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On optimal velocity during cycling
OCZl-9290/M
J. Bmnechonrs Vol. 21, No. 2. PP. 1X05-213, 1994. Printed in GreatBritain
ON OPTIMAL
0
VELOCITY RYSZARD
1993
16.00 + .oO
Pergamon PressLtd
DURING CYCLING
MAROI;ISKI
Institute of Aeronautics and Applied Mechanics, ul. Nowowiejska 24, 00-665 Warsaw, Poland Abstract-This paper focuses on the solution of two problems related to cycling. One is to determine the velocity as a function of distance which minimizes the cyclist’s energy expenditure in covering a given distance in a set time. The other is to determine the velocity as a function of the distance which minimizes time for fixed energy expenditure. To solve these problems, an equation of motion for the cyclist riding over arbitrary terrain is written using Newton’s second law. This equation is used to evaluate either energy expenditure or time, and the minimization problems are solved using an optimal control formulation in conjunction with the method of Miele [Optimization Techniques with Applications to Aerospace Systems, pp. 69-98 (1962) Academic Press, New York]. Solutions to both optimal control problems are the same. The solutions are illustrated through two examples. In one example where the relative wind velocity is zero, the optimal cruising velocity is constant regardless of terrain. In the second, where the relative wind velocity fluctiates, the optimal cruising velocity varies. NOMENCLATURE initial point final point prescribed constant aerodynamic coefficient of air resistance aerodynamic drag cyclist’s external energy expenditure (external work) rolling and bearing resistance gravitational acceleration profile of route performance index mass of cyclist with bicycle maximal power output normal reaction of the ground radius of curvature frontal projected area of the cyclist propulsive force time velocity wind velocity coordinate (approximately the covered distance)
local inclination angle angle describing cyclist’s position relative to initial point border of admissible domain power setting Lagrange multiplier known function of x and v known function of x and o density of the air fundamental function derivative. INTRODUCTION
Over the last few years the interest of many scientists has been focused on mathematical analysis of races and especially on those relating distance to time. The time may be calculated after integrating the equations of racer’s motion. However, in such an approach, we face a difficulty-these equations depend on unknown functions representing the fact that the racer may
spend his energy in different ways; therefore, we can talk about the tactics of the race. This difficulty may be overcome in the following manner (Townend, 1984): -for short-distance races it is assumed that the competitor moves using his maxima1 abilities, -for long-distance races it is assumed that his velocity is constant. The first assumption is not questionable. The second one may be brought in question because the typical race contains acceleration immediately after the start of the race and at the end, where at these junctures the velocity certainly varies. Furthermore, the wind velocity and wind heading cannot be treated as constant because the trace may change its direction. The local slope of the trace may vary too. All these facts induce us to suppose that the racer’s optimal velocity varies with the distance. Evaluation of these variations is the topic of this work. In this paper, it is assumed that the most important measure of physiological cost is the work done against the external resistive forces over the whole distance of the race. The energy can be supplied by the competitor with variable intensity via proper variation of the propulsive force magnitude, and this propulsive force (or the actual power output) may be treated as a control of the system. The cyclist’s tactics during the race depends on controlling the propulsive force magnitude to achieve the highest mean speed for a given amount of propulsive energy, or to minimize the time of covering the whole distance. In this paper other equivalent formulations will also be given. The optimal speed variations of the competitor are determined by formulating and solving a mathematical problem in optima1 control theory. Equations of the rider’s motion are based on Newton’s second law. MODEL
The main assumptions of the mode1 of rider-bicycle motion are:
Accepted in final form 4 May 1993.
205
206
R. MARO~~SKI
(1) The competitor with the cycle is regarded as a particle (Fig. 1). This assumption results from the fact that linear dimensions of the rider and the cycle are small in comparison with the distance. (2) The profile of the route may be described by a known smooth function h(x). The angle y between the tangent to this curve and the horizontal is small, and the radius of curvature of the curve h(x) is large at every point of the path. (3) The maximal power output N is a function of the actual rider’s velocity u. During the course of the race, an athlete makes a large number of crank revolutions. In the model, the fluctuations of the magnitude of the maximal power output due to different crank angles are neglected, and N takes an average value over one crank cycle. This magnitude should be determined experimentally on the cycle ergometer for each individual. (4) The air resistance is a function of the competitor’s velocity and the velocity of the wind. It is well known that the aerodynamic drag is minimal when the cyclist sits on the saddle and adopts the conventional crouched racing position. Other positions are nonoptimal. However, variations of rider’s position during climbing, finishing, etc., may be included into the model. (5) The rolling and bearing resistance F is a given function of the coordinate x (covered distance). The rolling resistance depends substantially on the inflation pressure of tires and on the characteristics of road surface and tires. For practical purposes, it is proportional to the overall weight of the cyclist with the bike and is often assumed to be constant (Whitt, 1971; Di Prampero et al., 1979; de Groot et al., 1990). The bearing resistance is small in comparison with the rolling resistance although it can be included under F (Pugh, 1974). In this paper, F depends on the coordinate x (covered distance) and therefore variations of the rolling resistance due to different road surfaces may be included.
(6) The cyclist does not use brakes for reducing his speed. This assumption is made for brevity. Using brakes is always connected with irreversible losses of energy, and this is nonoptimal. However, including braking into the model is possible. FORMULATION OF THE PROBLEMS
The cyclist of mass m (with the cycle) moves on the route. The profile of the route is described by the function h(x). It joins the initial point A and the final point B. The competitor moves in the presence of the wind. The wind velocity V, may be constant or a given function of the coordinate x. Two problems are formulated. Problem 1: For the given time tB of covering the whole distance from point A to point B, we have to find a velocity of the cyclist u(x), that minimizes the cyclist’s external energy expenditure E, during the race. Problem 2: For the given amount of energy EB, which can be used for covering the distance, we have to find a velocity u(x) that minimizes the time tg. It should be noted that in both problems there are some constraints: the time tB in problem 1 and the energy EB in problem 2 are necessary constraints for proper formulation of both problems. In calculus of the problems like these are called variations, isoperimetric problems. Equations of the cyclist’s motion in the natural coordinate system (7, n) are dv
mdt=T-D-F-mgsiny, V2
m-=mgcosy-R, r
(14
(lb)
where m is the mass of the cyclist with the bicycle, v is the velocity, t is the time, T is the propulsive force, D is the aerodynamic drag, F is the rolling and bearing resistance, g is the gravitational acceleration, y is the local inclination angle of the route, r is its radius of curvature, and R is the normal reaction of the ground. Under assumption (2) the equations of motion (1) may be simplified because the left-hand side of equation (lb) approaches zero and the following approximations are valid: sinyztgy=g,
cosyxl.
(2b)
Equation (lb) reduces to Rzmg.
Fig. 1. Particle model of a cyclist. The forces exerted on the rider-bicycle system are: D, aerodynamic drag; F, rolling and bearing resistance; R, normal reaction of the ground; T, propulsive force; rug, weight of the cyclist with the bicycle. Other symbols denote: y, local inclination angle of the route; v, cyclist’s velocity; h(x), function describing the profile of the route; (r, n). natural coordinate system.
(3)
The propulsive force Tin equation (la) is a function of maximal power output N(u) and the power setting V(O, l>,
207
Optimal velocity during cycling For the variable wind pattern, the aerodynamic drag D apearing in equation (la) is a function of the riders velocity v, the wind velocity V, and the coordinate x (van Ingen Schenau and Cavanagh, 1990). For the wind velocity vector parallel to the direction of movement it is given in the form D=d(x)(v-
I’,,,(x))~,
d(x) =OSp SC&),
(5b)
and (v- V,) is the velocity of the cyclist relative to air (cf. Di Prampero, 1979; Ward-Smith, 1985b). An assisting wind velocity V, aligned with the direction of cycling has a positive value, a velocity of the hsad wind-a negative value. The velocity V, and the rolling and bearing resistance F may be given functions of the coordinate x. The aerodynamic coefficient d may also be a given function of x. A situation like this occurs when the cyclist changes his position during the race. The horizontal component of the cyclist’s velocity along x axis is as follows: cosy.
This equation, applying the simplification be rewritten in the form
(2b), may
dx -=v. dt
!py”_
VW(X))2
[
F(x) -~--V
mg dh v dx
“hd=“li,
dx
”
and the time tB from the integral Bl - dx.
tg= s
(11)
A”
Now, problems 1 and 2 may be formulated in a formal way. Problem 1: For the given value of tB expressed by integral (1 l), we have to find the cyclist’s velocity v(x) satisfying the state equation (8) with the boundary conditions (9), which minimizes integral (10). Problem 2: For the given value of EB expressed by the integral (lo), we have to find the cyclist’s velocity v(x) satisfying the state equation (8) with the boundary conditions (9), which minimizes integral (11). Both problems are typical in optimal control (or calculus of variations). Searching for the optimal solution is carried out under the assumption that one of the two integrals (time or energy) is constant, and the second one is minimized. The solution method is based on the fact that both problems 1 and 2 may be reduced to the extremization of linear integrals depending on two variables x and v, thus Miele’s method may be applied.
1
“A,
It follows from the Appendix that both problems 1 and 2 may be solved applying Miele’s method. The solution will be discussed for problem 1. Since the solution of problem 2 is similar, detailed discussion of such a case will be omitted. Only the basic results will be given. Problem 1
We can use the state equation (8) for eliminating the control variable r] from integral (10). Now the minimized energy expenditure takes the form
Boundary conditions for the state equation are v(O)=
A
SOLUTION
Now, we can rearrange the equation of the cyclist’s motion (la). Applying equations (4), (5a), (2a) and (7) we have the so-called state equation ;;_; ---
s
B-N(“h
Eg=
(5a)
where
dx X=v
integral
(94
Eg=
dx + $(V) da,
(12)
0(x,v)=d(x)Cv-V,(x)12+F(x)+mg~,
(13)
4(x,
V)
(W
where the initial (vA) and final (vg) velocities are given. We assume that the velocity v is greater than zero everywhere. It follows from the fact that for velocity approaching zero the right-hand side of equation (8) becomes singular. The final velocity vg is the competitor’s velocity at finish and it is usually relatively high. It is caused by the fact that athletes avoid excessive exertion in the early stages and in the middle of the race. Excessive exertion activates the glycolytic mechanism, characterized by the production of lactic acid. It is associated with considerable physical discomfort. Therefore, moving with high speed is delayed to the final stage of the race. The energy needed to overcome the external resistive forces over the given distance results from the
where
$(v)=mv.
(14)
The isoperimetric constraint (11) may be rewritten in the form rri= ;&(u)dx+$r(X.r)dv, s
(15)
where 1 41(v)=;’
J/,(x,
e=o,
(16)
(17)
R. MAROASKI
208
and the value of tB is given. Applying Miele’s method we can find that, for relatively low initial cyclist’s velocities, the distance of the race may be divided into three stages. (1) The early stage of the race (acceleration), where the competitor moves with his maximal power output (q = 1). On the xv-plane this stage is represented by the arc on the border of admissible domain which joins the initial point A with the curve w,(x, o, A)=0 (cf. segment AF in Fig. 6). (2) The middle stage of the race (cruise), where the cyclist moves using his partial power output (0~ q < 1). On the xv-plane it is represented by the curve 0*(x, u, a)=0 (cf. segment FEG in Fig. 6). (3) The final stage of the race (finish), where for a given relatively high velocity us, the competitor moves again with his maximal power output (n= 1). On the xu-plane this stage is represented by the arc on the border of admissible domain which joins the curve w,(x, u, I) = 0 with the final point B (cf. segment GB in Fig. 6). Since the cruise predominates, we will discuss the solution during the second stage. Optimal velocity of the cyclist during the cruise may be obtained from the equation eJ*(x, u, &)=O,
(18)
where o* is the augmented fundamental function, given by equation (Al@, and II is the constant Lagrange multiplier for problem 1. Then the following equation is valid: 2d(x)u2[u-
V,(x)]-&=O.
(19)
The optimal velocity during the cruise should satisfy such an equation. This is an algebraic equation of the third order depending on the variable x (covered distance); therefore, the optimal cruising velocity may be a function of x. Problem 2
Solution of the minimum-time problem for the given amount of cyclist’s external energy expenditure is similar to the solution of problem 1. Differences appear in the form of the performance index [equation (1l)] which is equal to tB= ;,(u)dx+$(x,u)du, s
(20)
where $(u)=$
(21)
l/%(x,u) = 0. The isoperimetric constraint EB=
)WvVx+MW, s
(22)
is (23)
where 0r(x,u)=d(x)
[u- Vw(x)]2+F(x)+mg~.
i+hl(u)=mu.
(24)
(25)
The optimal cruising velocity should now satisfy the equation 2d(x)u2[u-
v,(x)]-;=o, 2
(26)
where 12 is the constant Lagrange multiplier for problem 2. Equations (19) and (26) enables one to formulating the following corollaries: (I) The form of optimal solution is the same for both problems 1 and 2 during the cruise. This means there exists such a data set such that the solution of problem 1 is the solution of problem 2 at the same time. (2) For the constant wind velocity [ VW(x)= const.] and for the constant aerodynamic coefficient [d(x)=const.], the optimal velocity of the cyclist is constant during the cruise. This property is independent of the profile of the route h(x). This implies that the optimal velolcity of the cyclist should be the same during climbing and descent. The property is independent of the parameters of the model like mass m, aerodynamic coefficient Cn, etc., and also the rolling and bearing resistance function F(x). The second corollary results from the fact that the Lagrange multipliers II, 12 appearing in equations (19) and (26) are constant. Therefore, for V,=const. and a’=const. the cyclist’s velocity u has to be constant too. Since equations (19) and (26) do not involve functions describing the profile of the route h(x) and rolling and bearing resistance F(x), the cruising velocity u is independent of the coordinate x and therefore of the actual slope of terrain, Consider the following problem as an example. The cyclist should cover, for still-air conditions, the given distance 40 km during 1 h minimizing the external energy expenditure (problem 1). The profile of the route is shown in Fig. 2. The remaining data for calculations are (cf. Soden and Adeyefa, 1979; Di Prampero et al., 1979): uA= 5 km h- ’ (I.39 m s-r), L)B=55kmh-‘(15.28ms-‘),~=750N,N=906W, d = 0.189 kg m - t, F = 5 N. The admissible domain for these data is shown in Fig. 3. The borders of this domain E(X,u)=O are obtained in the manner described in Miele (1962) after integration of equation (8) forward (from xA to xa) and backward (from xg to x~) for q = 0 or n = 1, respectively. The solutions of the state equation (8) for all admissible values of qo( 0, 1) lie in this domain. It follows from the applied method that the optimal solution consists of three parts (bold line in Fig. 3):
209
Optimal velocity during cycling
Fig. 2. Profile of the route. It is described by equation (cos Znx/x, - 1). For h,., =400 m the maximal slope angle is less than 1.8” and the minimal radius of curvature r is greater than 200 km and therefore the assumptions of the model are satisfied. h = OSh,.,
Fig. 4. Wind pattern in relation to the trace. Symbol 6 denotes the angle describing the cyclist’s position relative to initial point A (start), II is the cyclist’s velocity and V. is the wind velcoity.
/ Admissible
/ 5 ms-1
0
10
20
30
vav = 40.0 km h-’
40
x tkml
Fig. 3. Optimal solution of problem 1 (minimization of cyclist’s external work for the given time of covering the distance, ta = 1 h). The undashed area is the admissible domain. Borders of this domain have been obtained after integrating the equation of cyclist’s motion forward and backward for maximal and minimal power setting (PI= 1 and 0, respcctively). It is assumed that the cyclist’s velocity is greater or equal to ~,,,~“=5km h-r everywhere. The optimal solution (bold line) consists of three parts: acceleration and finish, on the border of admissible domain where competitor uses his maximal abilities (q = I), and cruise, where cyclist’s velocity is constant. Corner points are switching points of the control function q.
-Acceleration, immediately after the start, where the cyclist moves with his maximal power output (rl= 1). -Cruise, where the optimal velocity of the cyclist is constant and his power output varies in the given interval. -Finish, where for relative high velocity tla the competitor moves again with his maximal power output (‘I = 1). The solutions have been shown (Fig. 3) for one value of the Lagrange multiplier 1, whereas the method generates a family of paths depending on 1. For this family, however, the isoperimetric constraint is also a function of 1, i.e. ta=ta(A).
(27)
Since the value of ta is known a priori (in the example
rs= 1 h), this equation allows us to determine the value of 1. It is clear from the diagram (Fig. 3) that for relatively long distances (say longer than 2Q km), the first and the third part of the solution may be neglected and the optimal velocity of the cyclist may be calculated dividing the distance by the time.
*IFig. 5. Optimal cruising velocities for the given wind pattern and different wind velocities VW. The optimal velocities of the cyclist are not constant here.
Consider another problem. Cyclist travels a circular horizontal trace with the given average velocity v,,=40.0kmh-’ (11.1 ms-‘). We assume that v~=v~=v.~ (this is a flying start or the competitor should make many laps). During the competition the wind blows with the constant given veloicity V,. The heading of the wind relative to the trace (but not to the cyclist) is constant too. The wind pattern is shown in Fig. 4. The question is how the cyclist should change his velocity to minimize the external work done during one lap. For d=0.189 kgm- ’ and for different values of the wind velocity V,, the optimal cruising velocities are shown in Fig. 5. (In the above example the value of d has been regarded as constant; however, it changes with the angle 6. The author did not have the necessary data of the aerodynamic coefficient of the air resistance Cn as a function of wind heading for the cyclist-bicycle system.) Optimal cruising velocities have been obtained from an equation similar to equations (19) and (26). This equation has been solved with respect to v for different values of 6. (It should be remembered that the air resistance vector is aligned to the relative wind velocity vector and therefore we should estimate this air resistance vector and then project it onto the tangent to the route.) This example visualizes the above statement that the optimal cruising velocity is constant only for the constant wind velocity VW(x)= const. (cf. corollary 2).
210
R. MARO~SKI
If the wind velocity varies, then it causes the variations of cruising velocity. DISCUSSION
In this paper two optimal control problems have been presented: minimizing the cyclist’s external energy expenditure for covering the given distance in the given time and minimizing the time for the given amount of the cyclist’s external energy expenditure. For both problems the optimal solutions consist of parts where the competitor moves either with the maximal power output or with the partial power output, The part corresponding to cruise predominates. Although the optimal control problems presented in the paper have been formulated for cycling, the results may be generalized for other human locomotion problems. For example, the following problem is considered in Sanderson and Martindale (1986). The rower goes by boat in the absence of streams. He should use the least amount of energy to travel the given distance during specified time (or, in other words, to maintain his average speed). It is assumed that the supplied energy must be equal to the energy dissipated due to the drag of water. The effect of inertia of the rower-boat system is neglected. Applying calculus of variations, it is proved that the boat moving with constant velocity uses the least amount of energy. This result may be obtained in a simple way from the presented model of motion including the inertia of the rower-boat system. We only need to put in the cyclist’s equation of motion, the functions h(x), F(x) and VW(x)identically equal to zero. The result obtained for the more general model of the rower-boat system is the same as that for the simple model. The constant boat velocity is optimal during the cruise. In Keller (1973, 1974) the minimum-time problem for a competitive running over a given distance is analyzed applying calculus of variations (cf. problem 2). In these papers the runner’s motion on a flat ground in the absence of wind is considered. The model of runner’s motion includes a resistive force linearly depending on the velocity. Instead of the isoperimetric constraint representing the given cyclist’s energy expenditure, the simple model of the oxygen balance is considered. In spite of the differences appearing in the equations of motion and in the boundary conditions (in Keller’s model the final velocity is not specified), the essential result is the same, i.e. the optimal velocity is constant during the cruise. [After reformulating of Keller’s problem it may also be solved applying Miele’s method (Marohski, 1992).] Behncke (1987) generalizes Keller’s model of a competitive run. The optimal control method is based on Filipov’s existence theorem (Filipov, 1962). From this method it follows that in general the distance may be divided into four (not three) parts. One of them is excluded from considerations based on intuition.
Such a fact cannot be deduced from the theory. Behncke has applied his model for prediction of world records in competitive running and swimming. Cooper (1990) adopted Behncke’s model and method for wheelchair athletics. In both papers, however, the variable trace slope and wind assistance were not considered. All models mentioned above are based on Newton’s second law. The equations obtained in such a manner have clear physical interpretation. However, they do not predict all performances observed during real races. For example, a sprinter running 100 m achieves peak velocity in the middle stages of the race, whereas from the models based on Newton’s law (and for constant propulsive force) it follows that the sprinter should attain maximal speed at the end of the race (Keller, 1973, 1974). To overcome this discrepancy Ward-Smith (1985a) proposed a new model based on the first law of thermodynamics. In another paper (Ward-Smith, 1985b), the analysis of running performance, originally derived for still-air conditions (Ward-Smith, 1985a) is extended to account for the effects of favorable and adverse winds. However, in both papers the analysis does not apply optimal control methods (calculus of variations). Some parameters of the run are assumed a priori and then the equations of motion are integrated. For a sprint it is assumed that the competitor moves at his maximal power output; for long distance events the speed is regarded to be constant. The velocity of assisting wind is constant too. The discussion which took place after publication of Ward-Smith’s model (van Ingen Schenau and Hollander, 1987; Ward-Smith, 1987) proved that creation of the reasonable competitor’s motion model, including the physiological side of the endurance activity, was a rather complex enterprise; needed a multidisciplinary approach. Inclusion of optimal control methods complicates the problem. Even for reasonable models, a smooth enough optimal solution may simply not exist. For example, in the minimum-fuel consumption problem the optimal controllers chatter between given values with the frequency approaching infinity (Gilbert, 1976; Zagalsky et al., 1971). Such control may give improved cruise performances in comparison with the steady-state cruise. It disagrees with intuitive understanding of the phenomenon. Moreover, there are some questions concerning the accuracy of the model working under such an unstable regime. In the problems presented in this paper, certain assumptions about the model of rider-bicycle motion have been made. Some of them are imposed by the applied method of the solution of the problems and they are very restrictive. Others are imposed for the sake of brevity of the paper and they may be attenuated in a simple way. For example, the variation of rider’s position [see assumption (4)] may be considered via the variations of the aerodynamic coefficient d. This coefficient should be given a priori as
I!11
Optimal velocity during cycling a function of the coordinate
x. It should be minimal when the cyclist adopts the conventional crouched racing position and it should increase when the athlete stands on the pedals. In such a case corollary 2 is not valid, but the method may be applied. The same refers to assumption (5), where the dependence of rolling and bearing resistance F on the cyclist’s velocity u has been neglected. Assuming that the rolling and bearing resistive force F is a function of the coordinate x and the velocity u (cf. Kyle, 1979), we have a new equation for the cruising velocity [cf. equation (19) for problem l] Zd(x)u*[o-Y,(x),-l,+gv*=O,
(28)
where aF/au is the partial derivative of the resistive force F with respect to u. If aF/au is independent of x, corollary 2 is valid. Including braking in the model seems to be also possible [see assumption (6)]. Let us consider the problem in which the rider’s velocity should be less than or equal to a given velocity, i.e. v < v,,,(x).
(29)
v,,,(x) is the maximal admissible velocity and it may be a given function of the coordinate x. In such a case, the borders of admissible domain on the xv-plane are obtained not only from integration of the equation of motion for q=O or q= 1. The upper part of the domain is limited by the curve u=umaX(x). Using brakes is necessary if this maximal velocity v,,,(x) is achieved from integration of equation of motion for coasting down, then r]=O. The admissible domain splits into two (or more) areas joined by the curve u=v,,,(x). The problem decouples into two (or more) problems with given initial and final conditions. The solution method of each of them is the same as the method presented in the paper. The performance index is the sum of the indices for these subproblems and the index for the subarc v=u,,,(x). [On the subarc v=u,,,(x) the performance index cannot be optimized. It results from the conditions u = urnaX( q = 0. The braking force may be calculated from the equation of motion.] In this paper an attempt has been made to solve two problems dealing with sport and recreational cycling: minimizing the amount of cyclist’s external energy expenditure for covering the given distance in the given time and minimizing the time for the given amount of cyclist’s energy expenditure. Miele’s method demonstrates that the solutions of these problems are similar. The optimal cruising velocity is constant and it is independent of the profile of the route and a function of rolling resistance. This result is consistent with the solutions given by other authors (Behncke, 1987; Cooper, 1990; Keller, 1973, 1974; Sanderson and Martindale, 1986). The proposed method enables one to solve these two problems for wind assistance. It seems that the results presented in
this paper are difficult to be foreseen using simple intuition or an experiment. Therefore, proper application of the mathematical theory is the best way to solve the problems like these. A complete treatment of the problems is not envisaged here. However, it is hoped that this treatment will prompt others to select models and methods necessary for further progress. REFERENCES
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Gilbert, E. G. (1976) Vehicle cruise: improved fuel economy by periodic control. Automatica 12, 159-166. de Groot, G., Aben, P. and Hoefnagels, K. (1990) Air friction and rolling resistance during cycling. Proc. XIllth dnt. Congress on Biomechanics. Book of Abstracts, 9-13 December, pp. 56-57. The University of Western Australia,
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Townend, M. S. (1984) Mathematics in Sport. pp. 23-28. Ellis Horwood, Chichester, U.K. Ward-Smith, A. J. (1985a) A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world class athletes. J. Biomechanics 18, 337-349. Ward-Smith, A. J. (1985b) A mathematical analysis of the influence of adverse and favourable winds on sprinting. J. Biomechanics 18, 351-357.
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Whitt, F. R. (1971) A note on the estimation of the energy expenditure of sporting cyclists. Ergonomics 14,419-424. Zagalsky, N. R., Irons, R. P. and Schultz, R. L. (1971) Energy state approximation and minimum-fuel fixed-range trajectories. J. Aircraft 8.488-490.
”
APPENDIX
The method of extremization of linear integrals by Green’s theorem was worked out by Miele (1962) at the beginning of the 1950s and successfully applied to many problems in dynamics of aircraft and rockets. It enables one to find, in a nonconventional way, optimal solutions for performance indices linear with respect to derivative. The simplified version of this method has been recalled in the paper dealing with skiing downhill, where the so-called fundamental function has the same sign everywhere within the admissible domain (Marotiski, 1990). Advantages of Miele’s method are noticeable for a more complicated case, when fundamental function changes its sign in the admissible domain. (In optimal control theory this case is equivalent to the problem with the so-called singular control.) A short description of this method will be given now. Consider the minimization problem of the integral which is linear with respect to derivative J=
s
1 C4k
WI dx,
d+W
(Al)
where 4 and $ are known functions of their arguments (x, u). Subscripts A and B refer to the initial and final points. Since u’=du/dx, integral (Al) may be rewritten in the form J=
s
:4(x,
u) dx+$(x,
u)du.
&(X,u)=O.
(A3) Furthermore, we assume that the initial point A and the final point B belong to the boundary of that region (see Fig. 6), UA)=&(XB,
c X
Fig. Al. Admissible domain for the case when the fundamental function o changes its sign. The curve ACEDB is an arbitrarily taken curve joining the initial point A and the final point B. AFEGB is the arc minimizing integral (Al), and FEG is the so-called singular arc.
where symbol
denotes cyclic integral. Notice that the
circuit ACEFA is travelled in the counterclockwise direction and the circuit EDBGE in the clockwise direction. Now, we can apply Green’s theorem to transform the cyclic integrals into the surface integrals (we assume that the functions I#, $ and their first partial derivatives are continuous everywhere in a and 8): JACEDB -JAFEGB
= J.r
642)
We assume that the class of investigated arcs (admissible domain) is contained within the region limited by the closed curve
E(XA,
I
us)=@
(A4) We additionally assume that admissible domain is divided into two regions by a curve described by the equation 0(x, u)=O,
(As) where w(x, u) is called fundamental function and it is defined as
-jI
J ACEDB - J AFEGB
=
ddx+$du c J ACEDB
-
c
odxdu,
(A@
P
where o(x, u) is fundamental function and it is given by equation (A6). Symbols a and /l denote areas bounded by the curves ACEFA and EDBGE, respectively. In the area a the inequality o > 0 is satisfied and w c 0, in the area 1; therefore, the following inequality is valid: J ACEDB >
JAFEGB . (A9) In conclusion, since the curve ACEDB has been taken arbitrarily, one can say that the curve AFEGB minimizes integral (A2) Cor WI. A simple modification of the previous problem is possible when the extremized integral (Al) [or equation (A2)] satisfies not only conditions (A3) and (A4) but also linear isoperimetric constraint C=
In both regions the sign of w is defined as in Fig. Al, negative to the left and positive to the right. We consider the admissible path ACEDB, which intersects the curve w=O at an arbitrarily specified point E (see Fig. Al). Now, we can find the difference between the line integrals (A2) for the paths ACEDB and AFEGB, which is given by
wdxdu D
I
:C~~(x.u)+II(x,u)dldx
WJ)
or equivalent constraint C=
B&(x.u)dx+$~(x,u)du, sA
(All)
where 4, and #i are known functions of variables x and u, and C is a prescribed constant. Condition (Al 1) is equivalent to the relationship o= s *cans
ddx+JIdu
$1 dx+JI, dus AFEGB
di dx+S,
do
J AFEGB
=&
J ACEFA
+i
=
+dx+$du,
J EDBGE
4,
I
4dx++du (A7)
ACEFA
dx+$,
du+
-~ i
& dx+JI, du.
(A121
EDBGE
Constraint (A12) may be satisfied after introducing an undetermined constant Lagrange multiplier I and combining equations (A7) and (A12) linearly. We can obtain the follow-
213
Optimal velocity during cycling ing result: V
J ACEDB - J AFEGB
=
4, dx+ti, PACEFA
+
I
t
dv
4, dx+ti, dv,
(A13)
+*=*+A*,.
(Af4)
EDBGE
where 9*=dJ+Ar#Q,
After employing Green’s theorem, the difference between the line integrals becomes I J .4CEDB - JAFEOB
w+(x.
= SI
L1
v, A)
dx dv
-
w,(x, v, I) dx do, (A15) ss B where w., denotes the augmented fundamental function
From the similarity of equations (AS) and (AlS), we can conclude that the previous results are applicable to the present problem. The differences appearing are listed below: (a) The fundamental function w must be replaced by the augmented function o*.
t x
Fig. A2. Optimal solution for the isoperimetric problem. The family of curves w.Jx, v, I)=0 represents the singular arcs for different given values of the constant Lagrange multiplier 1. (b) In the xv-plane the equation w*(x, u, I)=0 of the isoperimetric problem represents not a curve but a family of curves, i.e. a curve for each value of the Lagrange multiplier I, (see Fig. A2). The particular value of 1 may be determined from the boundary conditions and the given isoperimetric constraint. For further details, see Miele (1962).
J. Bmnechonrs Vol. 21, No. 2. PP. 1X05-213, 1994. Printed in GreatBritain
ON OPTIMAL
0
VELOCITY RYSZARD
1993
16.00 + .oO
Pergamon PressLtd
DURING CYCLING
MAROI;ISKI
Institute of Aeronautics and Applied Mechanics, ul. Nowowiejska 24, 00-665 Warsaw, Poland Abstract-This paper focuses on the solution of two problems related to cycling. One is to determine the velocity as a function of distance which minimizes the cyclist’s energy expenditure in covering a given distance in a set time. The other is to determine the velocity as a function of the distance which minimizes time for fixed energy expenditure. To solve these problems, an equation of motion for the cyclist riding over arbitrary terrain is written using Newton’s second law. This equation is used to evaluate either energy expenditure or time, and the minimization problems are solved using an optimal control formulation in conjunction with the method of Miele [Optimization Techniques with Applications to Aerospace Systems, pp. 69-98 (1962) Academic Press, New York]. Solutions to both optimal control problems are the same. The solutions are illustrated through two examples. In one example where the relative wind velocity is zero, the optimal cruising velocity is constant regardless of terrain. In the second, where the relative wind velocity fluctiates, the optimal cruising velocity varies. NOMENCLATURE initial point final point prescribed constant aerodynamic coefficient of air resistance aerodynamic drag cyclist’s external energy expenditure (external work) rolling and bearing resistance gravitational acceleration profile of route performance index mass of cyclist with bicycle maximal power output normal reaction of the ground radius of curvature frontal projected area of the cyclist propulsive force time velocity wind velocity coordinate (approximately the covered distance)
local inclination angle angle describing cyclist’s position relative to initial point border of admissible domain power setting Lagrange multiplier known function of x and v known function of x and o density of the air fundamental function derivative. INTRODUCTION
Over the last few years the interest of many scientists has been focused on mathematical analysis of races and especially on those relating distance to time. The time may be calculated after integrating the equations of racer’s motion. However, in such an approach, we face a difficulty-these equations depend on unknown functions representing the fact that the racer may
spend his energy in different ways; therefore, we can talk about the tactics of the race. This difficulty may be overcome in the following manner (Townend, 1984): -for short-distance races it is assumed that the competitor moves using his maxima1 abilities, -for long-distance races it is assumed that his velocity is constant. The first assumption is not questionable. The second one may be brought in question because the typical race contains acceleration immediately after the start of the race and at the end, where at these junctures the velocity certainly varies. Furthermore, the wind velocity and wind heading cannot be treated as constant because the trace may change its direction. The local slope of the trace may vary too. All these facts induce us to suppose that the racer’s optimal velocity varies with the distance. Evaluation of these variations is the topic of this work. In this paper, it is assumed that the most important measure of physiological cost is the work done against the external resistive forces over the whole distance of the race. The energy can be supplied by the competitor with variable intensity via proper variation of the propulsive force magnitude, and this propulsive force (or the actual power output) may be treated as a control of the system. The cyclist’s tactics during the race depends on controlling the propulsive force magnitude to achieve the highest mean speed for a given amount of propulsive energy, or to minimize the time of covering the whole distance. In this paper other equivalent formulations will also be given. The optimal speed variations of the competitor are determined by formulating and solving a mathematical problem in optima1 control theory. Equations of the rider’s motion are based on Newton’s second law. MODEL
The main assumptions of the mode1 of rider-bicycle motion are:
Accepted in final form 4 May 1993.
205
206
R. MARO~~SKI
(1) The competitor with the cycle is regarded as a particle (Fig. 1). This assumption results from the fact that linear dimensions of the rider and the cycle are small in comparison with the distance. (2) The profile of the route may be described by a known smooth function h(x). The angle y between the tangent to this curve and the horizontal is small, and the radius of curvature of the curve h(x) is large at every point of the path. (3) The maximal power output N is a function of the actual rider’s velocity u. During the course of the race, an athlete makes a large number of crank revolutions. In the model, the fluctuations of the magnitude of the maximal power output due to different crank angles are neglected, and N takes an average value over one crank cycle. This magnitude should be determined experimentally on the cycle ergometer for each individual. (4) The air resistance is a function of the competitor’s velocity and the velocity of the wind. It is well known that the aerodynamic drag is minimal when the cyclist sits on the saddle and adopts the conventional crouched racing position. Other positions are nonoptimal. However, variations of rider’s position during climbing, finishing, etc., may be included into the model. (5) The rolling and bearing resistance F is a given function of the coordinate x (covered distance). The rolling resistance depends substantially on the inflation pressure of tires and on the characteristics of road surface and tires. For practical purposes, it is proportional to the overall weight of the cyclist with the bike and is often assumed to be constant (Whitt, 1971; Di Prampero et al., 1979; de Groot et al., 1990). The bearing resistance is small in comparison with the rolling resistance although it can be included under F (Pugh, 1974). In this paper, F depends on the coordinate x (covered distance) and therefore variations of the rolling resistance due to different road surfaces may be included.
(6) The cyclist does not use brakes for reducing his speed. This assumption is made for brevity. Using brakes is always connected with irreversible losses of energy, and this is nonoptimal. However, including braking into the model is possible. FORMULATION OF THE PROBLEMS
The cyclist of mass m (with the cycle) moves on the route. The profile of the route is described by the function h(x). It joins the initial point A and the final point B. The competitor moves in the presence of the wind. The wind velocity V, may be constant or a given function of the coordinate x. Two problems are formulated. Problem 1: For the given time tB of covering the whole distance from point A to point B, we have to find a velocity of the cyclist u(x), that minimizes the cyclist’s external energy expenditure E, during the race. Problem 2: For the given amount of energy EB, which can be used for covering the distance, we have to find a velocity u(x) that minimizes the time tg. It should be noted that in both problems there are some constraints: the time tB in problem 1 and the energy EB in problem 2 are necessary constraints for proper formulation of both problems. In calculus of the problems like these are called variations, isoperimetric problems. Equations of the cyclist’s motion in the natural coordinate system (7, n) are dv
mdt=T-D-F-mgsiny, V2
m-=mgcosy-R, r
(14
(lb)
where m is the mass of the cyclist with the bicycle, v is the velocity, t is the time, T is the propulsive force, D is the aerodynamic drag, F is the rolling and bearing resistance, g is the gravitational acceleration, y is the local inclination angle of the route, r is its radius of curvature, and R is the normal reaction of the ground. Under assumption (2) the equations of motion (1) may be simplified because the left-hand side of equation (lb) approaches zero and the following approximations are valid: sinyztgy=g,
cosyxl.
(2b)
Equation (lb) reduces to Rzmg.
Fig. 1. Particle model of a cyclist. The forces exerted on the rider-bicycle system are: D, aerodynamic drag; F, rolling and bearing resistance; R, normal reaction of the ground; T, propulsive force; rug, weight of the cyclist with the bicycle. Other symbols denote: y, local inclination angle of the route; v, cyclist’s velocity; h(x), function describing the profile of the route; (r, n). natural coordinate system.
(3)
The propulsive force Tin equation (la) is a function of maximal power output N(u) and the power setting V(O, l>,
207
Optimal velocity during cycling For the variable wind pattern, the aerodynamic drag D apearing in equation (la) is a function of the riders velocity v, the wind velocity V, and the coordinate x (van Ingen Schenau and Cavanagh, 1990). For the wind velocity vector parallel to the direction of movement it is given in the form D=d(x)(v-
I’,,,(x))~,
d(x) =OSp SC&),
(5b)
and (v- V,) is the velocity of the cyclist relative to air (cf. Di Prampero, 1979; Ward-Smith, 1985b). An assisting wind velocity V, aligned with the direction of cycling has a positive value, a velocity of the hsad wind-a negative value. The velocity V, and the rolling and bearing resistance F may be given functions of the coordinate x. The aerodynamic coefficient d may also be a given function of x. A situation like this occurs when the cyclist changes his position during the race. The horizontal component of the cyclist’s velocity along x axis is as follows: cosy.
This equation, applying the simplification be rewritten in the form
(2b), may
dx -=v. dt
!py”_
VW(X))2
[
F(x) -~--V
mg dh v dx
“hd=“li,
dx
”
and the time tB from the integral Bl - dx.
tg= s
(11)
A”
Now, problems 1 and 2 may be formulated in a formal way. Problem 1: For the given value of tB expressed by integral (1 l), we have to find the cyclist’s velocity v(x) satisfying the state equation (8) with the boundary conditions (9), which minimizes integral (10). Problem 2: For the given value of EB expressed by the integral (lo), we have to find the cyclist’s velocity v(x) satisfying the state equation (8) with the boundary conditions (9), which minimizes integral (11). Both problems are typical in optimal control (or calculus of variations). Searching for the optimal solution is carried out under the assumption that one of the two integrals (time or energy) is constant, and the second one is minimized. The solution method is based on the fact that both problems 1 and 2 may be reduced to the extremization of linear integrals depending on two variables x and v, thus Miele’s method may be applied.
1
“A,
It follows from the Appendix that both problems 1 and 2 may be solved applying Miele’s method. The solution will be discussed for problem 1. Since the solution of problem 2 is similar, detailed discussion of such a case will be omitted. Only the basic results will be given. Problem 1
We can use the state equation (8) for eliminating the control variable r] from integral (10). Now the minimized energy expenditure takes the form
Boundary conditions for the state equation are v(O)=
A
SOLUTION
Now, we can rearrange the equation of the cyclist’s motion (la). Applying equations (4), (5a), (2a) and (7) we have the so-called state equation ;;_; ---
s
B-N(“h
Eg=
(5a)
where
dx X=v
integral
(94
Eg=
dx + $(V) da,
(12)
0(x,v)=d(x)Cv-V,(x)12+F(x)+mg~,
(13)
4(x,
V)
(W
where the initial (vA) and final (vg) velocities are given. We assume that the velocity v is greater than zero everywhere. It follows from the fact that for velocity approaching zero the right-hand side of equation (8) becomes singular. The final velocity vg is the competitor’s velocity at finish and it is usually relatively high. It is caused by the fact that athletes avoid excessive exertion in the early stages and in the middle of the race. Excessive exertion activates the glycolytic mechanism, characterized by the production of lactic acid. It is associated with considerable physical discomfort. Therefore, moving with high speed is delayed to the final stage of the race. The energy needed to overcome the external resistive forces over the given distance results from the
where
$(v)=mv.
(14)
The isoperimetric constraint (11) may be rewritten in the form rri= ;&(u)dx+$r(X.r)dv, s
(15)
where 1 41(v)=;’
J/,(x,
e=o,
(16)
(17)
R. MAROASKI
208
and the value of tB is given. Applying Miele’s method we can find that, for relatively low initial cyclist’s velocities, the distance of the race may be divided into three stages. (1) The early stage of the race (acceleration), where the competitor moves with his maximal power output (q = 1). On the xv-plane this stage is represented by the arc on the border of admissible domain which joins the initial point A with the curve w,(x, o, A)=0 (cf. segment AF in Fig. 6). (2) The middle stage of the race (cruise), where the cyclist moves using his partial power output (0~ q < 1). On the xv-plane it is represented by the curve 0*(x, u, a)=0 (cf. segment FEG in Fig. 6). (3) The final stage of the race (finish), where for a given relatively high velocity us, the competitor moves again with his maximal power output (n= 1). On the xu-plane this stage is represented by the arc on the border of admissible domain which joins the curve w,(x, u, I) = 0 with the final point B (cf. segment GB in Fig. 6). Since the cruise predominates, we will discuss the solution during the second stage. Optimal velocity of the cyclist during the cruise may be obtained from the equation eJ*(x, u, &)=O,
(18)
where o* is the augmented fundamental function, given by equation (Al@, and II is the constant Lagrange multiplier for problem 1. Then the following equation is valid: 2d(x)u2[u-
V,(x)]-&=O.
(19)
The optimal velocity during the cruise should satisfy such an equation. This is an algebraic equation of the third order depending on the variable x (covered distance); therefore, the optimal cruising velocity may be a function of x. Problem 2
Solution of the minimum-time problem for the given amount of cyclist’s external energy expenditure is similar to the solution of problem 1. Differences appear in the form of the performance index [equation (1l)] which is equal to tB= ;,(u)dx+$(x,u)du, s
(20)
where $(u)=$
(21)
l/%(x,u) = 0. The isoperimetric constraint EB=
)WvVx+MW, s
(22)
is (23)
where 0r(x,u)=d(x)
[u- Vw(x)]2+F(x)+mg~.
i+hl(u)=mu.
(24)
(25)
The optimal cruising velocity should now satisfy the equation 2d(x)u2[u-
v,(x)]-;=o, 2
(26)
where 12 is the constant Lagrange multiplier for problem 2. Equations (19) and (26) enables one to formulating the following corollaries: (I) The form of optimal solution is the same for both problems 1 and 2 during the cruise. This means there exists such a data set such that the solution of problem 1 is the solution of problem 2 at the same time. (2) For the constant wind velocity [ VW(x)= const.] and for the constant aerodynamic coefficient [d(x)=const.], the optimal velocity of the cyclist is constant during the cruise. This property is independent of the profile of the route h(x). This implies that the optimal velolcity of the cyclist should be the same during climbing and descent. The property is independent of the parameters of the model like mass m, aerodynamic coefficient Cn, etc., and also the rolling and bearing resistance function F(x). The second corollary results from the fact that the Lagrange multipliers II, 12 appearing in equations (19) and (26) are constant. Therefore, for V,=const. and a’=const. the cyclist’s velocity u has to be constant too. Since equations (19) and (26) do not involve functions describing the profile of the route h(x) and rolling and bearing resistance F(x), the cruising velocity u is independent of the coordinate x and therefore of the actual slope of terrain, Consider the following problem as an example. The cyclist should cover, for still-air conditions, the given distance 40 km during 1 h minimizing the external energy expenditure (problem 1). The profile of the route is shown in Fig. 2. The remaining data for calculations are (cf. Soden and Adeyefa, 1979; Di Prampero et al., 1979): uA= 5 km h- ’ (I.39 m s-r), L)B=55kmh-‘(15.28ms-‘),~=750N,N=906W, d = 0.189 kg m - t, F = 5 N. The admissible domain for these data is shown in Fig. 3. The borders of this domain E(X,u)=O are obtained in the manner described in Miele (1962) after integration of equation (8) forward (from xA to xa) and backward (from xg to x~) for q = 0 or n = 1, respectively. The solutions of the state equation (8) for all admissible values of qo( 0, 1) lie in this domain. It follows from the applied method that the optimal solution consists of three parts (bold line in Fig. 3):
209
Optimal velocity during cycling
Fig. 2. Profile of the route. It is described by equation (cos Znx/x, - 1). For h,., =400 m the maximal slope angle is less than 1.8” and the minimal radius of curvature r is greater than 200 km and therefore the assumptions of the model are satisfied. h = OSh,.,
Fig. 4. Wind pattern in relation to the trace. Symbol 6 denotes the angle describing the cyclist’s position relative to initial point A (start), II is the cyclist’s velocity and V. is the wind velcoity.
/ Admissible
/ 5 ms-1
0
10
20
30
vav = 40.0 km h-’
40
x tkml
Fig. 3. Optimal solution of problem 1 (minimization of cyclist’s external work for the given time of covering the distance, ta = 1 h). The undashed area is the admissible domain. Borders of this domain have been obtained after integrating the equation of cyclist’s motion forward and backward for maximal and minimal power setting (PI= 1 and 0, respcctively). It is assumed that the cyclist’s velocity is greater or equal to ~,,,~“=5km h-r everywhere. The optimal solution (bold line) consists of three parts: acceleration and finish, on the border of admissible domain where competitor uses his maximal abilities (q = I), and cruise, where cyclist’s velocity is constant. Corner points are switching points of the control function q.
-Acceleration, immediately after the start, where the cyclist moves with his maximal power output (rl= 1). -Cruise, where the optimal velocity of the cyclist is constant and his power output varies in the given interval. -Finish, where for relative high velocity tla the competitor moves again with his maximal power output (‘I = 1). The solutions have been shown (Fig. 3) for one value of the Lagrange multiplier 1, whereas the method generates a family of paths depending on 1. For this family, however, the isoperimetric constraint is also a function of 1, i.e. ta=ta(A).
(27)
Since the value of ta is known a priori (in the example
rs= 1 h), this equation allows us to determine the value of 1. It is clear from the diagram (Fig. 3) that for relatively long distances (say longer than 2Q km), the first and the third part of the solution may be neglected and the optimal velocity of the cyclist may be calculated dividing the distance by the time.
*IFig. 5. Optimal cruising velocities for the given wind pattern and different wind velocities VW. The optimal velocities of the cyclist are not constant here.
Consider another problem. Cyclist travels a circular horizontal trace with the given average velocity v,,=40.0kmh-’ (11.1 ms-‘). We assume that v~=v~=v.~ (this is a flying start or the competitor should make many laps). During the competition the wind blows with the constant given veloicity V,. The heading of the wind relative to the trace (but not to the cyclist) is constant too. The wind pattern is shown in Fig. 4. The question is how the cyclist should change his velocity to minimize the external work done during one lap. For d=0.189 kgm- ’ and for different values of the wind velocity V,, the optimal cruising velocities are shown in Fig. 5. (In the above example the value of d has been regarded as constant; however, it changes with the angle 6. The author did not have the necessary data of the aerodynamic coefficient of the air resistance Cn as a function of wind heading for the cyclist-bicycle system.) Optimal cruising velocities have been obtained from an equation similar to equations (19) and (26). This equation has been solved with respect to v for different values of 6. (It should be remembered that the air resistance vector is aligned to the relative wind velocity vector and therefore we should estimate this air resistance vector and then project it onto the tangent to the route.) This example visualizes the above statement that the optimal cruising velocity is constant only for the constant wind velocity VW(x)= const. (cf. corollary 2).
210
R. MARO~SKI
If the wind velocity varies, then it causes the variations of cruising velocity. DISCUSSION
In this paper two optimal control problems have been presented: minimizing the cyclist’s external energy expenditure for covering the given distance in the given time and minimizing the time for the given amount of the cyclist’s external energy expenditure. For both problems the optimal solutions consist of parts where the competitor moves either with the maximal power output or with the partial power output, The part corresponding to cruise predominates. Although the optimal control problems presented in the paper have been formulated for cycling, the results may be generalized for other human locomotion problems. For example, the following problem is considered in Sanderson and Martindale (1986). The rower goes by boat in the absence of streams. He should use the least amount of energy to travel the given distance during specified time (or, in other words, to maintain his average speed). It is assumed that the supplied energy must be equal to the energy dissipated due to the drag of water. The effect of inertia of the rower-boat system is neglected. Applying calculus of variations, it is proved that the boat moving with constant velocity uses the least amount of energy. This result may be obtained in a simple way from the presented model of motion including the inertia of the rower-boat system. We only need to put in the cyclist’s equation of motion, the functions h(x), F(x) and VW(x)identically equal to zero. The result obtained for the more general model of the rower-boat system is the same as that for the simple model. The constant boat velocity is optimal during the cruise. In Keller (1973, 1974) the minimum-time problem for a competitive running over a given distance is analyzed applying calculus of variations (cf. problem 2). In these papers the runner’s motion on a flat ground in the absence of wind is considered. The model of runner’s motion includes a resistive force linearly depending on the velocity. Instead of the isoperimetric constraint representing the given cyclist’s energy expenditure, the simple model of the oxygen balance is considered. In spite of the differences appearing in the equations of motion and in the boundary conditions (in Keller’s model the final velocity is not specified), the essential result is the same, i.e. the optimal velocity is constant during the cruise. [After reformulating of Keller’s problem it may also be solved applying Miele’s method (Marohski, 1992).] Behncke (1987) generalizes Keller’s model of a competitive run. The optimal control method is based on Filipov’s existence theorem (Filipov, 1962). From this method it follows that in general the distance may be divided into four (not three) parts. One of them is excluded from considerations based on intuition.
Such a fact cannot be deduced from the theory. Behncke has applied his model for prediction of world records in competitive running and swimming. Cooper (1990) adopted Behncke’s model and method for wheelchair athletics. In both papers, however, the variable trace slope and wind assistance were not considered. All models mentioned above are based on Newton’s second law. The equations obtained in such a manner have clear physical interpretation. However, they do not predict all performances observed during real races. For example, a sprinter running 100 m achieves peak velocity in the middle stages of the race, whereas from the models based on Newton’s law (and for constant propulsive force) it follows that the sprinter should attain maximal speed at the end of the race (Keller, 1973, 1974). To overcome this discrepancy Ward-Smith (1985a) proposed a new model based on the first law of thermodynamics. In another paper (Ward-Smith, 1985b), the analysis of running performance, originally derived for still-air conditions (Ward-Smith, 1985a) is extended to account for the effects of favorable and adverse winds. However, in both papers the analysis does not apply optimal control methods (calculus of variations). Some parameters of the run are assumed a priori and then the equations of motion are integrated. For a sprint it is assumed that the competitor moves at his maximal power output; for long distance events the speed is regarded to be constant. The velocity of assisting wind is constant too. The discussion which took place after publication of Ward-Smith’s model (van Ingen Schenau and Hollander, 1987; Ward-Smith, 1987) proved that creation of the reasonable competitor’s motion model, including the physiological side of the endurance activity, was a rather complex enterprise; needed a multidisciplinary approach. Inclusion of optimal control methods complicates the problem. Even for reasonable models, a smooth enough optimal solution may simply not exist. For example, in the minimum-fuel consumption problem the optimal controllers chatter between given values with the frequency approaching infinity (Gilbert, 1976; Zagalsky et al., 1971). Such control may give improved cruise performances in comparison with the steady-state cruise. It disagrees with intuitive understanding of the phenomenon. Moreover, there are some questions concerning the accuracy of the model working under such an unstable regime. In the problems presented in this paper, certain assumptions about the model of rider-bicycle motion have been made. Some of them are imposed by the applied method of the solution of the problems and they are very restrictive. Others are imposed for the sake of brevity of the paper and they may be attenuated in a simple way. For example, the variation of rider’s position [see assumption (4)] may be considered via the variations of the aerodynamic coefficient d. This coefficient should be given a priori as
I!11
Optimal velocity during cycling a function of the coordinate
x. It should be minimal when the cyclist adopts the conventional crouched racing position and it should increase when the athlete stands on the pedals. In such a case corollary 2 is not valid, but the method may be applied. The same refers to assumption (5), where the dependence of rolling and bearing resistance F on the cyclist’s velocity u has been neglected. Assuming that the rolling and bearing resistive force F is a function of the coordinate x and the velocity u (cf. Kyle, 1979), we have a new equation for the cruising velocity [cf. equation (19) for problem l] Zd(x)u*[o-Y,(x),-l,+gv*=O,
(28)
where aF/au is the partial derivative of the resistive force F with respect to u. If aF/au is independent of x, corollary 2 is valid. Including braking in the model seems to be also possible [see assumption (6)]. Let us consider the problem in which the rider’s velocity should be less than or equal to a given velocity, i.e. v < v,,,(x).
(29)
v,,,(x) is the maximal admissible velocity and it may be a given function of the coordinate x. In such a case, the borders of admissible domain on the xv-plane are obtained not only from integration of the equation of motion for q=O or q= 1. The upper part of the domain is limited by the curve u=umaX(x). Using brakes is necessary if this maximal velocity v,,,(x) is achieved from integration of equation of motion for coasting down, then r]=O. The admissible domain splits into two (or more) areas joined by the curve u=v,,,(x). The problem decouples into two (or more) problems with given initial and final conditions. The solution method of each of them is the same as the method presented in the paper. The performance index is the sum of the indices for these subproblems and the index for the subarc v=u,,,(x). [On the subarc v=u,,,(x) the performance index cannot be optimized. It results from the conditions u = urnaX( q = 0. The braking force may be calculated from the equation of motion.] In this paper an attempt has been made to solve two problems dealing with sport and recreational cycling: minimizing the amount of cyclist’s external energy expenditure for covering the given distance in the given time and minimizing the time for the given amount of cyclist’s energy expenditure. Miele’s method demonstrates that the solutions of these problems are similar. The optimal cruising velocity is constant and it is independent of the profile of the route and a function of rolling resistance. This result is consistent with the solutions given by other authors (Behncke, 1987; Cooper, 1990; Keller, 1973, 1974; Sanderson and Martindale, 1986). The proposed method enables one to solve these two problems for wind assistance. It seems that the results presented in
this paper are difficult to be foreseen using simple intuition or an experiment. Therefore, proper application of the mathematical theory is the best way to solve the problems like these. A complete treatment of the problems is not envisaged here. However, it is hoped that this treatment will prompt others to select models and methods necessary for further progress. REFERENCES
Behncke, H. (1987) Optimization models for the force and energy in competitive sports. Math. Meth. appl. Sci. 9, 298-311. Cooper, R. A. (1990) A force/energy optimization model for wheelchair athletics. IEEE Trans. Systems Man Cyhernet. 20, 444-449.
Di Prampero, P. E., Cortili, G., Mognoni, P. and Saibene, F. (1979)Equation of motion of a cyclist. J. appl. Physiol. 47, 201-206.
Filipov, A. F. (1962). On certain questions in optimal control. SIAM J. Control 2, 76-84.
Gilbert, E. G. (1976) Vehicle cruise: improved fuel economy by periodic control. Automatica 12, 159-166. de Groot, G., Aben, P. and Hoefnagels, K. (1990) Air friction and rolling resistance during cycling. Proc. XIllth dnt. Congress on Biomechanics. Book of Abstracts, 9-13 December, pp. 56-57. The University of Western Australia,
Perth. van Ingen Schenau, G. J. and Cavanagh, P. R. (1990)Power equations in endurance sports. J. Biomechanics 23, 865-881.
van Ingen Schenau, G. J. and Hollander, A. P. (1987) Comment on “a mathematical theory of running” and the applications of this theory. J. Biomechanics 20, 91-93. Keller, J. B. (1973) A theory of competitive running. Phys. Today 26,42-47.
Keller, J. B. (1974) Optimal velocity in a race. Am. Math. Mon. 81, 474-480. Kyle, C. R. (1979) Reduction of wind resistance and power output of racing cyclists and runners travelling in groups. Ergonomics 22, 387-397.
Maroxiski, R. (1990) On optimal running downhill on skis. J. Biomechanics 23, 435-439.
Maroxiski, R. (1992) On “the theory of competitive running”. Zeszyty Naukowe Politechniki Slqskiej Mechanika z. 107, 25 l-256 (in Polish).
Miele, A. (1962) Extremization of linear integrals by Green’s theorem. In Optimization Techniques with Applications to Aerospace Systems (edited by Leitmann. G.), pp. 69-98. Academic Press, New York. Pugh, L. G. C. E. (1974) The relation of oxygen intake and speed in competition cycling and comparative observations on the bicycle ergonometer. J. Physiol. 241.795-808. Sanderson, B. and Martindale, W. (1986) Towards optimizing rowing technique. Med. Sci. Sports Exert. l&454-468. Soden, P. D. and Adeyefa, B. A. (1979) Forces applied to a bicycle during normal cycling. J. Biomechanics 12, 527-541.
Townend, M. S. (1984) Mathematics in Sport. pp. 23-28. Ellis Horwood, Chichester, U.K. Ward-Smith, A. J. (1985a) A mathematical theory of running, based on the first law of thermodynamics, and its application to the performance of world class athletes. J. Biomechanics 18, 337-349. Ward-Smith, A. J. (1985b) A mathematical analysis of the influence of adverse and favourable winds on sprinting. J. Biomechanics 18, 351-357.
Ward-Smith, A. J. (1987) Reply to “comment on a mathematical theory of running” and the applications of this theory”. J. Biomechanics 20, 94-95.
R. MAROASKI
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Whitt, F. R. (1971) A note on the estimation of the energy expenditure of sporting cyclists. Ergonomics 14,419-424. Zagalsky, N. R., Irons, R. P. and Schultz, R. L. (1971) Energy state approximation and minimum-fuel fixed-range trajectories. J. Aircraft 8.488-490.
”
APPENDIX
The method of extremization of linear integrals by Green’s theorem was worked out by Miele (1962) at the beginning of the 1950s and successfully applied to many problems in dynamics of aircraft and rockets. It enables one to find, in a nonconventional way, optimal solutions for performance indices linear with respect to derivative. The simplified version of this method has been recalled in the paper dealing with skiing downhill, where the so-called fundamental function has the same sign everywhere within the admissible domain (Marotiski, 1990). Advantages of Miele’s method are noticeable for a more complicated case, when fundamental function changes its sign in the admissible domain. (In optimal control theory this case is equivalent to the problem with the so-called singular control.) A short description of this method will be given now. Consider the minimization problem of the integral which is linear with respect to derivative J=
s
1 C4k
WI dx,
d+W
(Al)
where 4 and $ are known functions of their arguments (x, u). Subscripts A and B refer to the initial and final points. Since u’=du/dx, integral (Al) may be rewritten in the form J=
s
:4(x,
u) dx+$(x,
u)du.
&(X,u)=O.
(A3) Furthermore, we assume that the initial point A and the final point B belong to the boundary of that region (see Fig. 6), UA)=&(XB,
c X
Fig. Al. Admissible domain for the case when the fundamental function o changes its sign. The curve ACEDB is an arbitrarily taken curve joining the initial point A and the final point B. AFEGB is the arc minimizing integral (Al), and FEG is the so-called singular arc.
where symbol
denotes cyclic integral. Notice that the
circuit ACEFA is travelled in the counterclockwise direction and the circuit EDBGE in the clockwise direction. Now, we can apply Green’s theorem to transform the cyclic integrals into the surface integrals (we assume that the functions I#, $ and their first partial derivatives are continuous everywhere in a and 8): JACEDB -JAFEGB
= J.r
642)
We assume that the class of investigated arcs (admissible domain) is contained within the region limited by the closed curve
E(XA,
I
us)=@
(A4) We additionally assume that admissible domain is divided into two regions by a curve described by the equation 0(x, u)=O,
(As) where w(x, u) is called fundamental function and it is defined as
-jI
J ACEDB - J AFEGB
=
ddx+$du c J ACEDB
-
c
odxdu,
(A@
P
where o(x, u) is fundamental function and it is given by equation (A6). Symbols a and /l denote areas bounded by the curves ACEFA and EDBGE, respectively. In the area a the inequality o > 0 is satisfied and w c 0, in the area 1; therefore, the following inequality is valid: J ACEDB >
JAFEGB . (A9) In conclusion, since the curve ACEDB has been taken arbitrarily, one can say that the curve AFEGB minimizes integral (A2) Cor WI. A simple modification of the previous problem is possible when the extremized integral (Al) [or equation (A2)] satisfies not only conditions (A3) and (A4) but also linear isoperimetric constraint C=
In both regions the sign of w is defined as in Fig. Al, negative to the left and positive to the right. We consider the admissible path ACEDB, which intersects the curve w=O at an arbitrarily specified point E (see Fig. Al). Now, we can find the difference between the line integrals (A2) for the paths ACEDB and AFEGB, which is given by
wdxdu D
I
:C~~(x.u)+II(x,u)dldx
WJ)
or equivalent constraint C=
B&(x.u)dx+$~(x,u)du, sA
(All)
where 4, and #i are known functions of variables x and u, and C is a prescribed constant. Condition (Al 1) is equivalent to the relationship o= s *cans
ddx+JIdu
$1 dx+JI, dus AFEGB
di dx+S,
do
J AFEGB
=&
J ACEFA
+i
=
+dx+$du,
J EDBGE
4,
I
4dx++du (A7)
ACEFA
dx+$,
du+
-~ i
& dx+JI, du.
(A121
EDBGE
Constraint (A12) may be satisfied after introducing an undetermined constant Lagrange multiplier I and combining equations (A7) and (A12) linearly. We can obtain the follow-
213
Optimal velocity during cycling ing result: V
J ACEDB - J AFEGB
=
4, dx+ti, PACEFA
+
I
t
dv
4, dx+ti, dv,
(A13)
+*=*+A*,.
(Af4)
EDBGE
where 9*=dJ+Ar#Q,
After employing Green’s theorem, the difference between the line integrals becomes I J .4CEDB - JAFEOB
w+(x.
= SI
L1
v, A)
dx dv
-
w,(x, v, I) dx do, (A15) ss B where w., denotes the augmented fundamental function
From the similarity of equations (AS) and (AlS), we can conclude that the previous results are applicable to the present problem. The differences appearing are listed below: (a) The fundamental function w must be replaced by the augmented function o*.
t x
Fig. A2. Optimal solution for the isoperimetric problem. The family of curves w.Jx, v, I)=0 represents the singular arcs for different given values of the constant Lagrange multiplier 1. (b) In the xv-plane the equation w*(x, u, I)=0 of the isoperimetric problem represents not a curve but a family of curves, i.e. a curve for each value of the Lagrange multiplier I, (see Fig. A2). The particular value of 1 may be determined from the boundary conditions and the given isoperimetric constraint. For further details, see Miele (1962).