IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 1237 Gait Simulation via a 6-DOF Parallel Robot With Iterative Learning Control Patrick M. Aubin, Matthew S. Cowley, and William R. Ledoux* Abstract—We have developed a robotic gait simulator (RGS) by leveraging a 6-degree of freedom parallel robot, with the goal of overcoming three significant challenges of gait simulation, including: 1) operating at near physiologically correct velocities; 2) inputting full scale ground reaction forces; and 3) simulating motion in all three planes (sagittal, coronal and transverse). The robot will eventually be employed with cadaveric specimens, but as a means of exploring the capability of the system, we have first used it with a prosthetic foot. Gait data were recorded from one transtibial amputee using a motion analysis system and force plate. Using the same prosthetic foot as the subject, the RGS accurately reproduced the recorded kinematics and kinetics and the appropriate vertical ground reaction force was realized with a proportional iterative learning controller. After six gait iterations the controller reduced the root mean square (RMS) error between the simulated and in situ vertical ground reaction force to 35 N during a 1.5 s simulation of the stance phase of gait with a prosthetic foot. This paper addresses the design, methodology and validation of the novel RGS. Index Terms—Gait simulation, iterative learning control, kinematics, kinetics, prosthetics. Fig. 1. The R-2000 Rotopod, with the (A) motors, (B) support frame, (C) prosthetic foot mounted to the frame, (D) force plate, and (E) mobile platform. The inset shows a drawing of the R-2000 removed from the surrounding steel frame. I. INTRODUCTION Dynamic in vitro gait simulators have been useful tools for biomechanical researchers studying the foot and ankle. In contrast to modeling with living subjects, in vitro models often provide direct access to the variables of interest. Measurement techniques too invasive for living subjects, such as using bone pins to quantify bony motion or strain gages to measure bone or fascia strain, are often employed. Beyond that of static models, dynamic simulators can provide additional insight into the functional role of the foot and ankle during locomotion. Several research teams have created dynamic in vitro foot and ankle models that recreate the kinematics and ground reaction forces (GRFs) of the stance phase of in vivo gait [1]–[4]. While greatly advancing the state of the art, these gait simulators all suffer from one or more of the following potential limitations: scaled gait simulation speeds (2–60 s) [1]–[4], simplified rigid body motion of the tibia [1]–[4] or scaled GRFs for cadaveric specimens (40–50%) [2]–[4]. Our research objective is to design and implement a novel in vitro robotic gait simulator (RGS) capable of simulating the kinematics and kinetics of the foot and ankle during the stance phase of gait. As an initial step, and for the scope of Manuscript received November 13, 2006. This work was supported in part by the Office of Research and Development, Rehabilitation Research and Development Service, Department of Veterans Affairs, under Grant numbers A2661C and A3923R. Asterisk indicates corresponding author. P. M. Aubin is with the VA RR&D Center of Excellence for Limb Loss Prevention and Prosthetic Engineering, VA Puget Sound Health Care System, Seattle, WA 98108 USA, and with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). M. S. Cowley is with the VA RR&D Center of Excellence for Limb Loss Prevention and Prosthetic Engineering, VA Puget Sound Health Care System, Seattle, WA 98108 USA (e—mail: [email protected]). *W. R. Ledoux is with the VA RR&D Center of Excellence for Limb Loss Prevention and Prosthetic Engineering, VA Puget Sound Health Care System, 1660 South Columbian Way, MS 151, Seattle, WA 98108 USA, and the Department of Mechanical Engineering and the Department of Orthopaedics and Sports Medicine, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TBME.2007.908072 this paper, we have focused on the simulation of transtibial amputee gait. This serves two purposes: first, it provides a simple test case in which questions regarding our methodologies and hardware can be investigated and answered, and second, it produces a system that can potentially be employed in the future to study and develop prosthetic feet, although this is not our intended primary goal. The conclusions drawn from the prosthetic gait simulations will be integrated into future cadaveric gait simulations. II. MATERIALS AND METHODS A. Living Subject Gait Data Collection One gait trial was collected from a healthy 59–year-old male transtibial amputee. The motion analysis system was a 12-camera Vicon system (Vicon; Lake Forest, CA) collecting at 250 Hz. The force plate (Bertec Corporation; Columbus, Ohio) recorded the GRF at 1500 Hz. A certified prosthetist replaced the subject’s prosthetic foot with a replacement foot (FS 1000 freedom, Freedom Innovations, Inc.; Irvine CA, i.e., the same model that the subject normally wore); this foot was used for the remainder of both the subject and robotic tests. A tibia coordinate system (TCS) was constructed with the x-axis pointing anteriorly, the y-axis pointing superiorly and z-axis pointing medially (for a left leg). During a gait trial, the motion of the TCS was recorded with respect to the lab coordinate system (LCS), an inert reference frame located on the ground next to the force plate. B. Robotic Gait Simulator The RGS consists of a force plate (Kistler Instrument Corporation; Amherst, NY) mounted to a 6-degree of freedom parallel robot named the R-2000 (Parallel Robotics Systems; Hampton NH), which is surrounded by a steel mounting frame (Fig. 1). The R-2000 design is similar to a classic Stewart platform, with a base, six legs and a mobile platform. Unlike a normal Stewart platform the R-2000’s legs are of 0018-9294/$25.00 © 2008 IEEE 1238 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 fixed length, and the mobile platform is positioned by moving the base of the legs around a circular steel ring. We described the pose of the mobile platform with six parameters, (x y z roll pitch yaw). These parameters state the translation and Cardan angle rotation of the tool frame (TF) with respect to the base frame (BF). The BF is a world reference frame fixed relative to both the laboratory and base of the robot. The TF is a body fixed coordinate system attached to the top center of the mobile force plate. The RGS uses relative motion to simulate the kinematics and kinetics of a transtibial amputee gait. During a simulation the pylon was held rigidly in place while the “ground” (a vertical force plate mounted to the robotic mobile platform) was moved by the R-2000 in order to create the physiologically correct relative pylon-ground motion. To simulate gait with the R-2000, the trajectory of the LCS (ground) with respect to the TCS (pylon) was solved for in terms of a trans~TCS LCS (n), lation vector from the TCS origin to the LCS origin, p and the triplet of ZXY Cardan angles, f (n); '(n); (n)g, where n denotes the sample number. The prosthetic foot was mounted relative to the R-2000 such that the TCS is identical to the BF. With this configuration the discrete time trajectory given by the set fp~TCS LCS (n); (n); '(n); (n)g will correctly position and rotate the TF of the mobile platform with respect to the BF in the same way the LCS translates and rotates with respect to the TCS during the in situ stance phase of gait. In order for the simulated vertical GRF to accurately match the in situ vertical GRF the TCS would have to be perfectly aligned to the R-2000’s BF. Prosthetic foot mounting inconsistencies and recorded living subject gait data positional noise contribute to simulated vertical GRF errors. Rather than manually fine tuning the position of the prosthetic foot, the RGS uses iterative learning control (ILC) [5] to adjust the kinematics of the R-2000 between gait iterations to achieve the desired GRF. Real time force feedback of the R-2000 is not feasible with the robot’s positional eight-move buffer system. During each gait simulation the R-2000’s motion was controlled via a closed loop PID position feedback controller. After a gait simulation the recorded vertical GRF was automatically analyzed and the motion of the R-2000 was adjusted with the purpose of altering the vertical GRF during the stance phase of the simulated gait. These new motion commands were saved and used in the next gait iteration. The only trajectory parameters that were changed between iterations by the ILC algorithm were p ~TCS LCS (n) (i.e., the x, y, and z translations of the TF), while (n); '(n); (n) remained constant. An increase in the vertical GRF at time step n was accomplished by changing the location of the TF origin at time step n. Incremental motion of the TF origin between iterations was prescribed only in the direction normal to the force plate surface. This normal direction was unique for each time step n. This technique decoupled changes to the vertical GRF from changes to the shear GRF. To prove stability and iteration domain tracking of the target vertical GRF during a simulation, we modeled the vertical component of the force developed between the force plate and the prosthetic foot by a simple time-varying linear spring 1 FGRE j = Kreal uj (2) where Kreal is a diagonal matrix defined as kreal (1) Kreal = 0 .. . 0 0 kreal (2) ... 0 ... 0 .. . 0 .. . ... (3) kreal (i) where i is the number of time steps in the gait simulation. The iterative control law is then written in the iteration domain as C. Iterative Learning Control FGRF (n) = kreal (n) 1 [ 0 where p ~BF TF (n) is a translation vector from the BF origin to the TF origin with respect to the BF. We define, p ~0 (n) by creating a special trajectory fp ~0 (n); (n); '(n); (n)g such that at every instant n the force plate is just touching the plantar surface of the prosthetic foot but not exerting any force. This trajectory uses the rotation angles ( (n); '(n); (n)) recorded from the living subject. These rotation angles also define the rotation matrix RTF BF (n), (i.e., RTF BF (n) = T T RBF TF (n) = (Rz ( )Rx (')Ry ( )) ). The spring constant of the prosthetic foot at time step n is given as kreal (n). The vertical GRF due to the compressed prosthetic foot at discrete time step n is FGRF (n). The error between the desired and actual vertical GRF at time step n is defined as e(n) = FGRF (n)target 0 FGRF (n)actual . Let u(n) = [ 0 1 0 ] 1 RTF BF (n) (p ~BF TF (n) 0 p~0 (n)) and further define, uj , FGRF j and ej as column vectors giving the complete history of the input u(n), output FGRF (n) and error e(n) for iteration j . This allows us to write the output vertical GRF at iteration j as 0] 1 RTF BF (n) 1 (p~BF TF (n) 0 p~0 (n)) (1) uj = uj 01 + Lej 01 (4) where L is the learning gain matrix. After manipulating (2) and (4) the iteration domain error in the vertical GRF is given as ej = (I 0 Kreal L)j e0 : (5) The requirement for iteration domain stability and vertical GRF tracking is based on the eigenvalues of [I 0 Kreal L], namely ji [I 0 Kreal L]j < 1: (6) Assuming the learning gain matrix L is diagonal reduces the stability criterion to j1 0 kreal (n)Lnn j < 1 for n = 1 . . . i: (7) An optimal control gain such as, Lnn = 1=kreal (n) was not attempted because we do not want the determination of kreal (n) to be a prerequisite for gait simulation. Rather a learning gain of Lnn 0:07 for all n was shown to be effective. D. Inertial Compensation During a simulation the force plate is accelerating and decelerating its mass in order to reproduce the in situ ground kinematics. Thus the output measured by the force plate is actually the sum of both the force from the prosthetic foot and forces resulting from the acceleration and deceleration of the force plate mass. Mathematically, the force plate output (Fmeasured j ) can be written as Fmeasured j = FGRF j + maj : (8) IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 Fig. 2. Comparison of the in situ vertical GRF to the simulated vertical GRF for a 1.5 s simulation. The GRF is always less than body weight due to abnormal center of mass acceleration caused by the prosthetic limb. Where FGRF j is a column vector containing the entire vertical GRF for iteration j ; m is the mass of the force plate; and aj is a column vector containing the y-axis component of the acceleration of the TF with respect to the BF as expressed in the TF coordinate system for iteration j . In order to track the target vertical GRF and perform the simulations at 1.5 s our approach was to first perform slow speed simulations at 6.0 s until good vertical GRF tracking with the ILC was observed (i.e., after N iterations). Then the kinematics were held constant and, with the ILC turned off, the simulation was sped up to 1.5 s and Fmeasured N +1 was obtained. After removing the prosthetic foot the same N +1 kinematics and kinetics were performed for iteration (N + 2) and the inertial force (maN +2 ) was measured repeatedly nine times and averaged. Because the target (N + 1)th kinematics are the same as the target (N + 2)th kinematics we can assume that, maN +1 maN +2 . Finally the vertical GRF for the (N + 1)th iteration simulation was determined by F GRF N +1 = Fmeasured N +1 0 maN +1 Fmeasured N +1 0 maN +2 : (9) The vertical GRF for the N th + 1 iteration simulation was then compared to the in situ vertical GRF and an RMS error calculated. III. RESULTS A. Living Subject Gait Data The living amputee GRF curve with three peaks all below body weight is somewhat unusual as compared to non-amputee gait (Fig. 2). Tracking this arbitrary shape demonstrates the RGS’s ability to simulate patient specific gait data. The sagittal, frontal and transverse angular rotations that describe the TCS with respect to the LCS were determined (Fig. 3). B. Simulation Results The living subject gait data (Fig. 3) were used as the inputs to the RGS. Due to motor velocity constraints the fastest possible simulation with our device in its current configuration was 1.5 s, rather than 0.75 s. During the six iterations of learning the RMS error was reduced from 78.73 N to 9.42 N (Fig. 4). The exponential decay of the error was expected and consistent with the derivation for the iteration domain 1239 Fig. 3. Rotation of the pylon during the stance phase of in situ gait. Positive sagittal rotation is defined as the proximal end of the tibia moving posteriorly. Positive frontal rotation is defined as the proximal end of the tibia moving medially. Positive transverse rotation is defined as the forefoot moving laterally (external rotation). Fig. 4. RMS error history for the iterative learning control (ILC). convergence of tracking, ej = (I 0 Kreal L)j e0 . Once the RMS error between the simulated and in situ vertical GRF was minimized, the kinematics were held constant and the simulation speed was increased to 1.5 s for iteration seven; the force plate output was again measured and the RMS was recalculated as 39.2 N (Fig. 2). This increase in RMS error from 9.42 to 39.2 N was due to increased inertial forces. After measuring the force plate output for iteration 7, the prosthetic foot was removed and nine inertia only force curves were recorded. The mean inertia only force curve was calculated (Fig. 5). The inter-iteration standard deviation of the inertia only force curves was between 1.0 N and 15.0 N, with a mean of 3.0 N. To estimate the vertical GRF (i.e., the force “felt” by the prosthetic without the inertial forces) the mean inertial force was subtracted from the force plate output as shown in (9). The RMS error between the estimated vertical GRF and the in situ vertical GRF was 35.0 N (Fig. 2). IV. DISCUSSION In order to compare and contrast different gait simulators it is useful to have a set of performance metrics. The performance characteristics of interest are: simulation DOF, velocity, inter-simulation kinematics adjustability and fidelity of the GRF. By fidelity of the simulated GRF, we mean whether or not the GRF is scaled and how well the simulated GRF tracks normative data. 1240 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 55, NO. 3, MARCH 2008 While the performance of the RGS is promising, we must acknowledge in our comparison that cadaveric gait simulation is far more complex then prosthetic gait simulation. Prosthetic gait simulation has a simple and repeatable system under load while cadaveric gait simulation has a complex and variable system that also requires muscle tendon actuation. V. CONCLUSION Fig. 5. Inertial force for a 1.5 s simulation. The zoomed in view provided as an inset shows the repeatability of the nine inertial force curves. Currently most gait simulators allow for two tibia translations and one sagittal plane tibia rotation [1], [3], [4]. Hurschler et al.’s simulator also has a total of 3-DOF, i.e., one translation, superior and inferior and two tibia rotations, sagittal and transverse [2]. In comparison, the RGS simulates three tibia translations and three tibia rotations. The working velocity of some gait simulators are much slower than are physiologically correct [1], [2], [4] making them quasi dynamic. Nester et al.’s gait simulator is the fastest, simulating the stance phase of gait in 2 s [3]. The RGS operates currently at 1.5 s, slower than what is physiologically correct, but an order of magnitude faster than most other simulators. Inter-simulation kinematics adjustability is important if one wants to simulate a variety of gait patterns. In order to do so, the gait simulator must have the ability to prescribe different tibia kinematics between simulation trials. Currently only the RGS and Hurschler et al.’s gait simulator [2] have the ability to do this easily because their kinematics are specified in software rather then dictated by hardware. (This issue has also been addressed with Sharkey et al.’s newest gait simulator, which no longer requires that the tibia move along a fixed cam profile [6].) The range of rotational motion of the R-2000, 615 in both the transverse and frontal plane, and 360 in the sagittal plane provides the ability to simulate a wide range of gait patterns. Some gait simulators have relatively good GRF fidelity for prosthetic feet [4] and for cadaveric feet [1], while other systems scale the GRF [2]–[4]. In some cases, this is a result of frail cadaveric specimens rather then hardware limitations [4] or due to a stated goal of matching kinematics rather than kinetics [3]. Gait simulators (including the RGS) that have performed experiments with prosthetic feet are able to prescribe full GRFs. The RGS’s vertical GRF fidelity is limited by our inertial cancellation method. At slow speeds, (i.e., 6.0 s) the vertical GRF fidelity is very high (9.42 N RMS, approximately twice the noise floor). As the simulation speeds increase our inertial estimate becomes worse resulting in larger vertical GRF RMS error. Close examination of the data reveals that the large inertial spikes at heel strike and toe off were accurately subtracted out, but that the error in midstance was increased. Also the inertial cancellation resulted in only a marginal reduction in the vertical GRF error, from 39.2 N RMS to 35.0 N RMS. In summary, an improved inertial estimation algorithm will result in improved GRF fidelity. The ability to perform a prosthetic gait simulation is a logical first step towards the more pursued goal of cadaveric gait simulation. The prosthetic gait simulations acted as a test bed to investigate the feasibility of our methodologies. Based on our simulation results, we conclude that if future cadaveric simulations are to be performed at simulation speeds faster than 1.5 s, then an improved inertial compensation algorithm is required. Another possibility would be to perform experiments at the slower speed and focus on effects that are not highly sensitive to loading rate in the cadaveric specimens. Although many aspects of ILC are advantageous including its fast implementation, simple stability criteria, and minimum required iterations until tracking, ILC cannot be the final force tracking control solution for cadaveric simulations because in contrast to the prosthetic foot GRF the cadaveric GRF cannot be prescribed by changing the ground kinematics alone but must be accompanied by a change in the applied extrinsic muscle tendon forces. However, our group did gain valuable experience implementing living subject data collection with embedded coordinate systems, configuring the R-2000 and prosthetic foot, and inverting the relative kinematics via a fixed tibia. REFERENCES [1] N. A. Sharkey and A. J. Hamel, “A dynamic cadaver model of the stance phase of gait: Performance characteristics and kinetic validation,” Clin. Biomech., vol. 13, no. 6, pp. 420–433. [2] C. Hurschler, J. Emmerich, and N. 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