontpage/0,11855,5-40109-70-357110630,00.html. http://dx.doi.org/10.1016/j.conengprac.2016.01.004 0967-0661/& 2016 Elsevier Ltd. All rights reserved.

use of passive prosthetic foot with fixed ankle can lead to asymmetric gait (Bateni & Olney, 2002; Kovac, Medved, & Ostojic, 2010), increased intact muscle contraction (Fey, Silverman, & Neptune, 2010), and higher metabolic energy expenditure in an individual (Torburn, Powers, Guiterrez, & Perry, 1995). Long-term health complications such as osteoarthritis, osteoporosis, back pain, and other musculoskeletal problems can also be linked to the poor fit and improper alignment of the prosthesis and can lead to poor overall quality of life of the individuals (Gailey, Allen, Castles, Kucharik, & Roeder, 2008). Research into amputee gait has shown that it is desirable that future prosthetic feet replicate the biomechanical characteristics of the biological ankle joint (Versluys et al., 2009). It is necessary for the prosthetic ankle joint to follow the typical joint displacement profile of a human ankle. Such profile helps absorbing shock due to the ground interaction and generating propulsion energy for the body to move forward, therefore guaranteeing the stability and mobility during gait. The displacement profile in Fig. 1 (Winter, 2009) is shown normalized with respect to the gait cycle and is related to the phases and events of the gait (Winter, 2009; Perry, 1992; Uustal & Baerga, 2004). The joint displacement profile is typical for all humans and is presumed to minimize the energy consumption during gait (Anderson & Pandy, 2001; Ackermann & van den Bogert, 2010).

2

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

Fig. 1. Typical human ankle displacement profile.

Next generation prosthetic feet are likely to be equipped with controlled actuators in order to provide variable stiffness and range of motion of the artificial ankle joint, and help achieve desired gait performance (Versluys et al., 2009). However, control of the ankle prosthesis to replicate the movement of a healthy ankle during gait is a difficult task due to the following reasons. a) The ideal ankle displacement profile depends on several factors such as user gait (e.g., stance time, swing time, step length, and stride length), selected walking speed, inclination of the terrain, and type of activity (e.g., walking on level ground, ascending/ descending stairs). Due to changes in gait and terrain and unknown intent of the user, the ideal ankle displacement profile cannot be specified a priori. b) During gait, the movement of the prosthetic foot is influenced by the reaction force resulting from the interaction of the foot with the terrain. This ground reaction force (GRF) plays a critical role in supporting the body weight, ensuring stability, and providing the necessary propulsion for the gait (Perry, 1992). GRF causes a reaction torque at the joint that has to be compensated in order to achieve proper tracking behavior. GRF is usually measured using motion tracking systems and force plates in a laboratory setting (Winter, 2009). Unfortunately, real-time measurement of GRF is not feasible during normal gait and therefore cannot be used in the control of the prosthetic joint. c) The dynamics of the foot are affected by the nonlinear coupling between the prosthetic ankle joint and the biological knee and hip joints of the amputee. These effects depend on anthropometric measurements of the human body and vary with gait. Neglecting these interactions will lead to larger tracking errors for a specified ankle displacement profile. Commercially available below-knee controlled prostheses such as Proprio Foot (Össur, 2015) and BIOM Ankle System (BIOM, 2015; Au, Weber, & Herr, 2009) are capable of manipulating the movement of the ankle joint and adapting to different gait and terrain (Versluys et al., 2009; Jiménez-Fabián & Verlinden, 2012). Improvement in gait performance with these controlled prostheses has been reported in several case studies (Wolf, Alimusaj, Fradet, Stegel, & Braatz, 2009; Ferris, Aldridge, Rábago, & Wilken, 2012; Grabowski & D’Andrea, 2013). Performance of other active foot prototypes such as SPARKy (Hitt, Sugar, Holgate, Bellman, & Hollander, 2009) and PPAMs (Versluys et al., 2008) in generating torque to manipulate the ankle movement is also promising. In general, control algorithms neglect the dynamics of the ankle joint, the interaction of the ankle with the remaining healthy

joints of the residual limb, and the effect of the ground reaction torque. These devices are based on the linearized dynamics of the joint and use proportional-derivative control with fixed control gains. While these controllers guarantee local stability, their performance might deteriorate quickly in the presence of unmodeled system dynamics and measurement noises. Advanced bionic feet in recent literature can vary the ankle stiffness and generate power to support the gait. These designs are in early development stage and researchers are aware of the challenges in achieving satisfactory performance when gait conditions change (Markowitz et al., 2011). Researchers have also shown that supervisory control schemes that implement different control strategies based on perceived gait help achieve desired performance of the foot (Au, Berniker, & Herr, 2008). However, the effect of gait and terrain changes, unmodeled dynamics and noise on the performance of these supervisory controllers have not been systematically studied (Jiménez-Fabián & Verlinden, 2012). In recent years, several researchers have investigated the use of artificial neural networks (ANNs) for recognizing the users' intent during gait (Au et al., 2008; Jin, Zhang, Zhang, Wang, & Gruver, 2000; Varol, Sup, & Goldfarb, 2008; Torrealba et al., 2010). However, the information gained was not used to directly control the prosthetic joint. In this paper, an artificial neural network-based control structure performs multiple tasks including learning the ankle dynamics, recognizing the varying gait intent, and generating an appropriate torque to drive the ankle joint along a desired displacement profile. The controller proposed in this paper is based on a hierarchical adaptive learning strategy and includes the following steps.


Generate an ankle joint displacement profile based on the measured gait data,


Estimate the ground reaction force during gait using a viscoelastic contact model, and


Implement an artificial neural network (ANN)-based control to approximate the ankle dynamics and track the generated ankle displacement profile. The closed-loop stability of the proposed approach is rigorously analyzed using Lyapunov stability theory and the robustness of the controller is studied using actual gait data collected from human subjects. Numerical simulations in the presence of noise, uncertainty in terrain, disturbance torques, and changes in gait are then used to demonstrate the tracking

Fig. 2. Link-segment diagram of the residual limb and the prosthetic foot.

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

performance and robustness of the control approach.

2. Dynamics of the prosthetic ankle joint In this section, the leg on the amputated side of a unilateral below-knee amputee is modeled by a link-segment diagram in the sagittal plane where most of the ankle joint movements occur during gait. This representation includes the biological knee and hip joints of the amputee and a computer controlled ankle joint on the prosthetic foot. Euler–Lagrange approach is used to derive the dynamics of this representation. By assuming the total human control of the biological joints, this section then concentrates on dynamics and control of the ankle joint. 2.1. Link-segment diagram

M (θ ) θ¨ + V ( θ , θ )̇ θ ̇ + G (θ ) + τd = τ ,

(1)

where 'θ' is the time-dependent joint variable (expressed in radians), ' τ ' is the torque generated by each joint (expressed in Newton-meters), and ' τd ' is the disturbance torque (expressed in Newton-meters). The joint variable θ (t ) = [ θa (t ) θk (t ) θh (t )]T ∈ R 3 is a vector comprising of the angular position of the prosthetic ankle joint (θa (t )), the biological knee joint (θk (t )), and the biological hip joint (θh (t )). Input torque to the model is τ = [ τa + τaG τk τh ]T ∈ R 3 with τa is the external torque generated by the actuator at the prosthetic ankle joint, τaG is the ground reaction torque (GRT) and caused by the interaction between the prosthetic foot and the ground during gait. τk and τh describe the internal torques generated by the biological knee and hip joints, respectively. The additional component τd = [ τad τkd τhd ]T ∈ R 3 represents the disturbance torque which results from the movement of Head–Arm–Trunk (HAT), i.e., the upper body, during gait. It is important to note that total human control is assumed at the biological knee joint and biological hip joint (Winter, 2009). Therefore, knee torque τk (t ) and hip torque τh (t ) are assumed to be generated by the amputee to compensate for the coupling effect from the ankle joint, effect of ground reaction, and disturbance torque from the HAT. Finally, nonlinear terms in (1) include the inertia matrix M (θ ), the vector of Coriolis and Centripetal forces V (θ , θ )̇ θ ,̇ and the vector representing gravitational forces G (θ ). Detailed dynamical equations and the parameters of the model are shown in Appendix A.1. 2.2. Ankle joint dynamics

3.1. Goals for the prosthetic ankle control As described in the Introduction, the prosthetic ankle is controlled by a torque generated by an external actuator and is expected to follow a displacement profile similar to that of a natural ankle (Fig. 1). The goals for the control system of the prosthetic ankle joint are listed below:


Recognize the type of gait and detect the gait events in real time using actual gait data measured from amputees,


Determine an ankle joint displacement profile corresponding to the selected gait of amputees,


Counter the effect of ground reaction τaG , and
Implement a control algorithm that can learn the dynamical interactions in (2) and generate a control torque τa that provides guaranteed tracking performance. Detailed implementation of the ankle prosthesis control system which achieves these goals will be described in Section 3.2–Section 3.5.

3.2. Gait-based displacement profiles In traditional control, the tracking error is computed as the difference between the ideal and actual displacement of the joint. In the case of prosthetic ankle, the ideal joint profile cannot be specified as the user's intent and the terrain are not known in advance. Therefore, as a first step, the gait of the user is recognized in real time and this information is used to specify the preliminary ankle joint displacement profile. Fig. 3 shows the block diagram of the controlled’‘residual limb– prosthetic foot’ system. During the gait, the Amputees' Intent Re-

Since total human control is assumed at the biological knee joint and the biological hip joint of the residual limb, this paper focuses on the dynamics and control of the ankle prosthesis. Dynamics of the prosthetic ankle joint can be extracted from (1) as follows:

UNKNOWN ANKLE DYNAMICS

Unknown dynamics of the prosthetic ankle joint (2) include the terms that depend only on the angular position of the ankle and its derivatives (Maa θ¨a + Vaa θȧ + Ga ) , the coupling between the ankle joint and the biological knee joint (Mak θ¨k + Vak θk̇ ), the coupling between the ankle joint and the biological hip joint (Mah θ¨h + Vah θḣ ). Maa , Vaa , Ga , Mak , Vak , Mah and Vah are nonlinear functions of the joint angles and are listed in Appendix A.1. The movement of the ankle is also affected by the ground reaction torque τaG and the HAT disturbance torque τad . Control torque τa is generated by a powered actuator to manipulate the movement of the prosthetic ankle joint. The control framework and problem formulation will be discussed in Section 3.

3. Control of the prosthetic ankle joint

Fig. 2 shows the link-segment representation of the leg on the amputated side of a unilateral below-knee amputee in the sagittal plane. The dynamics of the combined limb and prosthesis is derived using the Euler–Lagrange approach (Winter, 2009; Amirouche, 1992) and can be expressed as:

′ M Vaa θȧ + G Mak θ′k + Vak θk̇ + Mah θ′h + Vah θḣ aa θa + a +      independent reflected dynamicsof knee joint refected dynamicsof hip joint ankle dynamics 

3

+

τad =   disturbance from the HAT

cognition block (Fig. 3) recognizes the intent of an amputee using the actual gait data measured from the prosthetic socket and the Residual Limb (Mai & Commuri, 2011). Then, the corresponding gait-based kinematic references

τa + 

τaG .  

control torque

ground reaction torgue

(2)

4

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

Fig. 3. Block diagram of the residual limb – controlled prosthetic foot system.

torque τaG at the prosthetic ankle joint is defined as

T T T⎤ ⎡ θ rg = ⎣ θ gT θ ̇ g θ¨ g ⎦ r r r

θrg (t )

g τ˜aG = τaG − τaG .

T = ⎡⎣ θarg (t ) θkrg (t ) θhrg (t )⎤⎦

Details of the empirical model and the bounds on τ˜aG will be presented in Section 4.

T g ̇ g (t ) θhr ̇ g (t )⎤⎦ θṙ (t ) = ⎡⎣ θaṙg (t ) θkr T g θ¨r (t ) = ⎡⎣ θ¨arg (t ) θ¨krg (t ) θ¨hrg (t )⎤⎦

(3)

are specifically generated for the recognized gait. The superscript ( . ) g indicates that these profiles are generated based on the determination of amputee gait. Let θ r = [ θrT θṙT θ¨rT ]T denotes the ideal kinematic profiles which are not available to the computation of the control input τa . Then, the difference between the ideal kinematic references and the gait-based references is defined as T θ˜ r = ⎡⎣ θ˜rT θ˜ṙT θ˜¨rT ⎤⎦

θ˜r = θr − θ˜ṙ = θṙ − θ˜¨ = θ¨ − r

r

θrg g

θṙ

g θ¨r .

(5)

3.4. Artificial neural network -based approximation of the ankle dynamics In this section, an artificial neural network will be used to approximate the unknown nonlinear terms (in Eq. (2)) that represent the interaction between the Prosthetic Ankle Joint and the biological knee and hip joints of the Residual Limb (Fig. 3). Inputs to the ANN are obtained from the angular displacement/rate of the ankle joint (θa, θȧ ) and the gait-based kinematic references, θ rg , which are specifically generated for the user's gait. It is important to note that, the control structure in Fig. 3 performs tracking of the gaitbased reference θarg instead of the predetermined trajectory θar as in a traditional ANN-based control approach.

(4)

In Section 4 of this paper, an approach to recognize the amputee gait through measurement of contact forces from the prosthetic socket will be described in more detail. Generation of the gait-based references θ rg and magnitudes of the differences θ˜ r will also be discussed therein.

3.4.1. Error dynamics Defining the (pseudo) ankle joint tracking error as

ea = θarg − θa,

(6)

the tracking error system can be represented as

Maa r ̇ = − Vaa r + f + τad − τa − τaG, 3.3. Gait-based ground reaction torque Calculation of the ground reaction torque τaG is usually attempted using kinematic tracking systems and force plates in a laboratory setting (Winter, 2009; Kirtley, 2006) where the participants walk a short distance with a limited number of steps. Under normal gait conditions, direct measurement of the ground reaction force is not feasible for the purpose of controlling the prosthetic foot, therefore τaG is not available for the calculation of the control input τa . In the control scheme shown in Fig. 3, the g gait-based ground reaction torque τaG is specifically generated for the recognized gait using an empirical model describing the footg is then included into the ground contact. The gait-based torque τaG control computation to compensate for the actual τaG . The differg ence between the gait-based τaG and the actual ground reaction

(7)

where r = eȧ + λea is the filtered tracking error and λ > 0 is a constant. ' f ' in (7) represents the unknown terms in the dynamics and can be represented as g g f = ⎡⎣ Maa θ¨ar + λeȧ + Vaa θaṙ + λea + Ga ( θa ) ⎤⎦

(

)

(

)

+ ⎡⎣ Mak θ¨k + Vak θ k̇ ⎤⎦ + ⎡⎣ Mah θ¨h + Vah θ ḣ ⎤⎦.

(8)

It is important to note that ea in (6) defines the (pseudo) tracking error similar to the traditional neural network control approach (Lewis, Jagannathan, & Yesilderek, 1999). However, the actual tracking error of the prosthetic ankle joint is calculated as the difference between the ideal displacement profile θar (which is not available to the control computation) and the actual ankle angular position θa as

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

eaactual = θar − θa.

(9)

3.4.2. ANN-based approximation of the ankle dynamics The physical constraints on the joints imply that (θa, θk, θh ) and their derivatives are finite. Therefore, the function f in (8) is a real function and is bounded on a compact region in R . Since human gait comprises of multiple gait cycles, f is also a periodic function. Therefore, f can be approximated by an artificial neural network with one hidden layer and ideal target weights Wf ∈ R Nh× 1, Vf ∈ R Nx × Nh as follows:

f (x) =

WTf σ

(

) + ε,

VTf x

(10)

in which, x ∈ R Nx × 1 is the vector of the neural network inputs, ε is the bounded function approximation error, i.e., ‖ε‖ < εB , σ ( . ) is a sigmoidal activation function, and Nh is the number of nodes in the hidden layer. The neural network input x is selected as follows:

̇ g θ¨krg θ g θhr ̇g x = ⎡⎣ ea eȧ θarg θaṙg θ¨arg θkrg θkr hr

g T θ¨hr ⎤⎦ .

(11)

The ideal weights Wf , Vf are unknown but they can be ap^ ^ proximated by adjustable weights Wf ∈ R Nh× 1, Vf ∈ R Nx × Nh with the network weight errors

5

of Head–Arm–Trunk is bounded, i.e., ‖τad ‖ ≤ Bd . Assumption 3.5.4. Ideal neural network weights are bounded, i.e., ‖Zf ‖F W ≤ BZ Θ where ‖. ‖F represents the Frobenius norm and f Zf = [ ]. Θ Vf Theorem 3.5.5. : Given the control structure in Fig. 3, let the ankle torque be computed as:

^ g τa = f − τaG + KV r − v,

(16)

Let the neural network weights be updated according to

^ ^T ^ Wf = Pσ^r − Pσ^′Vf xr − kP r Wf

(

^ ^ Vf = Qx σ^′Wf r

T

)

^ − kQ r Vf ,

(17)

(18)

and the robustifying term be selected as v = − KZ (‖Z^f ‖F + BZ ) r, where KV , P, Q and k are design parameters. Then, the actual tracking error eaactual defined in (9) and the neural network weight ˜ f , V˜f are uniformly ultimately bounded. errors W 3.6. Stability analysis

^ ˜ f = Wf − W W f ^ V˜ f = Vf − Vf .

(12)

Then, the unknown function f is approximated by

^ g +v Maa r ̇ = − ( Vaa + KV ) r + f − f + τad − τaG + τaG

^ ^ T ⎛ ^T ⎞ f (x) = Wf σ ⎜ Vf x⎟, ⎝ ⎠

(13)

with the ANN approximation error given by

)

(14)

^T Using Taylor series expansion around (Vf x ) for a given x , (14) can be expressed as

⎤ ⎡ ⎛ ^T ⎞ ⎛ ^ T ⎞⎛ ^T ⎞ f˜ = WTf ⎢ σ ⎜ Vf x⎟ + σ ′ ⎜ Vf x⎟ ⎜ VTf x − Vf x⎟ + ( H . O. T ) ⎥ ⎠ ⎠⎝ ⎝ ⎠ ⎦ ⎣ ⎝

^T VTf x = V f x

(19)

˜ T σ^ + WT σ^′V˜ T x Maa r ̇ = − ( Vaa + KV ) r + W f f f + ⎡⎣ WTf ( H . O. T ) + ϵ + τad − τ˜aG ⎤⎦ + v T ⎞ ⎛ ^ T ^′ ˜ T ˜ T ⎜ σ^ − σ^′V^ x⎟ + W = − ( Vaa + KV ) r + W f f f σ Vf x ⎠ ⎝

˜ T σ^′VT x + […] + v. +W f f

^ T ⎛ ^T ⎞ ˜ T σ^ + WT σ^′V˜ T x + WT ( H . O. T ) + ε, − Wf σ ⎜ Vf x⎟ + ε = W f f f f ⎠ ⎝ ^T where σ^ = σ (Vf x ), σ^′ = σ ′(VTf x )

= − ( Vaa + KV ) r + f˜ + τad − τ˜aG + v. Substituting f˜ from (15), (19) can be written as

^ ^ T ⎛ ^T ⎞ f˜ = f − f = WTf σ VTf x − Wf σ ⎜ Vf x⎟ + ε ⎠ ⎝

(

3.6.1. Closed-loop error dynamics With the control signal (16), the closed loop error dynamics (7) can be expressed as

(20)

Defining the total uncertainty ' δ ' as

(15)

is the Jacobian matrix of the T

T ^ activation function with respect to its inputs, V˜ f x = VTf x − Vf x , and (H . O. T ) represents all high order terms in the Taylor series expansion of σ (VTf x ).

˜ T σ^′VT x + ⎡⎣ WT ( H . O. T ) + ϵ + τad − τ˜aG ⎤⎦, δ=W f f f

(21)

the closed-loop error dynamics can be expressed as T ⎞ ⎛ ^ T ^′ ˜ T ˜ T ⎜ σ^ − σ^′V^ x⎟ + W Maa r ̇ = − ( Vaa + KV ) r + W f f f σ V f x + δ + v. ⎠ ⎝

(22)

Lemma 3.6.1. : Bound on the neural network input x 3.5. Control algorithm

The neural network input x is bounded by

The following assumptions are made for the stability analysis of the control algorithm. Details and justification of these assumptions are provided in the Discussion.

‖x‖ ≤ C2 + C3 ‖r‖,

Assumption 3.5.1. The ideal kinematic references, gait-based kinematic references, and their differences are bounded, i.e., ‖ θ r ‖ ≤ Bθr , ‖ θ rg ‖ ≤ B θrg , and ‖ θ˜ r ‖ ≤ B θ˜r .

Proof. : Given in Appendix A.2.

Assumption 3.5.2. The difference between the modeled ground reaction torque and the actual ground reaction torque experienced at the prosthetic ankle joint is bounded, i.e., ‖τ˜aG ‖ ≤ BG . Assumption 3.5.3. The disturbance torque τad due to the motion

(23)

where r is the filtered tracking error and C2, C3 are positive constants.

Lemma 3.6.2. Bound on the high order terms in neural network approximation (H . O. T ) The high order terms in the neural network approximation are bounded by T T ‖H . O. T‖ ≤ 1 + C2 Bσ ‖Z˜ f ‖F + C3 Bσ ‖Z˜ f ‖F ‖r‖,

(24)

6

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

⎡W ˜f Θ⎤ ⎥, σ^′ ≤ Bσ , and where r is the filtered tracking error, Z˜f = ⎢ ⎢⎣ Θ V˜f ⎥⎦ C2, C3 are positive constants.

L ̇ = − rKV r + k‖r‖ tr

{ Z˜ ( Z − Z˜ ) } + r ( δ + v). T f

f

f

(32)

T Since tr {Z˜ f (Zf − Z˜f )} ≤ ‖Z˜f ‖F (BZ − ‖Z˜f ‖F ) and δ is bounded as in C (25), by selecting KZ > C6 , C7 = BZ + k5 , then

Proof. : Given in Appendix A.3. Lemma 3.6.3. : Bound on the uncertainty δ

2 ⎧ ⎫ ⎡ ⎪ ⎪ kC2 C ⎤ L ̇ ≤ − ‖r‖ ⎨ KV ‖r‖ + k ⎢ ‖Z˜f ‖F − 7 ⎥ − 7 − C4 ⎬ . ⎪ ⎪ ⎣ ⎦ 2 4 ⎩ ⎭

The uncertainty term δ is bounded by T

and (29) becomes,

T

‖δ‖ ≤ C4 + C5 BZ ‖Z˜ f ‖F + C6 BZ ‖Z˜ f ‖F ‖r‖,

(25)

(33)

(33) indicates that L ̇ will be negatively definite on the region

⎡W ˜f Θ⎤ ⎥, ‖Zf ‖F ≤ BZ , and where r is the filtered tracking error, Z˜f = ⎢ ⎢⎣ Θ V˜f ⎥⎦ C4, C5, C6 are positive constants.

Therefore, using LaSalle's (Slotine & Li, 1991) extension, the filtered tracking error r and the neural network weight error Z˜f will be

Proof. : Given in Appendix A.4.

uniformly ultimately bounded (UUB) when

Proof of Theorem 3.5.5. With the selection of the finite neural network input x in (11), the ideal neural network WTf σ (VTf x ) is defined in a compact set Sx = {x, ‖x‖ < Bx } and therefore provides a bounded approximation for the nonlinear function f . The proof of the theorem is attempted in two parts. First, the boundedness of the (pseudo) tracking error ea in (6) will be shown using a procedure inspired by the traditional neural network control approach (Lewis et al., 1999). In the second step, eaactual in (9) which describes the overall tracking error of the prosthetic ankle joint will be shown to be bounded. ^ ^ T ^T The neural network f (x ) = Wf σ (Vf x ) is defined for x on a compact set Sx = {x, ‖x‖ < Bx } to approximate the unknown nonlinear function f in (8). From the inequality (23), define the compact set Sr = {r , ‖r‖ < Br } in which Br = (Bx − C2 ) /C3, then the universal approximation property of the neural network holds for all r ∈ Sr . Now, consider the Lyapunov function

‖r‖ > br ≡

L=

1 1 rMaa r + tr 2 2

{ W˜ P W˜ } + 21 tr { V˜ Q V˜ }, T f

−1

T f

f

−1

f

(26)

˜ f , V˜f in (12), and P, Q in (17), (18), respectively. with W Taking derivatives of L with respect to time,

1 L ̇ = rMaa r ̇ + rṀ aa r + tr 2

{

˜ T P −1W ˜̇f W f

} + tr {

}

T V˜ f Q−1V˜ ḟ ,

where

KV ‖r‖ −

kC72 4

− C4 > 0 or

k [‖Z˜f ‖F −

C7 2 ] 2

C4 + kC72/4 , KV

−

kC72 4

− C4 > 0.

(34)

or

‖Z˜f ‖F >

C5 + 2

⎛ C5 ⎞2 C2 ⎜ ⎟ + . ⎝ 2⎠ k

(35)

Conditions (34)–(35) define the compact set outside which the derivative of the Lyapunov function L ̇ < 0. From (34), in order to satisfy br < Br , then the control gain KV has to be selected as

KV >

(

)

C3 C4 + kC72/4 C4 + kC72/4 . = Br Bx − C2

(36)

Finally, since r is bounded i.e., r ∈ L∞ then ea ∈ L∞ and eȧ ∈ L∞ (Dawson, Hu, & Burg, 1998), or the tracking error of prosthetic ankle joint displacement is bounded, i.e., ‖ea ‖ ≤ εa . Since the actual tracking error in (9) can be written as eaactual = (θar − θa ) = (θar − θarg ) + (θarg − θa ) then

‖eaactual ‖ = ‖ ( θar − θarg ) + ( θarg − θa ) ‖ ≤ ‖ ( θar − θarg ) ‖ + ‖ ( θarg − θa ) ‖ ≤ ‖θ˜ar ‖ + εa,

(27)

(37)

or the actual tracking error is bounded.

then substituting for Maa r ̇ from (22),

1 r ( Maa − 2Vaa ) r 2 ⎤ ⎡ T⎛ T ⎞ ^ T ^′ ˜ T ˜ ⎜ σ^ − σ^′V^ x⎟ + W + r⎢W f f f σ V f x + δ + v⎥ ⎠ ⎝ ⎦ ⎣

4. Discussion

L = − rKV r +

+ tr

{ W˜ P W˜ } + tr { V˜ Q V˜ }. T f

−1

T f

f

−1

f

4.1. Structure of the prosthetic ankle control system

(28)

The term r (Ṁ aa − 2Vaa ) r vanishes because Maa is a constant and the choice of the reference frame makes Vaa = 0. Then (28) can be simplified as,

⎧ T⎛ T ⎞⎫ ˜ ⎜ P −1W ˜ f + σ^r − σ^′V^ xr ⎟ ⎬ L = − rKV r + tr ⎨ W f f ⎠⎭ ⎝ ⎩ ⎧ T⎛ ^ T ⎞⎫ + tr ⎨ V˜ f ⎜ Q−1V˜ f + xrWf σ^′⎟ ⎬ + r ( δ + v) ⎠⎭ ⎩ ⎝

(29)

Since the ideal neural network weights Wf , Vf are constant, T ^ ^ ^ ˜ f = Wf − W ^ ^′ ^ W f = − Wf = − Pσ r + Pσ Vf xr + kP r Wf

(

^ ^ ^ V˜ f = Vf − Vf = − Vf = − Qx σ^′Wf r

T

)

^ + kQ r Vf ,

(30)

(31)

The controller proposed in this paper has a hierarchical structure. At the lowest level, the controller is responsible for generating a control torque τa which drives the ankle joint along a gait-based reference trajectory θarg . This control action uses an ANN to approximate the unknown ankle dynamics. Terrain interaction is also accounted for through the use of the gait-based g . As a result, commonly ignored terms in ground reaction torque τaG the dynamics of ankle prosthesis are accounted for thereby improving the tracking performance. In the higher level of the controller, the Amputees' Intent Recognition block (Fig. 3) processes the gait data measured in real time from amputees to detect gait events and recognize the type of gait. Appropriate gait-based kinematic references θ rg and ground reaction torque approximation g are then generated based on the amputees' gait. τaG The control approach in Fig. 3 can also be viewed as a multiloop structure. The inner most nonlinear loop comprises of an ANN that is responsible for approximating the unknown ankle dynamics. The outer tracking loop reduces the tracking error eaactual

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

7

Fig. 4. (a) Gait monitoring device; (b) Ultra-thin force sensor; (c) Prosthetic socket; (d) Unilateral below-knee amputee during gait.

and finally, the monitoring loop processes the actual gait data to recognize the amputee's intent. Compared to PD control of prosthetic ankle, this structure requires smaller control gains and is robust to external disturbances and changes in gait.

4.2. Recognition of gait and detection of gait events Recognition of gait and detection of gait events can be done using actual gait data such as residuum socket interface force (RSI) measured from the prosthetic socket of below-knee amputees. The RSI forces that the residual limb applies on the prosthetic socket during gait of an amputee can be captured using ultra-thin FlexiForces sensors (Tekscan, 2015) and a gait monitoring device as seen in Fig. 4a. These sensors (Fig. 4b) can be embedded inside the socket (Fig. 4c) to provide non-intrusive force measurement in real time and during daily activities outside the laboratory setup (Fig. 4d). It has been shown that the recorded RSI forces relate to the gait (Mai et al., 2012) and can be used to distinguish between different gait types (normal self-paced walk, brisk walk, ascending/descending stairs, walking on a ramp, etc.), and to detect temporal gait parameters (heel strike, mid stance, toe off, stance time, etc.) for each detected gait (Mai & Commuri, 2011). The relationship between the RSI forces and the gait events that are detected using foot switches is shown in Fig. 5.

Fig. 5. Normalized socket contact force and actual gait events detected using foot switches.

4.3. Generation of gait-based kinematic references The gait-based kinematic references θ rg satisfying the bound condition ‖ θ rg ‖ ≤ B θrg can be generated for each gait type. Since the

8

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

human gait and joint movements are cyclic in nature, the gaitbased kinematic references can be generated using Fourier series as follows (Peasgood, Kubica, & McPhee, 2007) 5

θrg (t ) = ag0 +

∑

k

k

ag g cos ( kg wg t ) + bg g sin ( kg wg t ), (38)

kg = 1 k

k

where ' t ' denotes the time. The parameters ag0, ag g , bg g , wg can be obtained using least square method to fit the model (38) to the actual displacement profile of the joints for a specific gait. Since the sine and cosine functions in (38) are bounded, the gait-based kinematic patterns θ rg are also bounded. The ideal kinematic references θ r are also bounded, i.e., ‖ θ r ‖ ≤ Bθr due to the cyclic nature of the gait and the human active control of the residual limb to follow specific joint patterns which help minimize the metabolic energy consumption during gait (Ackermann & van den Bogert, 2010). Finally, since θ r , θ rg are both bounded and θ˜ r = θ r − θ rg , the kinematic reference error θ˜ r is also bounded. Fig. 6. Approximated ground reaction force and loading profile from actual foot pressure sensors.

4.4. Gait-based ground reaction torque The actual ground reaction torque τaG (Fig. 2) is approximated by g τaG (t ) = daZ (t ) FX (t ) + daX (t ) FZ (t ),

(39)

where t is the gait time, FX [N], FZ [N] are the horizontal and vertical ground reaction forces, and daX [m], daZ [m] are the horizontal and vertical distances between ankle joint and the center of pressure (contact point) of the foot during gait. These quantities depend on the height and weight of the individual, the type of gait, and the nature of the terrain. During gait on level ground, ground reaction forces FX , FZ can be modeled by a combination of a spring and a position-dependent damper as follows (Peasgood et al., 2007; Millard, McPhee, & Kubica, 2008; Klute, Berge, & Segal, 2004): e¯ ̇ FZ = k¯ ( z¯PEN ) + c¯ d z¯PEN

(40)

FX = μ¯ FZ sgn ( x¯ Ṗ ) ,

(41)

̇ [m /s] are the penetration and penetrain which z¯PEN [m] , z¯PEN tion rate of the foot into the ground, k¯ [N /m] , e¯ are spring coefficient and spring exponent, c¯d [N /(m /s )] is the damping coefficient, μ¯ is the friction coefficient, x¯Ṗ [m /s] is the horizontal velocity of the contact point with respect to the ground, and sgn ( . ) is the sign function. These parameters can be obtained using experiments similar to the works in (Peasgood et al., 2007) and (Klute et al., 2004). It is noted that different body weights and walking speeds result in variations in the penetration and penetration rate into the ground. Deviations between the estimation of these quantities and their actual values will contribute to the error in approximation of the ground reaction torque τ˜aG . The distances daX , daZ depend on the gait-based references θ rg , temporal gait parameters, and the contact profiles assumed for the prosthetic foot. g in (39) is bounded because FX , FZ in It can be seen that τaG (40)–(41) are modeled using passive mechanical components and finite penetration, and daX , daZ are functions of the bounded kinematic references. The actual ground reaction torque θ rg is also bounded due to its cyclic nature and the finite human weight. g is bounded. Therefore, the difference τ˜aG = τaG − τaG The dependence of the vertical ground reaction force generated by the empirical model on the gait cycle (40) is confirmed by the actual foot loading profile measured during the gait on level ground of an able bodied individual. The normalized foot loading profile in Fig. 6 is calculated as the summation of the forces

measured by force sensors (FlexiForces A201, (Tekscan, 2015)) placed under the heel, outer ball of the foot, and toe of the foot. The typical double peak shape of the modeled ground reaction force also agrees with ground reaction force captured using force plates and reported in the literature (Winter, 2009; Kirtley, 2006). 4.5. Bound of the disturbance torque τad It has been shown that the upper body movement and arm swing during gait are passive and mainly powered by the movement of the lower body (Pontzer, Holloway, Raichlen, & Lieberman, 2009). These movements are the result of an individual to further stabilize the gait and reduce metabolic energy consumption during gait (Collins, Adamczyk, & Kuo, 2009). Calculation of the Head– Arm–Trunk torque from the quantitative gait data in (Winter, 2009) also confirms that this is a bounded quantity. In the ankle dynamics (2), the external disturbance τad represents the HAT torque, therefore is bounded.

5. Experimental gait study and numerical simulation 5.1. Experimental gait study The recognition of gait type and detection of gait events are based on the gait data collected from a group of 9 unilateral below-knee amputees (1.83 70.05 (m) height, 94.5 715.5 (kg) weight, 40 712 years of age, and at least 6 months following the amputation procedure) at the University of Oklahoma Health Science Center. All subjects were capable of walking at different speeds without using any assistive devices except their own prostheses. The study included three gait types during which each subject walked on the level ground for 2 min with three self-selected speeds corresponding to ‘slow’, ‘normal’, and ‘fast’ walking. The gait monitoring device (Fig. 4a) used force sensors placed inside the prosthetic socket (Fig. 4b) to measure the residuum socket interface forces in real time. These forces were used to distinguish the gait type and detect gait events. Based on the recognized gait type and detected gait events, the updated gait-based kinematic references θ rg and approximated g were then specifically generated for ground reaction torque τaG the recognized gait type and were activated at the moment when

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

9

Table 1 Control and neural network parameters. Parameters

Value

Control gain KV Design parameter λ Number of nodes in input layer Nx Number of nodes in hidden layer Nh Number of nodes in output layer

2 5 11 22 1

ground contact (Heel Strike) was observed. The proposed control approach used these gait-based quantities to calculate an appropriate control torque for the recognized gait. 5.2. Simulation setup The performance of the controller designed in the previous section was studied during normal gait of an unilateral belowknee amputee. Performance of the proposed control approach was evaluated through simulation in Matlab/Simulink environment. The simulation model included the link-segment diagram dynamics (1) with both prosthetic ankle joint and biological joints of the residual limb. A fully connected artificial neural network with random initial weights was trained online during gait with inputs from the gait-based kinematic references θ rg and the iṅ stantaneous pseudo tracking errors (ea, ea ) to approximate the unknown nonlinear ankle dynamics f in (8). The step size was 0.005 s and the neural network was iteratively trained using tuning rules (17), (18). Gait-based ground reaction torque was generated using (39) from the reaction forces (40), (41). The ankle torque of the ANNþPD approach was then calculated by (16). A classical PD-only control g τaPD = KVPD λ PDea + eȧ − τaG

(

)

was also simulated for comparison. Table 1 lists the parameters used in the control algorithms. The control approaches were simulated for different simulation scenarios (Table 2) under the effect of noisy measurement, θa, noisy = (1 + nθa ) θa ; noisy control, τa, noisy = (1 + n τa ) τa ; ground reaction torque compensation error, τ˜aG ; disturbance torque, τad ; error in detection of gait event ΔtHS , which was defined as the temporal difference between the occurrence of the actual heel strike and the heel strike detected by the RSI force (Fig. 5); and changes in the type of gait (changes in walking speed). During walking on level ground, three gait types with different walking speeds were considered. The simulation test criteria included 5 gait cycles of which the

Table 2 Simulation scenarios. Scenario

n θa

S1 – Ideal S2 – Noisy S3 – GRT error S4 – Disturbance torque S5 – HS detection error S6 – Gait change

nτa

τ˜aG

τad

ΔtHS

0 0 0 ±[5 ÷ 10]% ±[5 ÷ 10]% 0 g 0 0 ±20%τaG

0 0 0

0 0 0

0

0

0

þ20 N m

0

0

0

0

0

±20ms

0

0

0

Varied with speed

0

Fig. 7. Ankle displacement under ideal conditions (S1).

first 3 gait cycles were simulated under the ideal condition S1 (see Table 2). Simulation scenarios S2–S6 then continued at the beginning of the 4th gait cycle (heel strike). In the noisy condition S2, magnitudes of the noises nθa and n τa was between 5% and 10% of the measured ankle displacement and generated ankle torque τa , respectively. In scenario S3, magnitude of the actual GRT experienced by the prosthetic joint was allowed to vary up to 20% of the GRT estimated using (39). A positive torque τad = 20Nm was added in scenario S4 to represents the disturbance created by HAT during gait. In scenario S5, uncertainty in the detection of heel strike was considered by introducing a detection error of ± 20 milliseconds. In scenario S6, the walking speed increased from ‘slow’ to ‘normal’ and ‘normal’ to ‘fast’, or decreased from ‘fast’ to ‘normal’ and from ‘normal’ to ‘slow’. All those changes occurred at the beginning of the 4th gait cycle. Gait-based kinematic references θ rg for these walking speeds were adapted from (Winter, 1991) and approximated by Fourier series with adjustable parameters as in (38). 5.3. Simulation results Initially, PD-only control with KVPD = 2 and λ PD = 5 did not yield stable performance. Parameters for the PD-only control were then increased to KVPD = 6 and λ PD = 20 so acceptable tracking performance was obtained. These parameters were then used for simulation of the PD-only control algorithm in other scenarios in Table 2. The ankle displacements resulted from the ANNþ PD and PDonly control approaches in the ideal condition S1 are shown in Fig. 7. The vertical dashed lines indicate the beginning of each gait cycle (heel strike). Without the presence of measurement/actuator noises, disturbances, and gait changes, both control approaches yielded satisfactory tracking performance. Tracking performance of the prosthetic ankle joint for scenarios S2–S5 are shown in Fig. 8. The longer vertical dashed lines indicate the beginning of the 4th gait cycle when the changes in gait condition occur. Shorter vertical dashed lines indicate the beginning of the 5th gait cycle after the neural network-based controller has learned the new dynamics and adapted to the changes. It can be seen that the ANN þPD control outperformed the PD-only control in ensuring robust tracking performance in the presence of unmodeled dynamics and external disturbances.

10

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

Fig. 8. Ankle displacement under the effects of noises (S2), ground reaction torque error (S3), disturbance torque (S4), and temporal error in detection of heel strike (S5).

Finally, when the walking speed changed (scenario S6), the artificial neural network was able to approximate the updated nonlinear ankle dynamics and adapt to the new gait. Fig. 9 shows the tracking performance of both control algorithms. It is important to note that under all non-ideal conditions (scenarios S2–S6), the ANN-based control approach needed at most one gait cycle to learn the new dynamics before providing good tracking. 6. Conclusions This paper presents a hierarchical control approach that incorporates the adaptation of the prosthetic ankle joint movement to

the amputees' intent during gait. Reference displacement for the ankle joint was specifically generated using real-time measurements collected during user gait. An artificial neural network was used to learn the nonlinear ankle dynamics, and the interaction between the foot and the walking terrain was compensated by an empirical model of the ground reaction force. Theoretical analysis was carried out to establish the bound of the actual tracking error and to ensure that the interconnection weights of the artificial neural network remain bounded. The control torque calculated in this method was applied to the prosthetic ankle joint to track the reference ankle displacement during gait.

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

11

Fig. 9. Ankle displacement with the changes in walking speed (S6).

Simulations carried out using data collected from amputees during gait showed that good tracking of the commanded ankle displacement was achieved in the presence of measurement/actuator noises, uncertainties in terrain interaction, disturbance torques, variations in gait parameters, and changes in walking speed. These results also showed that the controller proposed in this paper can improve the tracking performance obtained using traditional PD controllers. While the mechanical design and fabrication of a prosthetic ankle was not addressed in this paper, the design of an intelligent prosthetic ankle and the validation of its performance in clinical settings are topics of ongoing research.

Acknowledgment The authors would like to thank their colleagues at the School of Electrical and Computer Engineering, and the Health Sciences Center at the University of Oklahoma, and especially the subjects for their contributions in the experimental gait study. We also thank the Oklahoma Center for the Advancement of Science and Technology (Project# HR11-097, IRB#15948) for funding the research on the gait study.

Appendix

⎡ Maa Mak Mah ⎤ ⎢ ⎥ M = ⎢ Mka Mkk Mkh ⎥ ⎢ ⎥ ⎣ Mha Mhk Mhh ⎦ ⎡ mf r 2f + I f ⎢ ⎢ m r L ⎢ f f s cos ( θa − ⎢ =⎢ ⎢ ⎢ mf r f L t cos ( θa − ⎢ ⎢⎣

θh )

(

(

)

)

⎡ Vaa Vak Vah ⎤ ⎥ ⎢ V = ⎢ Vka Vkk Vkh ⎥ ⎢⎣ V V V ⎥⎦ ha hk hh ⎡ 0 mf r f L s θ k sin ⎢ ⎢ ( θa − θk ) ⎢ ⎢ − m r L θ sin θ − θ 0 ( ) f f s a a k ⎢ = ⎢⎢ ⎢ ⎢ − mf r f L t θa sin ( θa − θ h ) − ( mf L s ⎢ L t + ms rs L t ) θ k ⎢ ⎢ sin ( θ k − θ h ) ⎣

⎤ ⎥ ( θa − θh ) ⎥⎥ ( mf Ls Lt + ms rs Lt ) ⎥⎥ ⎥ θ h sin ( θ k − θ h ) ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎦ mf r f L t θ h sin

⎤ ⎡ Ga ⎤ ⎡ mf r f g sin θa ⎥ ⎢ ⎥ ⎢ ⎥ G = ⎢ Gk ⎥ = ⎢ ( mf L s + ms rs ) g sin θ k ⎢⎣ Gh ⎥⎦ ⎢⎢ m L + m L + m r g sin θ ⎥⎥ s t t t) h⎦ ⎣( f t 2

Appendix A.1 Link segment and ankle joint dynamics.

θk )

mf r f L s cos ( θa − θ k ) mf r f L t cos ( θa − θ h ) ⎤⎥ ⎥ mf L s2 + ms r s2 + Is mf L s L t + ms rs L t ⎥ ⎥ cos ( θ k − θ h ) ⎥ ⎥ ⎥ mf L s L t + ms rs L t mf L t2 + ms L t2 ⎥ ⎥⎦ + mt rt2 + It cos ( θ k − θ h )

2

2

I f = mf ( αf L f ) ; Is = ms ( αs L s ) ; It = mt ( αt L t ) ;

12

A. Mai, S. Commuri / Control Engineering Practice 49 (2016) 1–13

mf ,

Lf ,

I f − mass, length, and inertia around the center of

mass of the foot ms , L s,

Is − mass, length, and inertia around the center of It − mass, length, and inertia around the center of

mass of the thigh (connect knee and hip) αf , s, t =

radius of gyration ; segment length

Lf 2

Ls 2

σ^ ′ VTf

F

F

x + WTf

( C2 + C 3

r

)

F

H . O . T + ϵB + Bd + B G

⎛ ⎞ T T r ⎟ + ϵB + Bd + B G + BZ ⎜ 1 + C2 Bσ Z˜ f + C 3 Bσ Z˜ f ⎝ ⎠ F F

T ≤ ϵB + Bd + B G + BZ + 2C2 Bσ BZ Z˜ f

αf = 0.475 (foot); αs = 0.302 (shank); αt

F

T + C 6 BZ Z˜ f

F

T + 2C 3 Bσ BZ Z˜ f

r F

r F

(46)

where

C4 = εB + Bd + BG + BZ C5 = 2C2 Bσ C6 = 2C3 Bσ

− distance from the ankle joint to the center of the

mass of the foot rs =

F

T ≤ C 4 + C5 BZ Z˜ f

= 0.323 (thigh) rf =

˜ T ≤ W f

T ≤ Bσ BZ Z˜ f

mass of the shank (connect ankle and knee) mt , L t ,

˜ T σ^ ′VT x + WT ( H . O . T ) + ϵ + τ − τ˜ δ = W aG f ad f f

(47)

− distance from the ankle/knee joint to the center of

the mass of the shank rt =

Lt 2

− distance from the knee/hip joint to the center of the

mass of the thigh

Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.conengprac.2016.01.004.

References Appendix A.2. Proof of Lemma 3.6.1 The physical/design constraints of the prosthetic ankle joint guarantee that the actual ankle angular position θa and velocity θȧ are always bounded. In addition, the gait-based kinematic referg g ences (θrg , θṙ , θ¨r ) are also bounded since they are approximated by finite Fourier series (Discussion). The tracking error (θarg − θa ) is therefore finite and since r = eȧ + λea , the filtered tracking error r is also finite, i.e., r ∈ L∞. By replacing eȧ in (11) by r − λea , one can show that

‖x‖ ≤ C2 + C3 ‖r‖

(42)

where C2 > 0 and C3 > 0 depend on the initial tracking error ea (0), the bound B θrg and design parameter λ . With the input vector x in a compact set x ∈ Sx ⊂ R Nx × 1, the approximation (10) holds.

Appendix A.3. Proof of Lemma 3.6.2 From Taylor series expansion of then

σ (VTf x ),

T (H . O. T ) = σ − σ^ − σ^′V˜ f x ,

T T H . O. T = σ − σ^ − σ^′V˜ f x ≤ σ − σ^ + σ^′ V˜ f T ≤ σ − σ^ + σ^′ V˜ f

F

( C2 + C3

r

)

x F

(43)

Since the sigmoid activation functions and their derivatives are bounded, i.e., ‖σ‖ < 1, ‖σ^‖ < 1, and σ^′ ≤ Bσ , then T T ‖H . O. T‖ ≤ 1 + C2 Bσ ‖V˜ f ‖F + C3 Bσ ‖V˜ f ‖F ‖r‖

(44)

⎡W ˜f Θ⎤ ˜ f ‖F ≤ ‖Z˜f ‖F , ‖V˜f ‖F ≤ ‖Z˜f ‖F , then ⎥, and ‖W With Z˜f = ⎢ ⎢⎣ Θ V˜f ⎥⎦ T T ‖H . O. T‖ ≤ 1 + C2 Bσ ‖Z˜ f ‖F + C3 Bσ ‖Z˜ f ‖F ‖r‖ (45)

Appendix A.4. Proof of Lemma 3.6.3 Since ‖ε‖ ≤ εB , ‖τad ‖ ≤ Bd , ‖τ˜aG ‖ ≤ BG ,

Ackermann, M., & van den Bogert, A. J. (2010). Optimality principles for modelbased prediction of human gait. Journal of Biomechanics, 43, 1055–1060. Amirouche, F. M. L. (1992). Computational Methods in Multibody Dynamics. Englewood Cliffs, NJ: Prentice Hall. Anderson, F. C., & Pandy, M. G. (2001). Dynamic optimization of human walking. Journal of Biomechanical Engineering, 123, 381–390. Au, S. K., Weber, J., & Herr, H. (2009). Powered ankle-foot prosthesis improves walking metabolic economy. IEEE Transactions on Robotics, 25, 51–66. Au, S., Berniker, M., & Herr, H. (2008). Powered ankle-foot prosthesis to assist levelground and stair-descent gaits. Neural Networks, 21, 654–666. Bateni, H., & Olney, S. J. (2002). Kinematic and kinetic variations of below-knee amputee gait. Journal of Prosthetics and Orthotics, 14, 2–13. BIOM (2015). BIOM – Personal Bionics. Available: 〈http://www.biom.com/〉. Brodzka, W. K., Thornhill, H., Zarapkar, S., Malloy, J., & Weiss, L. (1990). Long-term function of persons with atherosclerotic bilateral below-knee amputation living in the inner city. Archives of Physical Medicine and Rehabilitation, 71, 895–900. Collins, S. H., Adamczyk, P. G., & Kuo, A. D. (2009). Dynamic arm swinging in human walking. Proceedings of the Royal Society B: Biological Sciences, 276, 3679–3688. Dawson, D. M., Hu, J., & Burg, T. C. (1998). Nonlinear Control of Electric Machinery. London, UK: Taylor & Francis. Ferris, A. E., Aldridge, J. M., Rábago, C. A., & Wilken, J. M. (2012). Evaluation of a powered ankle-foot prosthetic system during walking. Archives of Physical Medicine and Rehabilitation, 93, 1911–1918. Fey, N. P., Silverman, A. K., & Neptune, R. R. (2010). The influence of increasing steady-state walking speed on muscle activity in below-knee amputees. Journals of Electromyography and Kinesiology, 20, 155–161. Gailey, R., Allen, K., Castles, J., Kucharik, J., & Roeder, M. (2008). Review of secondary physical conditions associated with lower-limb amputation and long-term prosthesis use. Journal of Rehabilitation Research and Development, 45, 15–29. Grabowski, A. M., & D’Andrea, S. (2013). Effects of a powered ankle-foot prosthesis on kinetic loading of the unaffected leg during level-ground walking. Journal of NeuroEngineering and Rehabilitation, 10, 1–12. Hitt, J., Sugar, T., Holgate, M., Bellman, R., & Hollander, K. (2009). Robotic transtibial prosthesis with biomechanical energy regeneration. International Journal of Industrial Robot, 36, 441–447. Jiménez-Fabián, R., & Verlinden, O. (2012). Review of control algorithms for robotic ankle systems in lower-limb orthoses, prostheses, and exoskeletons. Medical Engineering & Physics, 34, 397–408. Jin, D., Zhang, R., Zhang, J., Wang, R., & Gruver, W. A. (2000). An Intelligent Aboveknee Prosthesis with EMG-based Terrain Identification (pp. 1859–1864). Kirtley, C. (2006). Clinical Gait Analysis – Theory and Practice. London, UK: Churchill Livingstone. Klute, G. K., Berge, J. S., & Segal, A. D. (2004). Heel-region properties of prosthetic feet and shoes. Journal of Rehabilitation Research and Development, 41, 535–546. Kovac, I., Medved, V., & Ostojic, L. (2010). Spatial, temporal and kinematic characteristics of traumatic transtibial amputees' gait. Collegium Antropologicum, 34, 205–213. Lewis, F. L., Jagannathan, S., & Yesilderek, A. (1999). Neural Network Control of Robot Manipulators and Nonlinear Systems. London, UK: Taylor & Francis. Mai, A., & Commuri, S. (2011). Gait identification for an intelligent prosthetic foot. In Proceedings of IEEE International Symposium on Intelligent Control (ISIC) (pp. 1341–1346). Denver, CO, USA. Mai, A., Commuri, S., Dionne, C. P., Day, J., Ertl, W. J. J., & Regens, J. L. (2012). Effect of prosthetic foot on residuum-socket interface pressure and gait characteristics

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in an otherwise healthy man with transtibial osteomyoplastic amputation. Journal of Prosthetics and Orthotics, 24, 211–220. Markowitz, J., Krishnaswamy, P., Eilenberg, M. F., Endo, K., Barnhart, C., & Herr, H. (2011). Speed adaptation in a powered transtibial prosthesis controlled with a neuromuscular model. Philosophical Transactions of the Royal Society B – Biological Sciences, 366, 1621–1631. Millard, M., McPhee, J., & Kubica, E. (2008). Multi-step forward dynamic gait simulation. Computational Methods in Applied Sciences, 12, 25–43. Össur (2015). Proprio Foot. Available: 〈http://www.ossur.com/?PageID¼ 12704〉. Peasgood, M., Kubica, E., & McPhee, J. (2007). Stabilization of a dynamic walking gait simulation. ASME Journal of Computational and Nonlinear Dynamics, 2, 65–72. Perry, J. (1992). Gait Analysis: Normal and Pathological Function (1st ed.). Thorofare, NJ: SLACK Incorporated. Pohjolainen, T., Alaranta, H., & Karkkainen, M. (1990). Prosthetic use and functional and social outcome following major lower limb amputation. Prosthetics and Orthotics International, 14, 75–79. Pontzer, H., Holloway, J. H., Raichlen, D. A., & Lieberman, D. E. (2009). Control and function of arm swing in human walking and running. Journal of Experimental Biology, 212, 523–534. Slotine, J. J. E., & Li, W. (1991). Applied Nonlinear Control. Englewood Cliffs, N.J.: Prentice-Hall, Incorporated. Tekscan (2015). FlexiForce Sensors – User Manual. Available: 〈http://www.tekscan. com/pdf/A201-force-sensor.pdf〉. Torburn, L., Powers, C. M., Guiterrez, R., & Perry, J. (1995). Energy expenditure during ambulation in dysvascular and traumatic below-knee amputees: a comparison of five prosthetic feet. Journal of Rehabilitation Research and Development, 32, 111–119. Torrealba, R. R., Perez-D’Arpino, C., Cappelletto, J., Fermın-Leon, L., Fernandez-Lopez, G., & Grieco, J. C. (2010). Through the development of a biomechatronic knee prosthesis for transfemoral amputees: mechanical design and manufacture, human gait characterization, intelligent control strategies and tests. In

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Proceedings of IEEE International Conference on Robotics and Automation (pp. 2934–2939). Anchorage, Alaska, USA. Uustal, H., & Baerga, E. (2004). Chapter 6: prosthetics and orthotics In: S. Cuccurullo (Ed.), Physical Medicine and Rehabilitation Board Review. New York: Demos Medical Publishing. Varol H. A., Sup, F., & Goldfarb, M. (2008). Real-time gait model intent recognition of a powered knee and ankle prosthesis for standing and walking. In: Proceedings of the 2nd Biennial IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics (pp. 66–72). Scottsdale, AZ, USA. Versluys, R., Beyl, P., Damme, M. V., Desomer, A., Ham, R. V., & Lefeber, D. (2009). Prosthetic feet: state-of-the-art review and the important of mimicking human ankle-foot biomechanics. Disability and Rehabilitation: Assistive Technology, 4, 65–75. Versluys, R., Desomer, A., Lenaerts, G., Damme, M. V., Beyl, P., Perre, G. V. D., Peeraer, L., & Lefeber, D. (2008). A pneumatically powered below-knee prosthesis: design specifications and first experiments with an amputee. In Proceedings of the 2nd Biennial IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechanics (pp. 372–377). Scottsdale, AZ, USA. Walker, C. R., Ingram, R., Hullin, M., & McCreath, S. W. (1994). Lower limb amputation following injury: a survey of long-term functional outcome. Injury, 25, 387–392. Winter, D. A. (2009). Biomechanics and motor control of human movement (4th ed.). Hoboken, N.J.: Wiley. Winter, D. A. (1991). The Biomechanics and Motor Control of Human Gait: Normal, Elderly and Pathological (2nd ed.). Waterloo, Ontario, Canada: University of Waterloo Press. Wolf, S. I., Alimusaj, M., Fradet, L., Stegel, J., & Braatz, F. (2009). Pressure characteristic at the stump/socket interface in transtibial amputees using an adaptive prosthetic foot. Clinical Biomechanics, 24, 860–865. Ziegler-Graham, K., MacKenzie, E. J., Ephraim, P. L., Travison, T. G., & Brookmeyer, R. (2008). Estimating the prevalence of limb loss in the United States: 2005 to 2050. Archives of Physical Medicine and Rehabilitation, 89, 422–429.