Evolutionary Optimization of Ground Reaction Force for a Prosthetic Leg Testing Robot

2014 American Control Conference (ACC) June 4-6, 2014. Portland, Oregon, USA Evolutionary Optimization of Ground Reaction Force for a Prosthetic Leg Testing Robot Ron Davis, Hanz Richter, Dan Simon, and Antonie van den Bogert Abstract Transfemoral amputees modify their gait in order to compensate for their prosthetic leg. This compensation causes harmful secondary physical conditions due to an overdependence on the intact limb and deficiencies of the prosthesis. Even with more advanced prostheses, amputees still have to alter their gait to compensate for the prosthesis. We present a novel way to quantify how much an amputee has to compensate for a prosthetic leg. We train a newly-developed prosthetic leg testing robot to walk with a prosthesis using an evolutionary algorithm called biogeography-based optimization (BBO). The robot is initially commanded to follow able-bodied hip and thigh trajectories, and BBO then modifies these reference inputs. We adjust the reference inputs to minimize the error between the ground reaction force (GRF) of ablebodied gait data, and that of the robot while walking with the prosthesis. Experimental results show a 62% decrease in the GRF error, effectively demonstrating the robot s compensation for the prosthesis. I. INTRODUCTION Prosthetic legs for transfemoral amputees are widely known to cause adverse ancillary health conditions in amputees [7], [8]. Walking with a prosthesis requires up to 65% more energy than able-bodied walking [3]. This results in higher strains and loading on the intact limb, leading to secondary conditions such as osteoarthritis, osteoporosis, back pain, and joint pain. A more advanced knee, such as the Otto Bock C-Leg, which is a variable-damping knee, requires less compensation from the amputee and consequently results in fewer side effects [3], [9]. Conversely, an older passive knee such as the Mauch SNS requires more compensation and results in more severe side effects [9], [20]. However, regardless of the prosthesis, amputees still alter their biomechanics to at least some extent to compensate for the prosthesis [23], [27]. Amputees incur a three-fold greater risk of osteoarthritis on their healthy side, and a significant decrease in neck bone density on their prosthesis side [8]. The goal of prosthetics is to mimic the function of human limbs; therefore, the less compensation that is required to achieve able-bodied GRF, the closer the prosthesis is to achieving its goal [19]. This work was supported by Grant 0826124 in the CMMI Division of the Engineering Directorate of the National Science Foundation, and by the Wright Center for Sensor Systems Engineering through the state of Ohio. Ron Davis was with Cleveland State University (CSU), Cleveland, OH, and is now with Casco USA (email rjdavis088@gmail.com). Hanz Richter and Antonie van den Bogert are with the Department of Mechanical Engineering at CSU (e-mail h.richter@csuohio.edu, a.vandenbogert@csuohio.edu). Dan Simon is with the Department of Electrical Engineering at CSU (corresponding author, e-mail d.j.simon@csuohio.edu). Many studies have found a higher GRF on the healthy leg of an amputee than on the prosthetic leg, and have hypothesized that equalizing the GRF of the two legs might help avoid degenerative health issues on the healthy leg [11], [15], [18], [25], [28], [29]. This research has been confirmed by clinical studies that show explicit correlations between GRF on the amputee s healthy side and degenerative health issues; see [5] and the many references therein. Prosthetic rehabilitation often includes strengthening exercises on the prosthesis side to improve long term health [10]. In view of these results, in this research we optimize the GRF of a prosthesis that is mounted on a robot in order to match able-bodied GRF as closely as possible. In order to optimize GRF, we adjust the vertical thigh motion and hip angle trajectory of the robot. This process emulates the amount of compensation required by a prosthesis user to match able-bodied GRF, and thus avoid degenerative side effects due to prosthesis use. The optimization program that we use in this research can be applied to various prostheses to test which ones are more or less likely to result in secondary health issues. Human subjects are often used to test prostheses. However, these tests are time consuming, pose liability risks, lack repeatability, and are tedious [17]. It can take up to three months for amputees to become acclimated to a new prosthesis [20]. There are liability and safety concerns due to the risk of falling or stumbling. The tests often require the use of a safety harness as a preventative measure [17]. Obtaining data from these tests is difficult because it requires the use of complex gait analysis labs where cameras, force plates, and other methods collect kinetic and kinematic data [2], [8], [9], [27]. These tests often produce limited results due to the lack of repeatability between subjects and because of the changing dynamics of human gait, which can be affected by factors such as height, weight, gender, age, general health, fatigue, or mood [16]. As part of our research, a hip robot (HR) was developed to test transfemoral prosthetic limbs [17]. The HR accurately simulates the motion of a human hip and can therefore test prostheses under realistic conditions. Unlike human tests, the HR provides a test bed in which prostheses can be tested under repeatable conditions. Liability and risk issues are avoided. The need for complex gait analysis labs is also circumvented since the HR and prosthesis can easily be fitted with standard sensors to collect relevant data [24]. The HR provides a test environment that cannot be obtained with human subjects. Unlike human subjects, the HR cannot fall, and can walk in any way desired, as long as relevant gait cycle data is available as a tracking reference. This opens up new possibilities for prosthetic tests under 978-1-4799-3271-9/$31.00 2014 AACC 4081

conditions that would either be unsafe or uncomfortable for human prosthesis users. Such tests can show the capabilities or potential performance levels of both new and existing prosthesis designs. In this research, we use the HR to simulate the hip motion of an able-bodied person, yet use a prosthetic leg on the HR. The goal of these tests, like the goal of any prosthesis, is to replicate able-bodied walking. If the prosthesis can successfully replicate able-bodied walking while mounted on the HR, than the potential exists that an amputee could do the same. Note that the replication of able-bodied walking with the HR/prosthesis combination does not guarantee that an amputee will be free from negative side effects. However, replication of able-bodied walking would show that an amputee at least has the potential to avoid negative side effects due to prosthesis use. Just as amputees alter their gait patterns to compensate for prosthesis deficiencies [2], [7], [9], the HR also needs to be able to compensate for the prosthesis. We use BBO as an indirect approach to GRF control. We use BBO to modify the gait pattern of the HR in order to adjust the GRF of the prosthesis. BBO optimizes the HR motion using the error between the GRF from able-bodied gait-cycle data, and the GRF from the HR/prosthesis combination. We use BBO for this research because it has performed well on many benchmark optimization problems, and it has also been applied to many real-world optimization problems, including aircraft engine sensor selection [21], control of mobile robots [13], and prosthetic knee control [24], [30]. In practice, many other evolutionary optimization algorithms besides BBO, such as genetic algorithms, could also be used for this work. The contribution of this work is not the application of BBO in particular, but the evolutionary learning of GRF in a prosthesis leg testing robot. II. HIP ROBOT (HR) DESIGN AND CONSTRUCTION The HR mimics the motion of a human hip. It operates with two degrees of freedom (DOF): vertical hip displacement, and thigh angle in the sagittal plane. Thigh movement is restricted to the sagittal plane because the knee has only one degree of freedom and is therefore not dynamically coupled to non-sagittal movement. A DC servo-motor, amplifier, and drive transmission controls each DOF. One DC motor drives a ball-screw linear actuator to control the hip s vertical displacement. The second DC motor drives a rotary stage to control thigh angle. Figure 1 shows a diagram of the HR. A treadmill attaches to the frame of the HR so the prosthesis has a moveable surface to walk on. To allow feedback control, both servo motors are equipped with encoders, which provide position and velocity feedback. A standard first-order sliding mode controller (SMC) [26] was developed for each drive stage. The encoder output signals are read using dspace software and hardware. We use dspace for simplicity and ease of integration with Simulink TM. Simulink software allows for the implementation of the SMC in a graphical programming interface. dspace allows real time manipulation, display, and comparison of all the values in the Simulink environment. This allows us to monitor reference angle tracking and perform control tuning without having to stop the system to observe outputs or change parameters. In order to emulate human hip motion, we use clinical human gait data as reference inputs to the HR. These data, provided by the Cleveland Clinic [27], include position data at all three reference points (hip, knee, and ankle) in three dimensions, moments, angles, and forces. We have data from multiple subjects performing a variety of movements, such as running, jogging, slow walk, normal walk, sitting, standing, and turning left/right. For this research we focus on normal walk data. The normal walk trajectories and drive stage tracking performance are shown later in Section V. Figure 1. Hip robot schematic [17] III. TEST CONFIGURATION To confirm that BBO can improve the GRF tracking of a prosthetic leg on the HR, we use standard prosthetic knee and foot components. We use an Össur Mauch Microlite S knee [14]. This prosthesis utilizes two settings for its mechanical damper, one setting each for flexion and extension of the knee. We configured the knee prosthesis with an Össur Flex-Foot [6], a widely used prosthetic foot. A photo of the assembled leg can be seen in Figure 2. A custom bracket was designed to mount a load cell between the prosthetic leg and foot. GRF was measured using the load cell mounted above the foot in Figure 2. Note that this does not measure GRF directly, but measures force along the lower shank of the prosthesis. In order to achieve GRF optimization, the measured force along the lower shank needs to be compared to force from the reference gait data. The force along the lower shank is not provided in the gait cycle data; however, the three-dimensional decomposition of GRF is provided. Using the combination of forces shown in Figure 3, 4082

Figure 2. Assembled prosthesis with leg, foot, and instrumentation. reference force along the lower shank was calculated as follows: F a = F x cos Θ + F y sin Θ (1) In Equation (1), F a is the force along the lower shank, F x the forward component of GRF, F y the vertical component of GRF, and Θ the angle of the lower shank. Θ is calculated from thigh angle h and knee angle k as follows: Θ = h k + π/2 (2) For simplicity of notation, and because F a was calculated from the reference GRF data, F a will be referred to as GRF for the remainder of this paper. IV. BIOGEOGRAPHY BASED OPTIMIZATION (BBO) A. Description of BBO Biogeography is the study of the speciation, migration, mutation, and extinction of organisms. BBO simulates this process with the goal of optimization [21], [22]. A habitat in nature contains various species. Immigration and emigration of species probabilistically occur between habitats based on how well a habitat can support life, as represented by habitat suitability index (HSI). Suitability index variables (SIVs) are the factors that influence HSI. Parameters such as rainfall, topography, and temperature are SIVs. Figure 4 shows immigration rate λ and emigration rate μ versus number of species s in a habitat. When s is low, λ is high and μ is low. This corresponds to a habitat with a low HSI. Conversely, when s is equal to its maximum possible value, λ is low and μ is high, corresponding to a habitat with high HSI. Due to overcrowding, individuals migrate to low HSI habitats, potentially strengthening those habitats and raising their HSI. In BBO, HSI is analogous to the inverse of the cost c of a candidate solution to an optimization problem. SIVs are factors that affect the cost of candidate solutions. The species that migrate between candidate solutions are analogous to the independent variables of an optimization problem solution. A habitat is analogous to a candidate solution in BBO. A population, or set of candidate solutions, is tested with respect to their performances on an Figure 3. Forces and angles of the human/prosthetic leg combination: h = thigh angle, k = knee angle, Θ = shank angle, F a = shank force. optimization problem. λ and μ are assigned to candidate solutions based on performance. Migration occurs probabilistically based on λ and μ. Our candidate solutions therefore share independent variables among themselves. We define λ and μ for each candidate solution as follows: λ i = f i /n μ i = 1 λ i for i [1, n], where n represents the total number of candidate solutions in the population (that is, population size); and f i is the cost rank of candidate solution i, where the best has a rank of 1 and the worst has a rank of n. As with other evolutionary algorithms, we also implement elitism and mutation. Elitism retains the best candidate solution(s) from one generation to the next. Mutation increases diversity in the population and injects new information into the population [22]. B. Application of BBO for GRF Control BBO is used in two distinct phases. The first phase optimizes the offsets of both the vertical displacement and thigh angle reference motion profiles. The second phase finds a continuous delta-displacement signal which is added to the vertical hip displacement reference profile. The deltadisplacement signal is discussed in detail in Section V. Before each optimization phase, the HR is operated with its initial parameter values until it reaches steady state. BBO then begins. Each test of a candidate solution (vertical offset Figure 4. BBO immigration and emigration rates versus number of species in habitat. (3) 4083

or reference motion profile) involves six strides. During the first two strides the HR tracks the original reference motion profiles. The third and subsequent strides use the altered reference motion profiles. GRF error is calculated during the fourth, fifth, and sixth strides. This is done because the HR requires at least one stride after altering the reference motion profiles to reach consistent, steady-state operation. Due to the use of a GRF load cell with inherent noise and experimental variations, testing the same candidate solution twice results in different GRF errors. When this occurs in different BBO generations, the errors are averaged. This also means that minimum error can increase from one generation to the next, even though elitism is used. For each candidate BBO solution, GRF error, referred to as cost, was calculated only during stance phase since GRF = 0 during swing. GRF error is calculated as the RMS error between the reference measured GRF: c = N i=1 (G di G ai ) 2 N where N is the total number of time samples, G di the reference GRF at time step i (Newtons), and G ai the experimental GRF of the HR at time step i (Newtons). Figure 5 depicts the operation of the optimization process. The reference inputs are passed to the robot controller (SMC), which generates motor torques. After six strides, the time history of GRF for each candidate solution is passed to BBO, which alters the reference inputs for each candidate solution via migration and mutation as it attempts to bring the experimental GRF closer to the reference GRF. Figure 5. Block diagram of HR optimization. (4) Figure 7. Bias BBO: Minimum cost per generation. V. EXPERIMENTAL RESULTS The HR was first run under normal operating conditions with no modification to the motion profiles as shown in Figure 6. Thigh angle and vertical displacement tracking from the motor controllers are very good. Knee angle is shown only for general interest since it is not actively controlled. GRF has poor tracking as it lacks a strong heel-strike peak. Calculated cost is c = 390.75 N. BBO was then run in its first phase to optimize the zero offsets of the motion profiles. Bias values were only adjusted slightly to keep the HR within safe operating conditions. Therefore, two BBO runs were performed, where the second run used the previous run s best solutions to define the domain for the search space. Test parameters for the second BBO run are shown in Table I, and performance results are shown in Figure 7. Note from Figure 7 that the cost at generation 0 is less than 320, which is lower than the initial GRF error of 390.75. That is, GRF error is lowered even with the initial random search at generation 0. BBO further lowers the GRF error to 197.61, or 49% lower than initial conditions. Figure 8 displays the HR operation after BBO was applied. Vertical displacement was lowered by 9.73 mm and thigh Table I. Bias BBO Test Parameters Population Size 10 Generations 12 Number of Elites 2 Number of 2 Independent Variables Min/Max Par Value [-0.1300 2.0265] [-0.1183 5.5000] Best Solution [-0.1253 4.8170] Best Cost 197.6100 Figure 6. No modification of reference motion profiles. GRF cost = 390.75 N. 4084

angle was biased by 0.1253 rad. GRF tracks fairly well except during the middle of stance when both heel and toe are in contact with the ground. We next execute BBO in its second phase. Since heelstrike and toe-off peaks fit well with the desired GRF after modifying the offsets of the reference profiles as described above in BBO phase 1, we do not modify the reference thigh angle trajectory any more in BBO phase 2. We instead modify the HR vertical displacement reference, with the initial condition set to the results from BBO phase 1. We modify the vertical displacement reference by adding a delta-displacement during stance phase. We represent deltadisplacement as a Fourier cosine series, and use its coefficients as the independent variables of the optimization process: u(t) = a 0 + M i=1 a i cos(2πit/t) The parameters for this BBO run are shown in Table II, and the convergence results are shown in Figure 9. BBO phase 2 lowered the GRF error to c = 148.33, or 62% lower than initial conditions. Figure 10 shows the final motion profiles and GRF plots. The differences between the original vertical displacement and thigh angle motion profiles show the amount by which the HR compensated for the prosthesis. VI. CONCLUSION A new method of prosthetic leg testing is proposed. Using a new prosthetic leg testing robot, which is designed to emulate a human hip, prostheses can be tested under conditions not possible with human subjects. Prostheses can be tested with able-bodied hip motions in order to test their potential to function in those conditions. This does not guarantee that an amputee will use the prosthesis in this manner, but it shows whether or not the prosthesis has the potential to function under those circumstances. We use BBO we optimize the hip trajectories of the robot to make the GRF of the prosthesis consistent with the GRF of an able-bodied person. This emulates the compensation that an amputee might use to overcome the shortcomings of a prosthesis. This provides data that can be used to evaluate (5) the prosthesis and provide insight as to how much an amputee needs to compensate for a prosthesis. An expanded version of this paper is available as a master s thesis [4]. A possible extension of this work is to optimize the hip robot to create rehabilitation exercises and strategies. If a prosthetists wants to train an amputee to replicate a given GRF, the hip robot could be optimized to learn the movement patterns, and those movement patterns could then be transferred to the amputee. Another possible extension of this work is determining if GRF is really the correct optimization metric. Other metrics of human walking could also be used [1]. We note that in most of the GRF plots, the experimental result typically have a time lag relative to the reference. Because the GRF rises and falls quickly, the tracking error is high on those slopes. Our Fourier series only has cosine terms, so it has no opportunity to phase shift the hip trajectory to adjust GRF timing. If we allow phase shifts in future work, we may be able to significantly reduce the GRF tracking error. REFERENCES [1] A. Ames, First steps toward automatically generating bipedal robotic walking from human data, Robot Motion and Control 2011, (K. Kozlowski, editor) pp. 89-116, Springer, 2012. [2] M. Bellman, T. Schmalz, and S. Blumentritt, Comparative Biomechanical Analysis of Current Microprocessor-Controlled Prosthetic Knee Joints, Archives of Physical Medicine & Rehabilitation, vol. 91, no. 4, pp. 644-652, 2010. [3] T. Chin, K. Machida, S. Sawamura, R. Shiba, H. Oyabu, Y. Nagakura, I. Takase, and A. Nakagawa, Comparison of Different Microprocessor Controlled Knee Joints on the Energy Consumption During Walking in Trans-Femoral Amputees: Intelligent Knee Prosthesis (IP) versus C-Leg, Prosthetics and Orthotics International, vol. 30, no. 1, pp. 73-80, 2006. [4] R. Davis, Design and Control of a Prosthetic Leg Testing Robot, Master s Thesis, Cleveland State University, 2014. [5] F. Farahmand, T. Rezaeian, R. Narimani, and P. Dinan, Kinematic and Dynamic Analysis of the Gait Cycle of Above-Knee Amputees, Scientia Iranica, vol. 13, pp. 261-271, 2006. [6] Flex-Foot Feet, http://www.ossur.com/?pageid=12639 [7] R. Gailey, K. Allen, J. Castles, J. Kucharik, and M. Roeder, Review of Secondary Physical Conditions Associated with Lower-Limb Amputation and Long-Term Prosthesis Use, Journal of Rehabilitation & Development, vol. 45, no. 1, pp. 15-20, 2008. [8] L. Graham, D. Datta, B. Heller, J. Howitt, and D. Pros, A Figure 8. After bias BBO run. Notice lowering of both thigh angle and vertical displacement. GRF cost = 197.61 4085

Comparative Study of Conventional and Energy-Storing Prosthetic Feet in High-Functioning Transfemoral Amputees, Archives of Physical Medicine and Rehabilitation, vol. 188, pp. 801-806, 2007. [9] J. Johansson, D. Sherrill, P. Riley, P. Bonato, and H. Herr, A Clinical Comparison of Variable-Damping and Mechanically Passive Prosthetic Knee Devices, American Journal of Physical Medicine & Rehabilitation, vol. 84, no. 8, pp. 563-575, 2005. [10] M. Jones, G. Bashford, and J. Mann, Weight bearing and velocity in trans-tibial and trans-femoral amputees, Prosthetics and Orthotics International, vol. 21, pp. 183-186, 1997. [11] S. Jones, P. Twigg, A. Scally, and J. Buckley, The gait initiation process in unilateral lower-limb amputees when stepping up and stepping down to a new level, Clinical Biomechanics, vol. 20, pp. 405-413, 2005. [12] G. Konidaris, S. Osentoski, and P. Thomas. Value function approximation in reinforcement learning using the Fourier basis, Twenty-Fifth Conference on Artificial Intelligence, San Francisco, California, pp. 1468-1473, 2011. [13] P. Lozovyy, G. Thomas, and D. Simon, Biogeography-Based Optimization for Robot Controller Tuning, in: Computational Modeling and Simulation of Intellect (B. Igelnik, editor) IGI Global, pp. 162-181, 2011. [14] Mauch Knee, http://www.ossur.com/?pageid=12644 [15] V. Michel and R. Chong, The strategies to regulate and to modulate the propulsive forces during gait initiation in lower limb amputees, Experimental Brain Research, vol. 158, pp. 356-365, 2004. [16] G. Nandi, A. Ijspeert, P. Chakraborty, and A. Nandi, Development of Adaptive Modular Active Leg (AMAL) Using Bipedal Robotics Technology, Robotics and Autonomous Systems, vol. 57, no. 6, pp. 603-616, 2009. [17] H. Richter, D. Simon, and W. Smith, Dynamic Modeling, Parameter Estimation and Control of a Leg Prosthesis Test Robot, Applied Mathematical Modelling, in print, 2014. [18] S. Rossi, W. Doyle, and H. Skinner, Gait initiation of persons with below-knee amputation: The characterization and comparison of force profiles, Journal of Rehabiliation Research and Development, vol. 32, pp. 120-127, 1995. [19] M. Schaarschmidt, S. Lipfert, C. Meier-Gratz, H.-C. Scholle, and A. Seyfarth, Functional gait asymmetry of unilateral transfemoral amputees, Human Movement Science, vol. 31, pp. 907-917, 2012. [20] A. Segal, M. Orendurff, G. Klute, M. McDowell, J. Pecoraro, J. Shofer, and J. Czerniecki, Kinematic and kinetic comparisons of transfemoral amputee gait using C-Leg and Mauch SNS prosthetic knees, Journal of Rehabilitation Research and Development, vol. 43, pp. 857 870, 2006. [21] D. Simon, Biogeography-Based Optimization, IEEE Transactions on Evolutionary Computation, vol. 12, no. 6, pp. 702-713, 2008. [22] D. Simon, Evolutionary Optimization Algorithms, Wiley, 2013. [23] F. Sup, A. Bohara, and M. Goldfarb, Design and Control of a Powered Transfemoral Prosthesis, International Journal of Robotics Research, vol. 27, no. 2, pp. 263-273, 2008. [24] G. Thomas, T. Wilmot, S. Szatmary, D. Simon, and W. Smith, Evolutionary Optimization of Artificial Neural Networks for Prosthetic Knee Control, Efficiency and Scalability Methods for Computational Intellect (B. Igelnik and J. Zurada, editors), pp. 142-161, IGI Global, 2013. [25] C. Tokuno, D. Sanderson, J. Inglis, and R. Chua, Postural and movement adaptations by individuals with a unilateral below-knee amputation during gait initiation, Gait Posture, vol. 18, pp. 158-169, 2003. [26] V. Utkin, J. Guldner, and J. Shi, Sliding Mode Control in Electro- Mechanical Systems, 2nd edition, CRC Press, 2009. [27] A. van den Bogert, S. Samorezov, B. Davis, and W. Smith, Modeling and Optimal Control of an Energy-Storing Prosthetic Knee, Journal of Biomechanical Engineering, vol. 134, no. 5, pp. 0510071-0510078, 2012. [28] M. van der Linden, S. Solomonidis, W. Spence, N. Li, J. Paul, A methodology for studying the e!ects of various types of prosthetic feet on the biomechanics of trans-femoral amputee gait, Journal of Biomechanics, vol. 32, pp. 877-889, 1999. [29] A. Vrieling, H. van Keeken, T. Schoppen, E. Otten, J. Halbertsma, A. Hof, and K. Postema, Gait initiation in lower limb amputees, Gait Posture, vol. 27, pp. 423-430, 2008. [30] T. Wilmot, G. Thomas, B. Montavon, R. Rarick, A. van den Bogert, S. Szatmary, D. Simon, W. Smith, and S. Samorezov, Biogeography-Based Optimization for Hydraulic Prosthetic Knee Control, Medical Cyber-Physical Systems Workshop, Philadelphia, PA, pp. 18-25, 2013. TABLE II. DELTA-DISPLACEMENT BBO PARAMETERS Population Size 25 Generations 30 Number of Elites 2 # of Ind. Vars. 7 Min/Max Par Value [0-0.72-0.72-0.72-0.72-0.72-0.72-0.72] [0 0.72 0.72 0.72 0.72 0.72 0.72 0.72] Best Solution [0 0.702 0.108-0.688 0.329 0.235 0.609] Best Cost 148.33 Figure 9. Delta Displacement BBO: Minimum cost per generation. Figure 10. After delta-displacement BBO run. Notice the change in the vertical displacement profile; the vertical displacement initial condition is the same as the vertical displacement final condition in Figure 8. The original biased signal is shown to reveal the effect of the delta-displacement signal. Thigh angle is not shown since it is the same as in Figure 8. GRF cost = 148.33. 4086