Active cadence braking (ACB) for safer recreational rehabilitation

Recreational rehabilitation is a proven way to provide a therapeutic experience and enhancement of quality of life for individuals with disabilities. The Mountain Chair is a device developed for individuals with high level traumatic spinal cord injuries (HL-SCI) to provide them with a system comparable to a downhill mountain bike. This system provides individuals with a chance to experience an extreme sport that otherwise would be impossible. One of the many goals of these programs is to enable individuals by providing the maximum amount of control of the system while maintaining the highest level of safety. Integrated control system design requires careful considerations of safety and an individual's ability to interface with the system. The authors developed an Active Cadence Braking (ACB) system to improve the safety and usability of the Mountain Chair. This project represents a collaborative effort between the Department of Mechanical Engineering Ergonomics and Safety Lab and the Therapeutic Recreation and Independent Lifestyle (TRAILS) program at the School of Medicine. The innovation of this project is a proportional, smart braking system to improve safety and redundancy for the current Mountain Chair device. A remote input device translates the patient's wrist rotation into an equivalent braking force. The system builds on traditional ABS to provide improved stopping distance while maintaining controllability for the patient.

ACTIVE CADENCE BRAKING (ACB) FOR SAFER RECREATIONAL REHABILITATION by John Lillquist A Senior Honors Thesis Submitted to the Faculty of The University of Utah In Partial Fulfillment of the Requirements for the Honors Degree in Bachelor of Science In Department of Mechanical Engineering Approved: ______________________________ _____________________________ Andrew Merryweather, PhD Tim Ameel, PhD Thesis Faculty Supervisor Chair, Department of Mechanical Engineering _______________________________ _____________________________ Kuan Chen, PhD Sylvia D. Torti, PhD Honors Faculty Advisor Dean, Honors College ABSTRACT Recreational rehabilitation is a proven way to provide a therapeutic experience and enhancement of quality of life for individuals with disabilities. The Mountain Chair is a device developed for individuals with high level traumatic spinal cord injuries (HL-SCI) to provide them with a system comparable to a downhill mountain bike. This system provides individuals with a chance to experience an extreme sport that otherwise would be impossible. One of the many goals of these programs is to enable individuals by providing the maximum amount of control of the system while maintaining the highest level of safety. Integrated control system design requires careful considerations of safety and an individual’s ability to interface with the system. The authors developed an Active Cadence Braking (ACB) system to improve the safety and usability of the Mountain Chair. This project represents a collaborative effort between the Department of Mechanical Engineering Ergonomics and Safety Lab and the Therapeutic Recreation and Independent Lifestyle (TRAILS) program at the School of Medicine. The innovation of this project is a proportional, smart braking system to improve safety and redundancy for the current Mountain Chair device. A remote input device translates the patient’s wrist rotation into an equivalent braking force. The system builds on traditional ABS to provide improved stopping distance while maintaining controllability for the patient. ii CONTENTS ABSTRACT.................................................................................................................................... ii TABLE OF FIGURES ................................................................................................................... iv INTRODUCTION .......................................................................................................................... 1 BACKGROUND ............................................................................................................................ 2 CONTROLLER BACKGROUND ................................................................................................. 3 CONTROLLER DESIGN .............................................................................................................. 4 MODEL DESIGN ......................................................................................................................... 10 SIMULINK MODEL EXECUTION ............................................................................................ 16 RESULTS ..................................................................................................................................... 19 CONCLUSION ............................................................................................................................. 23 REFERENCES ............................................................................................................................. 25 iii TABLE OF FIGURES Figure 1: The current braking mechanism. ......................................................................................3 Figure 2: Friction factor as a function of wheel slip: (a) dry asphalt; (b) wet asphalt; (c) gravel; and (d) packed snow. (Olson, Shaw, & Stepan, 2003) ....................................................................4 Figure 3: First step of the controller, speed error applied to PID controller then added to gravity compensation. This force is then converted to a plunger force. ......................................................5 Figure 4: Second section of controller: applies a ratio to the desired force based on the dynamic distribution. ......................................................................................................................................6 Figure 5: The state machine for the front and back wheels. The figure shows the inputs and outputs. .............................................................................................................................................6 Figure 6: Inside the state machine for the front wheels, the rear wheel state machine is the same but has its own set of variables. There are three states: free, locked, unbraked (forced unlocked). Inputs: (1) FrontState- whether the wheels are locked or free currently, (2) F_front- the desired force on the wheels, time- the current time in order to start timers. Outputs: Force- the force to be applied to the wheels. .......................................................................................................................7 Figure 7: This figure relates the force on the plunger as a function of servo displacement. The values for this function are found experimentally. ..........................................................................8 Figure 8: Servo FBD, the displacement can be found using Eq. 4 from the servo angle. ...............8 Figure 9: This section of the controller takes the force from the state machine and is used to solve for the servo angle that applies the desired force. ............................................................................9 iv Figure 10: FBD for brake plungers ................................................................................................10 Figure 11: FBD of the brake rotor. ................................................................................................11 Figure 12: FBD of one of the chair wheels. ...................................................................................12 Figure 13: Free body diagram for mountain chair. ........................................................................13 Figure 14: Free body diagram for the force due to gravity ............................................................14 Figure 15: The fit to the Cavalier braking data. .............................................................................15 Figure 16: Simulink subsystem for the Mountain Chair dynamics. The force on the plunger is input and converted to an acceleration...........................................................................................16 Figure 17: The comparison of the maximum allowable braking force to desired stopping force on the wheel. .......................................................................................................................................17 Figure 18: The case where both wheels are locked. ......................................................................18 Figure 19: The case where both wheels are free. ...........................................................................18 Figure 20: The entire Simulink model. ..........................................................................................19 Figure 21: The theoretical user input used to test the model. ........................................................19 Figure 22: This plot shows the deceleration when wheels are locked as a function of weight to tire width ratio. The blue stars represent experimental data, while the red o is the mountain chair model..............................................................................................................................................20 Figure 23: A simulation of the ACB. The ACB prevents the chair from exceeding acceptable speed and matches the desired velocity. ........................................................................................21 Figure 24: Plot of deceleration rate as a function of brake ratio for various unlock forces. The lines represent a variation in the amount of force applied to the wheels. 250 N was the upper limit to ensure that the wheels do in fact unlock. Note: the standard for safe deceleration is 4.5 v m/s2 and is denoted by the solid line. The dashed line represents the deceleration of standard ABS. ...............................................................................................................................................22 Figure 25: Plot of stopping distance of the mountain chair for fully locked wheels, standard ABS, and ACB. Note the significant improvement over ABS using ACB. The ACB also increases controllability over locked wheels. ................................................................................................23 vi 1 INTRODUCTION Recreational rehabilitation is shown to enhance quality of life for those with serious spinal cord injuries. The Mountain Chair is a gravity powered off-road adaptive sport device designed and built at the University of Utah. The intended audience is individuals with tetraplegia and limited upper body and trunk mobility. A major challenge with this novel technology is providing the users with a safe and efficient way to control the Mountain Chair on rugged terrain. The purpose of this project was to improve safety and control of the Mountain Chair. The mountain chair is a mobile platform that can be modeled as an automobile; therefore the system was inspired from existing automobile braking models. Braking, especially in situations that require additional skill (e.g., gravel roads, snow, wet) require special attention to improve control while still managing an efficient braking strategy. Automotive engineers have reduced the necessary user skill required to safely brake by implementing anti-lock brake systems (ABS). ABS reduces the braking distance and improves control in most situations without requiring any user training. Users simply apply maximum braking force and the ABS system takes control to minimize wheel locking. Some limitations of ABS performance are the inability to modify function based on terrain differences. For example, in gravel, ABS does not take advantage of the "damming" effect of loose road surfaces. (Burg & Blazevic, 1997) The damming effect occurs when gravel stacks up during wheel lock, increasing the resistance, resulting in better braking response. On the other 2 hand, if ABS is disabled and the (Aly, Zeidan, Hamed, & Salem, 2011) A hybrid system, designed to maximize controllability while achieving an adequate stopping force is necessary to improve deceleration. The proposed system is called an Active Cadence Braking (ACB) system. ACB combines the decreased stopping distance of locked wheels with the increased control of ABS. ACB builds upon the traditional ABS, improving stopping distance while maintaining directional control in gravel. This allows users to be able to stop safely and effectively even without prior experience. BACKGROUND The ACB system improves on ABS specifically in off-road situations where the Mountain Chair will be used. This system relies on a cadence braking technique. Instead of a binary pulse of ABS, cadence braking has a ratio of time when the wheels are locked to unlocked. ACB is also able to keep a minimum braking force, whereas ABS operates between brakes “on” or “off”. The current mechanical braking system is shown in Figure 1. A servo is used to release the braking force constantly applied by the spring. By using the servo to counteract the spring, the system defaults to having the brakes applied in the event of power loss or electronics malfunction. Based on this current mechanical design ACB outputs a servo control. For user control, the mountain chair uses a single degree of freedom wrist actuated joystick. This takes a user input and determines the desired velocity. This velocity can be mapped to each patient to limit the maximum velocity and to handle all ranges of wrist motion. ACB is designed to match the speed of the Mountain Chair to the users desired speed while maintaining maximum directional control. 3 Braking plunger Braking spring Servo Figure 1: The current braking mechanism. A Simulink model was implemented to quantify the effectiveness of the ACB system. The major factors that were analyzed were the pulse lengths from the controller and the force applied to the brakes while in the “unbraked” state. The unbraked state is the state of the brakes when, based on the user's input, the brakes would be locked but the ACB unlocks them. The deceleration for every case was compared to the industry proposed standard for safe deceleration of 4.5 m/s2. (Vehicle Stopping Distance and Time) CONTROLLER BACKGROUND The initial literature review uncovered many options for traction control in vehicles. The majority of braking methods used "preventing wheel slip" as the main objective and basis for the controller. The wheel slip is the ratio between the vehicle speed and tire speed. (Aly, Zeidan, Hamed, & Salem, 2011) Figure 2 shows friction as a function of wheel slip for various road surfaces. A common idea is to maximize the friction that relies on keeping the wheel slip in a positive slope region. (Burg & Blazevic, 1997) This method is ineffective in soft surfaces as the 4 slip function is always positive. Control in gravel surfaces is easier as there is no optimal zone for the slip. The braking force is maximized when the wheels are completely locked. Figure 2: Friction factor as a function of wheel slip: (a) dry asphalt; (b) wet asphalt; (c) gravel; and (d) packed snow. (Olson, Shaw, & Stepan, 2003) The controller design for the ACB can be summarized in three main aspects: minimizing stopping distance, maximizing control, and maintaining desired speed. All of these aspects combine to create the safest experience for patients. CONTROLLER DESIGN The ACB controller is a PID state machine controller. The controller starts with an input of speed error, the difference between the user’s desired speed and the actual speed of the Mountain Chair. Figure 3 shows the error input applied to the PID and then added to the gravity compensation. The PID gains are tuned and the final values can be found in the results section. The PID and gravity compensation combine to output a desired braking force which is then converted to a plunger force at the brake master cylinder. The gravity compensation algorithm is based on Eq. 1. The gravity compensation relies on real time reading of the slope of the hill, 5 where mc is the mass of the Mountain Chair. The brake to plunger block is based on Eq. 15. This block converts the desired stopping force to a desired total force on the brake plungers. Figure 3: First step of the controller, speed error applied to PID controller then added to gravity compensation. This force is then converted to a plunger force. = sin( )∗ ∗ Eq. 1 Figure 4 shows the dynamic distribution being used to determine how much of the total desired braking force will be applied to the front and rear wheels. This ratio is based on the deceleration of the vehicle and is derived from Eq. 13 and Eq. 14. This term saturated to a maximum of 1 and minimum of 0.5 because the vehicle is always in deceleration, meaning the rear wheels should never be braking harder than the front wheels. If the Mountain chair is decelerating fast enough, then it is possible that all the weight will be on the front wheels. Figure 5 shows the inputs and outputs to the state machine. The inputs are braking forces, state of the wheels (wheel states), the ratio of lock time to unbraked time (brake_ratio), the maximum time that the wheels can be locked (lock_max), and the force on the brakes when they are unbraked (max_unlock). The state machine outputs the final force that the controller has determined; this force is limited to values between 0 and Fspring in the saturation block. 6 Figure 4: Second section of controller: applies a ratio to the desired force based on the dynamic distribution. Figure 5: The state machine for the front and back wheels. The figure shows the inputs and outputs. 7 Figure 6 shows inside the state machine. The inputs are FrontState- weather the wheels are locked or free currently, F_front- the desired force on the wheels, time- the current time in order to start timers. The output is Force- the force to be applied to the wheels. The wheels can be in a free, locked, or forced free state. The state goes from free to locked when the state input shows that the wheels have locked or goes back when the state input shows the wheels are unlocked. The state goes from locked to unbraked when the wheels have been locked for the specified amount of time, brake_ratio*lock_max. The state goes from unbraked to locked after the specified amount of time, lock_max. In the locked and free state the desired brake force is fed through the state machine to the output. In the unbraked state the output force is a defined force, max_unlock. Figure 6: Inside the state machine for the front wheels, the rear wheel state machine is the same but has its own set of variables. There are three states: free, locked, unbraked (forced unlocked). Inputs: (1) FrontState- whether the wheels are locked or free currently, (2) F_front- the desired force on the wheels, time- the current time in order to start timers. Outputs: Force- the force to be applied to the wheels. 8 The last feature of the ACB controller is defining the servo angle based on the desired braking force. Eq. 2-Eq. 3 determine the displacement necessary to achieve the desired braking force. This equation is found using Figure 7, an experimentally determined relationship between the brake piston force and the servo displacement. Figure 7: This figure relates the force on the plunger as a function of servo displacement. The values for this function are found experimentally. + = =1 − ∗ Eq. 2 Eq. 3 Figure 8 is used with Eq. 4 to find the servo angle based on servo displacemnt. Figure 8: Servo FBD, the displacement can be found using Eq. 4 from the servo angle. 9 Θ = ( 1 ) Eq. 4 Eq. 4 and Eq. 3 are combined in Eq. 5. This equation is used to solve for the servo angle based on desired force, where dmax is found experimentally based on the displacement of the servo when there is no force on the brake caliper. = asin( 1 − ∗ ∗ 1 ) Eq. 5 Figure 9 applies Eq. 5 to the ACB controller. This section of the controller requires the assumption that dmax is half of L1. It also includes a saturation limit so the input to asin is between -1 and +1. Finally, there is a speed limiter and a quantizer because the servo can only move 9.52 rad/s and the servo can only take increments of degrees. Figure 9: This section of the controller takes the force from the state machine and is used to solve for the servo angle that applies the desired force. Based on this controller design, the needed inputs are: speed error, wheel state, acceleration, and hill slope. The outputs are: servo angle, which relates to the second output, force on the piston by the empirically derived relationship in Eq. 2 and Eq. 3. Based on these inputs and outputs, the 10 dynamic model was designed to use force on the piston as an input and return the desired outputs. MODEL DESIGN The calculations require that the system start with the desired plunger force from the ACB controller. Figure 10: FBD for brake plungers = ∗ 2 1 Eq. 6 The force on the calipers is a simple piston ratio. A1 is the area of the smaller piston and A2 is the area of the larger piston on the wheel. These values can be found experimentally from a force ratio or taken from the brake manufacturers specifications. Once the piston force is converted to the force on the caliper, the next step is to determine the force that this caliper force exerts on the wheels (see Figure 11). 11 Figure 11: FBD of the brake rotor. = ∗ , Eq. 7 The force of the brake rotors is calculated as the product of force on the caliper and the friction factor between the calipers and rotors, usually around 0.6, where r2 is the distance from the average caliper contact to the center of the wheel. Once the force that the caliper exerts on the wheel is determined, the model calculates the stopping force that the wheel imposes on the Mountain Chair. 12 Figure 12: FBD of one of the chair wheels. , = = ∗ , Eq. 8 ∗ 2 1 Eq. 9 The maximum stopping force is the dynamic weight distribution V, derived in Eq. 10-Eq. 14, multiplied by the rolling friction factor, , (0.3-0.5 depending on tire and road surface). The friction factor in the model is 0.4. The actual stopping force is the force on the brake rotor multiplied by the ratio of rotor radius to wheel radius, r1. In order to determine the maximum stopping force, the dynamic weight distribution needs to be determined. 13 Figure 13: Free body diagram for mountain chair. ∗ = Eq. 10 ∗ = Eq. 11 Eq. 10Eq. 11 are used to find the static distribution of the weight on the front and rear wheels. = ∗ ∗ ∗ Eq. 12 In Eq. 12, WT is the dynamic weight transfer from the rear to the front wheel, where a is the deceleration of the chair, hCG is the height of the center of gravity, and g is the acceleration due to gravity (9.81 m/s2). , , = + = − Eq. 13 Eq. 14 14 Eq. 13 and Eq. 14 are the final dynamic distribution of the mountain chair's weight on the front and rear tires. These equations take the static weight distribution and add the dynamic weight transfer to the front wheels and subtract it from the rear, this is for deceleration. The signs are reversed for acceleration. Eq. 15 is derived from Eq. 6-Eq. 9. The equation for the dynamics define force required to stop. ∗ 2∗ 2∗ 1∗ 1 = Eq. 15 Fstopping is the stopping force of the front and rear pair of wheels, independently. Figure 14: Free body diagram for the force due to gravity = ∗ ∗ ( _ ) Eq. 16 Eq. 16 describes the force of gravity on the chair. Where mchair is the mass of the chair and theta_hill is the angle of the hill. 15 = / =( + , + . )/ Eq. 17 In Eq. 17, Newton’s second law defines the total acceleration of the chair. However, this is only valid when the wheels are not locked. When the wheels are locked, the deceleration is modeled from empirical data describe in the literature from a Chevy Cavalier. (Macnabb, Ribartis, Mortimer, & Chafe, 1998) Figure 15 shows the Cavalier declaration data from three trials with a cubic spline fit applied. Eq. 18 is the cubic fit and is implemented in the dynamic model of the chair. The validity of this model is Deceleration (g's) explored in the results section. Figure 15: The fit to the Cavalier braking data. = 1.45 6 ∗ − 1.91 4 ∗ − 24.9 ∗ − .00377 Eq. 18 16 SIMULINK MODEL EXECUTION The dynamic model was then implemented into Simulink. Figure 16 shows the first step of the model where a desired plunger force is converted to braking force for the front and rear wheel independently. This corresponds to Eq. 15. Figure 16: Simulink subsystem for the Mountain Chair dynamics. The force on the plunger is input and converted to an acceleration. The stopping force in each pair of wheels is then compared to the maximum stopping force dictated by the dynamic distribution of the weight. There are four cases, as seen in Figure 17: both stopping forces exceed the maximum force in which case they both lock as seen in Figure 18, only the front exceeds the maximum in which case only the front set is locked, only the rear wheels exceed the maximum in which case only the rear wheels lock, or both set of wheels remain unlocked as seen in Figure 19. Figure 18 shows the acceleration when both wheels are locked, it is modeled off the Cavalier data. The function for the deceleration when the wheels are locked is supplied a constant of 0.01 because this is the size of the step used in the calculation of this simulation. In an ideal system 17 this function would ramp up to the value at 0.01 but the difference is neglected in this model for simplicity. Figure 19 shows the acceleration when both wheels are free, the forces on the wheels are the forces output by the ACB. The stopping force from each wheel is added together and then subtracted from the force generated by multiplying mass by gravity as seen in Eq. 17. This total force is divided by the mass of the Mountain Chair to determine the total acceleration. Figure 17: The comparison of the maximum allowable braking force to desired stopping force on the wheel. 18 Figure 18: The case where both wheels are locked. Figure 19: The case where both wheels are free. The final system can be seen in Figure 20. The input is either a step input, used to determine the stopping distance from 10 m/s, or a theoretical user input as seen in Figure 21. This input is compared to the velocity of the dynamic model and the error is fed back to the ACB. In this simulation, theta is not fed to anything, while in the final implementation, this signal will be fed to the servo motor that drives the brakes. 19 Figure 20: The entire Simulink model. Figure 21: The theoretical user input used to test the model. RESULTS Figure 22 shows the deceleration rates of various vehicles as a function of the ratio of weight to tire width. (Macnabb, Ribartis, Mortimer, & Chafe, 1998) The mountain chair model fits the line well, and is actually below the line meaning that the performance could be slightly better on the physical chair. 20 0.8 Vehicle Data linear Mountain Chair Results Deceleration (g) 0.75 0.7 0.65 0.6 0.55 0.5 3 4 5 6 7 8 9 10 11 12 Ratio of weight to tire width (kg/mm) Figure 22: This plot shows the deceleration when wheels are locked as a function of weight to tire width ratio. The blue stars represent experimental data, while the red o is the mountain chair model. The first step to improve the controller response was to tune the PID gains. The final gains were set as: D = 300; I = 5; P = 500. With the tuned gains, the model produced the velocity as seen in Figure 23. This is a simulation of a possible user input as the Mountain Chair descends a trail. The vehicle does not exceed the desired speed, an important aspect of the safety of the vehicle. 21 Figure 23: A simulation of the ACB. The ACB prevents the chair from exceeding acceptable speed and matches the desired velocity. Once the PID gains were tuned, the simulation was run for various combinations of brake ratio and unbraked forces as seen in Figure 24. It is clear that ACB can produce safe deceleration rates and improves the performance over a standard ABS controller. 22 0 0N 62.5 N 125 N 187.5 N 250 N -1 -2 -3 -4 -5 -6 -7 0 1 2 3 4 5 6 Brake Ratio Figure 24: Plot of deceleration rate as a function of brake ratio for various unlock forces. The lines represent a variation in the amount of force applied to the wheels. 250 N was the upper limit to ensure that the wheels do in fact unlock. Note: the standard for safe deceleration is 4.5 m/s2 and is denoted by the solid line. The dashed line represents the deceleration of standard ABS. Finally, using an unbraked force of 125 N and a brake ratio of 1.5 the stopping distance of ACB, ABS, and fully locked wheels are compared in Figure 25. ACB can stop in 63% of the distance as traditional ABS. ACB could provide even lower stopping distances but any decrease in stopping distance will come at the expense of controllability. The final value of brake ratio and unbraked force will have to be determined with the actual Mountain Chair. 23 Figure 25: Plot of stopping distance of the mountain chair for fully locked wheels, standard ABS, and ACB. Note the significant improvement over ABS using ACB. The ACB also increases controllability over locked wheels. CONCLUSION The ACB system developed for off-road braking improvements proved to decrease the estimated stopping distance over traditional ABS by taking advantage of the damming effect of gravel. This new feature also provides users assured directional control while braking by maintaining significant rotation in the wheels even under extreme braking maneuvers. Much like the introduction of ABS into automobiles, ACB will provide safer operation of the Mountain Chair for all users. Moving forward, physical implementation of this controller on the mountain chair will require a tilt sensor, GPS, and tachometer. The controller will be easily converted to C code and implemented into the custom made onboard microcontroller. Once ACB is programmed on the 24 microcontroller, the controller and accuracy of the dynamic model will be put to the test as real world trials begin. Based on this simulation the ACB system will provide a safer recreational rehabilitation experience for the intended users. 25 REFERENCES Aly, A. A., Zeidan, E.-S., Hamed, A., & Salem, F. (2011). An Antilock-Braking Systems (ABS) Control: A Technical Review. Intelligent Control and Automation, 186-195. Burg, J., & Blazevic, P. (1997). Anti-Lock Braking and Traction Control Concept for All-Terrain Robotic Vehicles. IEEE. Macnabb, M. J., Ribartis, S., Mortimer, N., & Chafe, B. (1998). ABS Performance on Gravel Roads. Proc. Int. Conf. Enhanced Safety Vehicles, 628-639. Olson, B. J., Shaw, S. W., & Stepan, G. (2003). Nonlinear Dynamics of Vehicle Traction. Vehicle System Dynamics, 377-399. Vehicle Stopping Distance and Time. (n.d.). Retrieved from http://nacto.org/docs/usdg/vehicle_stopping_distance_and_time_upenn.pdf nacto.org: