Developing a Functional Roller Coaster Optimizer. Ernest Lee. April 20, Abstract

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1 Developing a Functional Roller Coaster Optimizer Josh Tsai Brigham Young University joshjtsai@gmail.com Ernest Lee Brigham Young University ernest.tylee@gmail.com April 20, 2017 Abstract Roller coasters are a popular and exciting form of entertainment, provided that the constraints of safety, cost, and ability to manufacture are met. While these principles are well-known to all roller coaster designers, much of the rest of the design is left to their creative discretion. With these variables in mind, we attempt to create the optimal experience for the roller coaster rider in which the ride is designed around the rider s maximum thrill experience. To maximize this thrill experience, our goal is to build a roller coaster that travels through a series of checkpoints as quickly as possible. In the optimization process, we keep in mind the safety of our riders by constraining forces acting on them. The product of this project is a working optimizer that maximizes the speed through any site design plan. Though this project clearly applies to the field of roller coaster design, the principles adhered to in the design process are applicable to all fields relating to similar non-powered transportation systems. For example, an escape system from a launch pad produced by NASA required similar variables of optimal height, speed, and safety. They are also apparent in sports such as snowboarding, where following a particular path produces certain forces on the body. I. Introduction THE job of a roller coaster engineer is to provide excitement, energy, and fun while keeping people from harm. Although the concept of thrill could be different to individuals, it can be agreed that those seeking it are not wishing to remain in their comfort zones. Thus the engineer guides the development of a roller coaster by allowing one to experience somewhat dangerous sensations under the guarantee of safety. Though thrill itself is a difficult term to define, we can help people know what it is a roller coaster provides by boasting specific features great height, great speed, or perhaps a certain kind of thrill element such as several loops or cobra rolls. With the right amount of assets, roller coaster designers can attempt to create the highest or fastest rides. Optimization problems such as these blend creativity with mathematical computation to balance the right amount of thrill with a safe ride. The simple mathematical problem of minimizing the time required for a frictionless particle to travel from one point to another under only the influence of gravity was introduced by Johann Bernoulli. The solution to this problem is known as the brachistochrone curve [2]. It can be solved using the principle of conservation of energy, from which the time needed to travel the curve can be solved and then minimized. Roller coasters can be modeled using the same principle, as they carry objects influenced only by gravity traveling down a track. The brachistochrone problem can serve as a somewhat crude model of how a series of cars will travel by optimizing the time is takes to travel from the beginning to end. The objective of interest then takes into account the average speed taken over the course of the ride, which 1

2 Figure 1: We looked at existing rides to get ideas for creating our own ground spline, such as this plan for Hershey Park s Skyrush [3]. is maximized over the optimum time traveled. Optimizing for several roller coaster sections and including more realistic factors such as initial velocity and friction can help us better understand how to further improve designs. II. Methodology We begin with the site plan for a roller coaster, a map which shows where on the given supposed park grounds the ride can be built. This basic information is imported into MATLAB as a series of splines connected by individually chosen points. The heights of the roller coaster at the specified points is also chosen to fit the design in mind. The cscvn function (Cubic Spline Curve Variational/Natural) produces splines in parametric equations which mimic the proposed site plan. In MATLAB, we express the objective of ride duration as a summation of many small, discrete sections. Our problem extends the brachistochrone problem to model full-length, three-dimensional roller coasters and incorporating further the variables of friction, initial velocity, and safety constraints. Thus the objective to minimize time remains the same, though certain variables have changed and several constraints applied. The standard which specifies restrictions on acceleration and other such limits is found in ASTM F2291. We program our optimizer in MATLAB uti- Figure 2: Bird s eye view of the sample roller coaster track using cscvn, with user selected poles (red circles). lizing the built-in solver fmincon, which minimizes a function subject to linear constraints. By default, this optimizer uses a BFGS algorithm. To more easily specify the constraints of velocity and acceleration, the track is modeled as a piecewise functions of cubic polynomials. This is made seamless using another MATLAB function pchip (Piecewise Cubic Hermite Interpolating Polynomial), which returns an interpolated cubic spline through a given series of values that ensures both a smooth curve and minimal oscillation. A matrix of coefficients is extracted from the cubic piecewise functions. The elements of this matrix are the design variables of this study. The optimizer runs as follows: 1. Select the desired x- and y-coordinates on the ground plane through which a cubic spline will travel and the roller coaster will follow. Select also the accompanying z- coordinates which will act as fixed heights along the path. 2. Input the initial velocity v 0, coefficient of friction µ k, and number of divisions n given to each spline segment separated by the coordinates selected in (1). 3. Each spline segment is flattened to a func- 2

3 Figure 3: The optimizer displays the curve with zero initial velocity, subject to κ.10 m 1, µ k =.018, and n = 40. tion of its height per unit length, thus the optimization occurs in two spatial dimensions. For each segment: (a) Produce a matrix of n 4 dimensions containing the coefficients of a line beginning at H and ending at 0. (A cubic polynomial has three coefficients and one constant, hence the 4). (b) Establish the constraints of continuity by setting A ( L n ) 3 + B ) 2 + C ) + D of one segment equal to D of the next. Similarly constrain the first and second derivatives at each point between segments. Provide additional constraints to acceleration and curvature. The initial height must equal that of the final height of the previous segment, as must the initial angle of descent/ascent. (c) Optimize the time function using fmincon, which recreates the piecewise polynomial according to the input matrix and solves for the minimum time taken to traverse the segment. 4. Plot the resulting roller coaster along with other desired information. The objective function in question calculated using the principle of conservation of energy is: [ xi n 1 +dx 1 + ( dy ] dx T = )2 i=1 2g(H y(x) µk x) dx x i Upon calculating total time traveled, the velocity, acceleration, and curvature of the track are also calculated. The piecewise polynomial is subjected to the constraints which follow. 1. DESIGN VARIABLE CONSTRAINTS: To establish track continuity: A i ) 3 + Bi ) 2 + Ci ) + Di = D i+1 To establish continuity in direction of motion: 3A i ) 2 + 2Bi ) + Ci = C i+1 To establish continuity in reaction forces on the riders: 3A i ) + Bi = B i+1 for i = {1 : n} Slope constraints at the beginning and ending, along with other desired poles with no slope (for example, the highest point of a loop): 3

4 Figure 4: With the conditions shown in the results section, a variety of track shapes is produced. Initial Condition Time (s) Original 4.21 Initial Velocity 1.55 Initial Angle 1.72 Curvature 8.42 Table 1: The shape of the resulting tracks is shown above. C i = 0 2. OBJECTIVE CONSTRAINTS: Curvature is restricted to ensure the roller coaster will not bend exceeding its designed minimum radius.: κ κ limit ASTM suggests [1] the following restriction in acceleration for the safety of the rider: a normal 6g a tangential 6g III. Results The code thus allows for multiple constraints and initial conditions, providing visual and numeric solutions to various desired inputs. With each individual segment a separate brachistochrone curve, several hills are created automatically to decrease runtime. The initial velocity, starting angle, and added curvature limit factors influence the final curve in various ways, as shown in Figure 4. i. The effect of no initial velocity The track without initial velocity is a brachistochrone curve subject to friction and curvature constraints is shown in Figure 3. The curve behaves just as one would expect with the brachistochrone curve, but with the ending slope constraint applied. ii. The effect of added initial velocity The addition of initial velocity to the curve greatly impacts the shape of the curve. Higher initial velocity results in a curve with less initial drop. Setting the initial velocity to 20 m/s 4

5 Figure 5: A 3D view of an optimized design. allows the same track to be ridden in a fraction of the original time. iii. The effect of initial angle Retaining an initial velocity of 20 m/s but changing the initial angle to 30 degrees above the x-axis results in a decreasing the travel time by a fraction.the roller coaster quickly overcomes the initial angle and drops sharply. v. The effect of the acceleration constraints The effects of acceleration constraints is harder to observe visually than the aforementioned effects. It mostly ensures that the riders are safe. However, with great speed summoned from the initial drop, it can be expected that the cars easily approach this constraint. Through our tests it has not yet been broken. vi. The transition to three dimensions iv. The effect of the curvature constraint With an adjusted curvature limit the initial velocity was adjusted to 8 m/s to accommodate visual accuracy. The curvature limit was raised to 15 m 1, increasing traversing time by several factors but making it a safe ride. Connecting the segments and plotting the optimized heights along the initial spline curve provides additional variation. Under the principle of conservation of energy with a low coefficient of friction typical of roller coasters, one can expect that in order for a car to successfully travel the roller coaster, it can never travel higher than the initial point. 5

6 IV. Conclusions While this process does not perfectly emulate the method in which roller coasters are optimized, it is a unique way to maximize the thrill of the roller coaster by optimizing its speed and acceleration. Because the duration of each segment is optimized, the average speed of the entire coaster is typically very short, but large variations of speed occur in each part, an important factor of thrill. For many parts of the roller coaster, the acceleration must be constrained to the prescribed 6g. Thus even though this presents a crude study of roller coaster physics, the results which reveal the relationship between various initial conditions and constraints provides additional optimization data. While any combination of these design protocols can create a fast roller coaster, the initial velocity and the velocity picked up at an initial drop are the most crucial. However, the fun factor for the riders is limited if the only feature of the roller coaster is speed. Forces acting on the rider must also be considered. Under the acceleration restraints provided and the variables chosen, an effective roller coaster section should have a balance of added initial velocity with a downward angle at a height similar to that chosen for this demonstration, similar to one labeled with initial velocity in Figure 4. The robustness of this optimizer provides insight to the way certain phenomena occur in real roller coasters. Hydraulic or pneumatic roller coasters which begin at high initial velocities do no experience the acceleration changes which occur naturally on hilly roller coasters, particularly at the beginning. Adjusting this parameter, either with greater height or by some other means, greatly affects the ride experience. It is also clear then that one may not want to optimize the speed of a roller coaster. We see from our various tests that great acceleration can be achieved quickly and often at a large range of speeds. Roller coasters do not slow down very much except due to differences in height because of the typical low coefficients of friction, so a wide range of speed is easily achievable with many designs. As stated earlier, the objective of thrill being subjective as it is, many different roller coasters exist to cater to the preferences of amusement park goers. However, we can conclude from this study that for a thrilling coaster, what one needs is good speed, leaving other factors at the discretion of the designer. References [1] ASTM International. (2016). ASTM F Standard Practice for Design of Amusement Rides and Devices. [2] Filobello-Nino, U., Vazquez-Leal, H., Pereyra-Diaz, D., Yildirim, A., Perez-Sesma, A., Castaneda-Sheissa, R.,... Hoyos-Reyes, C. (2013). A generalization of the Bernoulli s method applied to brachistochronelike problems. Applied Mathematics and Computation, 219(12), doi: 017 [3] [The site plan for Skyrush, a 200- foot tall roller coaster that will open in 2012.]. (2011, August 2). Retrieved May 19, 2017, from x.ssf/2011/08/hersheyparkrevealsplansfo r.html [4] Xu, G. N., Xin, H. J., Lu, F. Y., Yang, M. L. (2011). Kinematics and Dynamics Simulation Research for Roller Coaster Multi-Body System. Advanced Materials Research,421, doi: / 276 [5] Zhou, Y., Wang, W. (2012). Design and Manufacturing of Large Amusement Devices for Performance and Reliability with Qualitative Reasoning. Advanced Materials Research, , doi: / 6