�

Abstract— This paper presents the analytical model of a DC-motor actuated active-knee transfemoral prosthesis operating within the mechanically dissipative but electrically regenerative region of the knee actuator’s performance space. A switching-based control scheme is developed that enables damping control within the limb’s regenerative region of operation. This control approach allows the limb to realize bounded mechanical dissipation using a strictly passive means, while at the same time generating electrical power to augment the limb’s power requirements in active modes of operation. Experimental characterizations of the performance and regenerative efficiency of the switching-based controller indicate that the passive switching controller can provide damping modulation, but at the expense of decreased efficiencies of regeneration as compared to the theoretical projections of the actuator model.

I. INTRODUCTION

RTIFCIAL limb technology has seen a number of advancements in recent years that portend the

commercial appearance of powered artificial limbs with improved functionality and human-like mechanical anthropometry. Improvements in electromechanical power and actuation have enabled the design of powered artificial limbs that can deliver human-scale power outputs within the volume and weight envelopes imposed on artificial prosthetic limbs. With respect to lower-extremity prosthetic devices, a number of battery-powered motor-actuated devices including actively-powered transtibial prostheses and transfemoral prostheses have demonstrated the feasibility of actively powered electromechanical artificial limbs [1]-[5]. Despite the promise offered by powered prosthetic legs, the long-term operation of these devices is limited by current state-of-the-art battery technology. While current electromechanical leg prototypes provide near-human outputs, they do so only for a fraction of the desired day’s worth of operation. In the absence of significantly improved source energy densities, improvements to the underlying energetic characteristics of electrically powered prosthetic legs are needed.

Legged locomotion entails a number of functions that require net dissipation of mechanical power, especially at the knee. Knee biomechanics for activities such as stair descent are almost entirely dissipative, with peak power

Manuscript received February 14, 2010. M. R. Tucker, formerly of Clarkson University, is with Raytheon

Company, Woburn, MA 01801 (email: tucker.michaelr@gmail.com). K. B. Fite is with Clarkson University, Potsdam, NY 13676 USA

(phone: 315-268-3809; fax: 315-268-6695; e-mail: kfite@clarkson.edu).

dissipation of as much as 400 W during the latter portion of controlled lowering [6]. Even for level walking, knee biomechanics over a full stride exhibit net dissipative characteristics [7]. Unlike the human musculoskeletal system which requires net metabolic energy consumption during contraction (whether concentric, eccentric, isometric, or isotonic), an actively powered artificial limb has the potential to leverage energy storage mechanisms for improved overall energetic performance. Some of the powered artificial legs incorporate passive mechanical energy storage components (i.e., springs) to augment the power requirements of the actuation system, thereby attaining improvements to the limb’s energetic performance for particular locomotive functions [1]-[2], [5]. In addition to incorporating auxiliary energy storage components within the artificial limb design, the presence of the electromechanical actuator in powered prostheses offers the fundamental physical capability bilateral power transformation. The ability of an electromechanical actuator to operate as an electrical power generator offers the potential to generate significant electrical power during mechanical dissipation and thereby extend the duration of self-contained battery-powered operation.

The idea of using an electromechanical actuator to passively dissipate mechanical power is not a new concept. The approach essentially involves using the actuator’s inherent coil resistance to dissipate generated electrical power as heat, thereby providing mechanical damping without any expenditure of electrical power. This capability has been exploited in the design of semi-active vehicle suspensions that provide variable damping characteristics [8]-[9]. Building upon the inherent passive dissipation characteristics of electromechanical actuators, other work has focused on approaches for storing the electrical power generated during passive dissipation of applied mechanical loads. The electrical power regenerated during the passive mode of operation can then be used to drive the actuator in an active power delivery mode or to power other auxiliary components within a particular system. Applications of note include regenerative braking for electrically powered automobiles [10] and active control of vehicle suspensions and building vibrations [11]-[12]. Applicable to the general problem of energy regeneration, researchers have developed a generalized actuator model to characterize an actuator’s potential for regenerating dissipated mechanical power [13]. Using a quasi-static model of an electromagnetic actuator, results based on optimizing the regeneration capability of

Mechanical Damping with Electrical Regeneration for a Powered Transfemoral Prosthesis

Michael R. Tucker and Kevin B. Fite, Member, IEEE

A

2010 IEEE/ASME International Conference onAdvanced Intelligent MechatronicsMontréal, Canada, July 6-9, 2010

978-1-4244-8030-2/10/$26.00 ©2010 IEEE 13

the system suggest that improved regeneration can be obtained by using a brake in combination with the regenerative actuator. Such an approach is effective when the desired dissipation lies outside of the actuator’s regeneration mode of operation. While no quantitative or experimental characterizations are presented, the modeling approach provides a general framework from which to investigate the regeneration capacity of a powered prosthetic leg during legged locomotion.

While the more noteworthy benefits of actively-powered artificial legs relative to commercially-available passive limbs lie in the ability to output net-positive mechanical power, the active limb must still operate in the electrically regenerative region of operation for mechanically dissipative gait functions. The ability to recover and reuse some of the generated electrical power may prove advantageous with respect to the energetic performance of the actively-powered artificial leg. The specific objectives of the work presented in this paper are characterization of the electrical regeneration region of a transfemoral prosthetic leg’s operation envelope and the development of a control methodology that provides for strictly passive limb operation with concomitant electrical energy recovery within this region of operation. Note that this work represents a small component of a larger effort in the development of a self-contained transfemoral prosthesis to be integrated with amputee subjects using a myoelectric control interface.

II. LUMPED PARAMETER MODEL OF THE TRANSFEMORALPROSTHESIS

The performance envelope of a motor-actuated system can be partitioned into two basic modes of operation, termed the active and passive modes. In the active mode, the actuator consumes electrical power to impose mechanical power at the joint of the artificial limb. In contrast, operation in the passive mode entails net dissipation of mechanical power at the joint with subsequent electrical power generation at the motor terminals. In order to analytically characterize these generative and absorptive modes in the context of lower-extremity prosthetic limb control, a lumped parameter model is derived for the transfemoral prosthesis prototype pictured in Fig. 1. The actuator of the prototype limb consists of a brushed DC motor (Maxon model RE40) coupled to a ball-screw transmission (Nook Industries model ECS-10020-RA). The resulting linear actuator is integrated into the limb with a three-bar linkage transmission which converts the translational mechanics of the actuator to rotational mechanics about the knee joint of the prosthesis prototype. Torque at the knee joint is measured via a uniaxial load cell (OMEGA model LCFD-5500) located in line with the motor. The revolute knee joint contains an integrated joint motion sensor (ALPS model RDC503) located on the lateral side of the knee joint. The knee actuator clamps to a

standard prosthesis shank (Otto Bock model 2R58) whose axial length can be varied such that the overall leg fits a given amputee’s anatomy. The limb uses pyramid connections at both the proximal and distal ends (i.e., the top of the knee and ankle) to accommodate standard prosthesis componentry for the socket interface and the foot. The foot pictured in Fig. 1 is a traditional solid-ankle cushioned heel (SACH) foot.

The dynamic model of the limb prototype was developed using bond graph theory, and the resulting bond graph is shown in Fig. 2. The model includes two exogenous inputs: the electrical voltage (V) imposed at the motor terminals and the torque (�ext) imposed at the knee. The DC motor model consists of the coil resistance (Rm), coil inductance (Lm),rotor inertia (Jm), and viscous damping (bm), coupled via the gyrator-like electromagnetic transduction constant (kt). The ball screw transmission is modeled by the ballnut mass (ml),viscous damping (bl), and lead of the screw transmission (kl). The bond graph model is completed with the configuration-dependent Jacobian for the three-bar linkage transmission (k�), rotational inertia of the limb about the knee joint (Jk), and viscous damping associated with the revolute knee joint. The Jacobian for the three-bar linkage transmission is derived using trigonometric functions for the linkage geometry shown in Fig. 3. In this figure, L1 and L2 represent links of constant length, and � is the enclosed angle between the two links. Using the law of cosines, a relationship between the effective length of the ball-screw actuator (x)and the enclosed angle (�) is given by:

���� cos2122

21

2 LL2LLx (1)

Fig. 1. DC-motor actuated transfemoral prosthesis prototype

Fig. 2. Bond graph for the knee joint of the prosthesis prototype.

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where the enclosed angle, as a function of knee angle, is given by:

ko ����� (2) where �k is the angle of the knee measured from full knee extension and �o is a constant representing the enclosed angle when �k = 0. Using the concept of virtual work as defined by:

kdFdx ���� (3) where F is the extensive force imposed by the ball screw actuator and � is the torque applied about the knee (with sign-convention such that positive � is in the direction of positive �k. The expression for actuator force as a function of knee angle is then given by:

�� �kF (4) where

�

����

���� sin

cos

21

2122

21k

LLLL2LL

dxd

k (5)

is obtained by applying the chain rule to eqs. (1) and (2). The bond graph of Fig. 2 has four energy-storage

elements (i.e., four generalized inertias), two of which are independent storage elements. As such, the overall model of the limb is second-order. To facilitate the investigation of the limb’s regenerative capability, the mechanical components of the model are collapsed into the simplified bond graph of Fig. 4. The mechanical model is represented by an equivalent inertia (Jeq) and viscous damping (beq)about the knee, actuated by the DC-motor through the equivalent gyrator transformation (keq), each of which are defined by:

m

2

ll

2

lkeq J

kk1m

k1JJ

���

��

���

���

�

(6)

m

2

ll

2

lkeq b

kk1b

k1bb

���

��

���

���

�

(7)

�

�kk

kk

l

teq (8)

The coupled pair of first-order equations of motion for the knee joint of the transfemoral prosthesis is then given by:

VkiRdtdiL keqmm ���� (9)

exteqkeqk

eq ikbdt

dJ �����

� (10)

These equations specify the open-loop electromechanical dynamics of the limb’s knee joint and provide a model for which the passive damping and electrical regeneration characteristics of the limb can be investigated.

III. PASSIVE DAMPING AND THE REGENERATION MANIFOLD

The model developed in Section II enables characterization of the region over which the limb can dissipate mechanical power while at the same time generate electrical power. Due to inherent losses in the system, an electromechanical actuator is unable to generate net electrical power over the entire region for which mechanical power is dissipated. In essence, the range of damping or dissipation that the limb can provide while generating electrical power is both upper-and lower-bounded due to fundamental physical constraints. The lower bound is determined by the nominal friction, modeled as viscous damping in the actuator and about the knee center of rotation, which acts in the case of zero current flow through the motor coil (i.e., when the motor terminal leads are open).

As seen from the bond graph of Fig. 4, the absence of current flow decouples the electrical and mechanical dynamics of the artificial knee, and the equation of motion for the knee reduces to:

extkeqkeq bJ ����� (11) The resulting mechanical damping depends solely on the equivalent physical damping defined by eq. (7). In this case, the motor terminal voltage would simply equal the induced back-emf voltage as a function of actuator speed. To achieve damping magnitudes below that of the nominal physical system, net-positive electrical power will have to be sourced to the DC motor. The motor would thus have to be actively driven in order to realize damping characteristics

Fig. 3. DC-motor actuated transfemoral prosthesis prototype.

Fig. 4. Reduced bond graph for the artificial knee joint.

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below the physical open-loop dissipation. The upper bound for mechanical dissipation with

electrical generation occurs when the voltage across the motor leads is zero (i.e., when the motor leads are shorted). At steady-state, the current generated in the coil is strictly a function of the induced back-emf voltage. For the voltage potential at the motor leads to be zero, the voltage across the motor’s lumped resistance must equal the negative of the back-emf voltage. The induced current flows in the opposite direction to the back-emf voltage, thereby providing a torque in opposition to the externally imposed mechanical power input at the knee joint. The magnitude of current induced is given by:

km

eq

Rk

i ��� (12)

Substituting eq. (12) into eq. (10), the equation of motion for the knee is:

extkm

2eq

eqk

eq Rk

bdt

dJ �����

�)( (13)

The exhibited mechanical damping includes a component associated with the coil resistance of the motor that scales with the square of the equivalent electromagnetic coupling coefficient. The total mechanical damping at this bound is then given by:

m

2eq

eqtot Rk

bb �� (14)

where btot represents the total damping about the knee at the upper bound for electrical generation. Realization of damping magnitudes in excess of that defined by eq. (14) requires net expenditures of electrical power at the motor terminals.

Visualization of the region of knee operation bounded by the upper and lower mechanical damping limts, beq and btot,is complicated by the fact that the three-bar linkage Jacobian of eqs. (2) and (5) depends on �k. Note that this angular dependency appears in both the upper and lower bounds for mechanical damping. The mechanically dissipative domain within which electrical power can be generated is thus

characterized by a volume bounded by surfaces that define the maximum and minimum resistive knee torques as a function of knee angle and velocity (�k and �k). This domain, termed the regeneration manifold, is shown in Fig. 5 and defined using the parameters of Table I. The parameters were experimentally validated with tests that characterized the bounds of the passive model with the foreleg swinging under the influence of a known load in a gravitational field. For purposes of visualization, the regeneration manifold shown in Fig. 5 only captures half of the artificial knee’s total region of regenerative operation. The other half of the regeneration manifold occurs for negative knee velocities for which the resulting resistive torque is also negative, but the shape of the bounds is otherwise unchanged. As indicated by the figure, maximum damping occurs near the midpoint of the knee’s range of motion and corresponds to the angle at which the magnitude of the normal vector between the applied ball screw force and the knee’s center of rotation is maximum.

TABLE IPARAMETERS FOR THE ARTIFICIAL KNEE MODEL

Parameter Value Unit L1 0.303 m L2 0.064 m �o 2.21 rad kt 6.03e-2 N·m/A kl 3.2e-4 m/rad Lm 3.29e-4 H Rm 1.6 �Jm 1.46e-5 kg·m2

ml 1.04 kg Jk 0.15 kg·m2

bm 6.2e-5 N·m·s/rad bl 0 kg/s bk 0 kg/s

IV. PWM-BASED DAMPING CONTROL WITHIN THE REGENERATION MANIFOLD

A switching-based control scheme offers a straightforward means for exploiting the mechanical dissipation of the artificial knee within the regeneration manifold. To modulate mechanical damping within the regeneration manifold, the system is switched between the upper and lower bounds of the manifold using a pulse-width modulated (PWM) command, thereby providing damping that is a function of the duty cycle of the PWM signal. An H-bridge operated in the braking mode enables pulse width modulation of the motor’s regenerative damping behavior. Figure 6 shows a schematic of the circuit used to experimentally characterize the motor’s behavior when operated in the PWM braking mode. The circuit consists of two n-channel MOSFETs which constitute half of a standard H-bridge, the drains of which are connected to each of the motor terminals. An ultracapacitor connected as shown provides a passive energy storage component for assessment of the regenerative characteristics of the PWM braking

0

0.5

1

1.5

0

2

4

6

0

100

200

300

400

500

600

angle (rad)speed (rad/s)

torq

ue (

N*m

)

Fig. 5. Bounds for knee regeneration manifold.

16

control scheme. A PWM command is input to each MOSFET, thereby switching the motor leads between shorted (i.e., the upper bound of the regeneration manifold) and open to the ultracapacitor. Note that in this configuration, the motor only charges the capacitor when the motor terminal voltage exceeds the capacitor voltage. Otherwise, the motor leads are effectively open, and the mechanical damping lies at the lower bound of the regeneration manifold. This scheme leverages the braking characteristics of the motor’s electrical dynamics to modulate damping at the knee, while at the same time leveraging the transient electrical dynamics during switching for electrical energy recovery.

Initial experiments were conducted to characterize the mechanical damping exhibited by the knee as a function of the duty cycle and switching frequency of the PWM braking command. The mechanical damping was measured as a function of duty cycle and switching frequency as the artificial limb was externally loaded by physically swinging the limb back and forth through its range of motion. Using the measured angle, velocity, and torque of the knee, the realized mechanical damping characteristics were computed for switching frequencies from 100 Hz to 25.6 kHz and duty cycles from 0% to 100% at 5% intervals. To remove the angular dependency of the knee’s damping behavior from the analysis, the damping characteristics of the knee were analyzed at the output of the ball screw actuator. Figure 7 shows the measured linear damping at the ball nut as a function of switching frequency and duty cycle. As seen in the figure, at low switching frequencies, the exhibited damping of the actuator varies linearly with duty cycle.

However, as the switching frequency increases, a nonlinear relationship between duty cycle and damping emerges. This can be attributed, in part, to the electrical dynamics of the DC motor. At low switching frequencies, the electrical time constant of the motor (�elec = 0.3 ms) is fast enough so as not to significantly affect resulting mechanical damping. As such, the relationship between duty cycle and mechanical damping is linear. However, as the switching frequency increases, the electrical dynamics become more significant to the motor’s braking characteristics, as evidenced by the change in shape of the surface of Fig. 7.

V. REGENERATION EFFICIENCY

While the ability to passively dissipate mechanical power has merit in and of itself (with respect to stability and the capability of passive operation without the need for external power), another important characteristic relates to the amount of electrical power that is generated by this control configuration. Theoretical projections for the quasi-static regeneration efficiency are obtained by computing the electrical and mechanical power at constant current and angular velocity. Mechanical power is defined by:

kextmechP ��� (14) where the external torque, obtained from eqs. (10) and (11) assuming constant current angular velocity, is given by:

VRk

Rk

bm

eqk

m

2eq

eqext ��

���

���� (15)

The electrical power at the motor leads is given by: ViPelec � (16)

where

� �keqm

kVR1i ��� (17)

The efficiency of electrical power generation within the regeneration manifold is now defined as the ratio of electrical to mechanical power as given by:

keq2k

2eq

eqm

keq2

VkR

kbR

VkV

���

���

��

���� (18)

where � is the regeneration efficiency. Within the regeneration manifold, the motor terminal voltage varies between the back-emf voltage (keq�k) and zero. The terminal voltage can thus be written as:

keqk1V ���� )( (19)

where � varies between zero and unity, and captures the range of mechanical damping between the lower and upper bounds of the regeneration manifold. Substituting eq. (19) into eq. (18), the resulting efficiency as a function of � is:

� �2eqeqm

2eq

kbRk1��

����� (20)

Q1 Q2

MAXON Re40

1 2

PWM Command

Fig. 6. PWM braking schematic.

0

50

100 010

2030

0

0.5

1

1.5

2

2.5

3

x 104

Switching Freq (kHz)Duty Cycle (%)

Dam

ping

(kg

/sec

)

Fig. 7. Surface plot of linear damping at the ball screw actuator versus switching frequency and duty cycle.

17

Figure 8 shows the theoretical regeneration efficiency as a function of �. Note that because all of the passive damping is lumped into the motor’s viscous damping (bm), the regeneration efficiency is not dependent upon the artificial knee’s Jacobian. The efficiency of regeneration peaks at 76% for � = 0.12 and converges to zero at the manifold bounds. For comparison, Fig. 9 shows the experimentally measured regeneration efficiency as characterized by the electrical power delivered to the ultracapacitor under the passive damping control scheme. This data was measured for a PWM switching frequency of 6.4kHz, and the maximum efficiency is only 18%. The large discrepancy between the theoretical and experimental efficiencies can be attributed in part to the approach by which damping control is achieved. The theoretical prediction assumes that the source voltage is held constant, in which case the current flowing through the motor during damping is the same as that flowing into the power source. Under the passive damping control approach, the current responsible for the knee’s exhibited mechanical dissipation is not fully captured by the power source. Instead, the experimental results indicate that much of the dissipated mechanical power is lost during the portion of the duty cycle for which the motor leads are shorted. Only a fraction of this power is available for regeneration when the MOSFETs close. Rather than regenerate electrical power while simultaneously providing mechanical damping, the passive damping control approach switches between passive dissipation and electrical regeneration but is unable to provide both concurrently. While the artificial knee possesses the potential to regenerate significant electrical power within its regeneration manifold, the PWM-based passive damping controller fails as a practical means for realizing the available potential of the actuator.

VI. CONCLUSION

This paper presented an analytical model describing the quasi-steady regeneration manifold within which a prototype active-knee transfemoral prosthesis is able to dissipate mechanical power while simultaneously generating electrical power. A switching-based braking control scheme was implemented to provide strictly passive damping modulation within the bounds of the regeneration manifold. Future work in this regard will focus on the development of a modified control architecture that explicitly considers and better leverages the regeneration potential of the limb’s passive domain of operation.

REFERENCES

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[2] S. K. Au, J. Weber, and H. Herr, “Powered Ankle-Foot Prosthesis Improves Walking Metabolic Economy,” IEEE Transactions on Robotics, vol. 25, no. 1, 51-66, 2009.

[3] D. B. Popovic, R. Tomovic, L. Schwirtlich, and D. Tepavac, “Control aspects of an active above-knee prosthesis,” International Journal of Man-Machine Studies, vol. 35, 751-767, 1991.

[4] K. Fite, J. Mitchell, F. Sup, and M. Goldfarb, “Design and control of an electrically powered knee prosthesis,” Proceedings of the 10th IEEE International Conference on Rehabilitation Robotics, pp. 902-905, 2007.

[5] F. Sup, H. A. Varol, J. Mitchell, T. J. Withrow, and M. Goldfarb, “Preliminary evaluations of a self-contained anthropomorphic transfemoral prosthesis,” IEEE/ASME Transactions on Mechatronics,vol. 14, no. 6, pp. 667-676, 2009.

[6] B. J. McFadyen and D. A. Winter, “An integrated biomechanical analysis of normal stair ascent and descent,” Journal of Biomechanics,vol. 21, no. 9, 733-744, 1988.

[7] D. A. Winter, Biomechanics and motor control of human movement,3rd ed. Hoboken, NJ: John Wiley & Sons, Inc., 2005.

[8] D. Karnopp, “Permanent magnet linear motors used as variable mechanical dampers for vehicle suspensions,” Vehicle System Dynamics, vol. 18, 187-200, 1989.

[9] D. Karnopp, “Active and Semi-Active Vibration Isolation,” 50thAnniversary of the ASME Design Engineering Division Special Combined Issue of the Journal of Mechanical Design and the Journalof Vibration and Acoustics, vol. 117, 177-185, 1995.

[10] P. Heister, S. Lawson, D. H. Sheffield, and D. Nelson, “Design and Development Process for the Equinox REV LSE E85 Hybrid Electric Vehicle,” SAE Paper 2006-01-0514, 2006.

[11] S. Kim and Y. Okada, “Variable resistance type energy regenerative damper using pulse width modulated step-up chopper,” ASME Journal of Vibration and Acoustics, vol. 124, no. 1, 110-115, 2002.

[12] J. T. Scruggs, A. A. Taflanidis, and W. D. Iwan, W. D., “Non-linear stochastic controllers for semiactive and regenerative systems with guaranteed quadratic performance bounds—Parts 1 and 2,” StructuralControl and Health Monitoring, vol. 14, 1101-1137, 2007.

[13] B. Seth and W. C. Flowers, “Generalized Actuator Concept for the Study of the Efficiency of Energetic Systems,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 112, no. 2, 233-238, 1990.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

12

14

16

18

effic

ienc

y (%

)

gamma

Fig. 9. Experimentally measured regeneration efficiency.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

effic

ienc

y (%

)

gamma

Fig. 8. Theoretical regeneration efficiency.

18