Original Article Proceedings of 7th ISEA CONFERENCE 2008

Biarritz, June 2-6, 2008

A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

Ankang Le1*, Lothar Litz2, Thomas Jaitner3

(1) (2) : Institute of Automatic Control, (3) : Department of Social Sciences, University of Kaiserslautern, Germany University of Kaiserslautern, Germany

+49 631 205 4459 / +49 631 205 4462 +49 631 205 4967 E-mail: {le, litz}@eit.uni-kl.de E-mail: jaitner@sowi.uni-kl.de

TOPICS: Bicycle, Virtual Reality & Computer application in Sports.

ABSTRACT: Determining the optimal exercise intensity is a crucial factor in cycling to improve performance and avoid overtraining. Novel sensor technologies allow to optimize the training not only for an individual cyclist but also for an entire group. A sensor-based Team Cycling Training System (TCTS) has been developed to optimize the group training in cycling. This system consists of three major parts: a hardware platform with basic sensors for training data acquisition, a wireless ad-hoc network that establishes the communication among multiple bicycles, and a control algorithm for the optimization of the training. The focus of this paper lies on the development of the control algorithm, a Model Predictive Controller (MPC). The MPC uses a cycling performance model to predict the physical work loads of the cyclists according to various conditions such as road profile, headwind, speed and position of cyclists within the group. Based on the predicted physical exercise loads, the MPC uses a cyclist individualized dynamic heart rate prediction model to determine the physiological load of each cyclist and regulates the group training by advising the cyclists to change the position in group, to adjust the group speed, or to split the group in such a way that each cyclist can meet his training plan as exactly as possible. Training sessions with two or four group members have been conducted under different conditions. The results of the trainings indicate that the TCTS with the MPC is an effective aid for the group training in cycling.

Key words: model predictive control, training optimization, cycling group training.

1- Introduction Determining the optimal exercise intensity is a crucial factor in cycling to improve performance. Low intensities will not result in the desired training effect, but too high intensities may cause overtraining or illness (Kuipers and Keizer, 1988). Therefore, it is important to monitor exercise intensities during training and competition. Typically, biomechanical parameters such as power, cadence and speed are used to quantify the external load. Among these parameters, the power exerted on the pedal can be considered as a direct and objective indicator of the external load (Coyle et al. 1991, MacIntosh et al. 2000, Stapelfeldt et al. 2006). To estimate the internal load or physical stress that results from an external load, the heart rate (HR) is a widely chosen parameter. The HR may change with the blood lactate concentration, hand (torso) position, temperature of the environment, altitude, training duration and so on. (Achten and Jeukendrup, 2003, Jeukendrup and van Diemen, 1998, Too 1990). However the study of Lucía et al. (2000) has confirmed that the values of the target HR generally remain stable for professional cyclists during the course of the season. Even though cycling is primarily known as an individual sport, teams play an important role in training and competition (Gregor and Conconi, 2000). In particular, team time trial (TTT) is a standard event in track and road races such as team pursuit and UCI Pro Tour TTT. Moreover, road cycling in groups is common in training. In typical team training, a group of cyclists covers a distance of up to 200 km with a varying road profile. The speed for all cyclists in the team is the same, but the power output depends on the position within the group. Due to the head wind the power output of the leading cyclist is up to 36% higher than the power output of subsequent cyclists (Neumann, 2000). To achieve the best training effects, each cyclist should ride with an individual exercise intensity that depends not only on factors such as cyclist’s physical capabilities and skills, bike aerodynamics, road surface and incline, head wind and temperature (Atkinson et al. 2003, Too 1990), but also on the position within the group. A sensor-based Team Cycling Training System (TCTS) has been developed to support the training of a group of cyclists (Litz et al. 2004, Jaitner et al. 2006, Le et al. 2007, Jaitner and Trapp, in print 2008). The objective of the TCTS is to optimize the group training such that each cyclist is as close to his individual predetermined exercise intensity as possible by changing the positions of the cyclists within the group or adjusting the group speed. The system consists of three major parts: a hardware platform with basic sensor technologies for training data acquisition, a communication middleware that establishes the

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7th ISEA CONFERENCE 2008 A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

communication among multiple bicycles, and a control algorithm for the optimization of the training. The focus of this paper lies on the control algorithm, a Model Predictive Controller (MPC). Based on a cycling performance model and a cyclist individualized HR prediction model (Le et al. in print 2008), the MPC predicts the physical and physiological loads of the cyclists online during training or competition. It optimizes the group training by minimizing the difference between actual and predicted HR values of the whole group. In section 2, we describe the TCTS system. The development of the MPC is presented in section 3. It is followed by the evaluation of the training results in section 4. We conclude this study in section 5.

2- The Team Cycling Training System (TCTS) The TCTS with four bicycles is shown in Figure 1. Each bicycle is equipped with an Ergomo™ Pro power meter (SG Sensortechnik GmbH & Co. KG, Germany) and an Ultra Mobile Personal Computer (UMPC). The communication between the Ergomo™ System and the UMPC is established via serial port (RS232). All UMPCs are connected to each other via an ad-hoc network formed by Wi-Fi technology (Jaitner and Trapp, in print 2008). The TCTS collects training status data dynamically and delivers them to a self-organizing automatic optimization algorithm which is running on the UMPC. Actual values as well as target values of heart rate, power, speed and cadence are displayed for each cyclist separately on his UMPC. The training instructions are also displayed on the panel guided by an audio signal simultaneously. These instructions may advise the cyclists to adjust the group speed or to change the position within the group. Additionally, all training data, instructions, information of group formation and training environment are stored on the UMPC for post-training analysis.

Sensor data: - Power - Speed - Cadence - Temperature - Air pressure - Altitude - Road grade

Wi-Fi

Ergomo RS232 UMPC

Sensor data HR sensor

HR

Ergomo RS232 UMPC

Sensor data HR sensor

HR

Ergomo RS232 UMPC

Sensor data HR sensor

HR

ErgomoRS232 UMPC

Sensor data HR sensor

HR

Figure 1: The team cycling training system with 4 bicycles

3- The Model Predictive Controller (MPC) In model predictive control, a dynamic process model is used for online prediction on a moving horizon of the future actions of the manipulated variables on the plant output. The future moves of the manipulated variables are determined by optimization with the objective of minimizing a defined cost function which comprises the predicted errors subject to operating constraints. The optimization is repeated at each sampling time based on updated information from the plant (García et al. 1989, Rawlings 2000). For the optimization of the cycling group training, the entire cyclist group is the control plant. The training intensities are the manipulated variables which are represented by the HR or power out of the cyclists. In order to optimize the training intensities of the cyclists by the MPC, a cycling performance model and a HR model are required to predict the physical and physiological loads of the cyclists during training. These models will be described in the following subsections.

2.1- The cycling performance model

In cycling, the power of a cyclist is required to overcome a complex interaction of resistive forces presented by the cycling environment. These forces include air resistance, rolling resistance, gravity, inertia and frictional losses from the drive chain and wheel bearings (Atkinson et al. 2003, Faria et al. 2005). Air resistance (FW) is the major resistive force at normal cycling speed. It increases with the square of speed and is given by

2WbpdW )(

21 vvAcF +⋅⋅⋅= ρ , (1)

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7th ISEA CONFERENCE 2008 A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

where cd represents the drag coefficient, Ap is the projected frontal area of cyclist and bike, ρ is air density, vb is steady-state bike speed and vW is head (positive) or tail (negative) wind speed (Olds et al. 1993). Cyclists may have the drafting effect by riding in the slipstream of the lead rider. The effect of drafting diminishes for the following cyclists on subsequent positions as the wheel spacing increases. A correction factor was derived by Olds (1998) as

2wwdraft 0452.00104.062.0 ddCF ⋅+⋅−= , (2)

where CFdraft is the ratio of air resistance under drafting conditions to that without drafting, and dw is the wheel-to-wheel distance between the preceding and the following cyclists. It is assumed that there is no benefit of drafting by cycling more than 3 m behind another rider (Olds 1998). The second major resistive force that must be overcome is rolling resistance (FR). It is expressed as

( ) (( )RbcRR arctancos GgMMcF ⋅⋅+⋅= )

)

, (3)

where GR is the road gradient, Mc and Mb are the masses of the cyclist and the bike respectively, cR is the coefficient of FR and g is the gravity acceleration (Martin et al. 1998). If the course is not flat, work is performed against or with the grade. The force (FG) due to the road gradient is related to the mass of rider and bike and is represented by

( ) (( )RbcG arctansin GgMMF ⋅⋅+= . (4)

Besides the physical resistances described above, work is also performed against or with the cycling direction while varying the bike speed. The force (FA) due to the speed variation is shown by equation 5.

(5) aMMF ⋅+= )( bcA

with acceleration a. Thus the power to overcome these resistances for the lead cyclist without drafting is calculated as

( ) bAGRWLN vFFFFP ⋅+++= . (6)

Additionally, when the wheels rotate, the spokes slice through the air like the blades of a fan. That causes also resistance (Martin et al. 1998). Finally, the frictional losses in wheel bearings and drive chain must also be considered. However, the wheel rotating resistance and frictional losses are relatively small and can be expressed together as a mechanical efficiency factor cm. Therefore the total power output PLT of the lead cyclist is modelled by

( ) mbAGRWLT cvFFFFP ⋅+++= . (7)

Similarly, the total power output PDT of cyclist riding with drafting is calculated as shown by equation 8.

( ) mbAGRdraftWDT cvFFFCFFP ⋅+++⋅= . (8)

If all of the model parameters can be accurately determined, these equations should provide a precise prediction of the necessary power to overcome all of the resistances in cycling. However the power output represents only the external physical load of cycling. A well-trained cyclist can perform a high power output, but the same power value might be too high for other cyclists with lower training conditions. Therefore, to prescribe the optimal exercise intensity for an individual cyclist, we need a HR model to predict the internal physiological load of the cyclist under certain power output.

2.2- The heart rate prediction model

Le and others have developed a dynamic HR prediction model and confirmed its validity under laboratory conditions (Le et al. in print 2008). This model predicts the HR of a cyclist based on the physical response of the HR to the exercise load online during training or competition. It considers the diverse HR responses to different exercise intensities according to the cyclist’s individualized blood lactate threshold. Eight parameters are used to calculate the future heart rate values from current values, training duration and training load by taking consideration of other effects such as exhaustion and recovery. Mathematically, the HR kinetics during training is modelled by the equations 9 and 10 (Le et al. in print 2008).

)()( S kHRHRkHR Δ+= , (9)

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7th ISEA CONFERENCE 2008 A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

[ ] [∑=

⋅− −⋅−⋅⋅+⋅−⋅+−Δ⋅+⋅=Δ1

iATiATA4/

321 )()()()e1()1()()( A

i

kT HRiHRHRiHRTKkPKkHRKkPKkHR στ ]−1k

[ ] [ ]∑−

=−⋅−⋅⋅+

1iATiATA5

on

)()(k

KjjHRHRjHRHRTK σ . (10)

where k is sampling step, HR(k) is HR value of k, HRS is HR value before the exercise start, and is the change of the HR due to the cycling workload. P(k) is the power output of the cyclist, TA is sampling time, and τ is a time constant related to training duration. HRiAT is the HR at cyclist’s individual anaerobic threshold. Kon is the time point, at which HR(k) is greater than HRiAT for the first time.

)(kHRΔ

()σ is the unit step function with

else ,0 ,

01

)(≥∀

⎩⎨⎧

=x

xσ . (11)

K1, K2, K3, K4 and K5 are model parameters which represent the ratio of the HR kinetics to training work load under different intensity levels. They were identified using the least-squares method by minimizing the loss function that is the sum of the squared differences between the measured data and the fitted functions. Based on the cycling performance model and HR model, the design process of the MPC algorithm can be started.

2.3- The control algorithm

The purpose of the MPC is to find the future values of the control signals by minimizing a predetermined cost function which comprises the predicted errors subject to operating constraints. For the group training optimization, each cyclist in the group should meet his training plan as exactly as possible. It means that the difference between the training HR and the predetermined reference value of each cyclist must be minimized. Therefore, the errors between the predicted HR and reference values from the training plan are comprised in the cost function. In group training, the group speed must be the same for all the cyclists. Due to the drafting effect, the training load of each cyclist can be optimized based on his individual performance profile and training plan by riding in different positions within the group. Thus, we take these three control signals for the training optimization: changing positions within the group, adjusting the group speed, and splitting the group. These control signals have different influences on the heart rates of the cyclists while regulating the training loads. The MPC must select the best way to regulate the load at each sampling step so that each cyclist can meet his training plan as exactly as possible. To achieve this objective, these control signals are taken into consideration while minimizing the cost function. Therefore, the cost function is defined to consist of the following two parts:

1. The errors between the predicted HR and reference values weighted by the position number within in the group. 2. Control signals: changing positions within group, adjusting group speed and splitting the group.

Mathematically, the cost function J is expressed by equation 12.

∑ ∑ ∑= = = ⎥

⎥⎦

⎤

⎢⎢⎣

⎡Δ+Δ+Δ+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

L

i

M

m

N

nGrpWSpdWPosWHRWJ

1GSP

1 1nn )(ε (12)

where L is the number of prediction steps, M is the minimal formation time. During the period of M*TA, the group formation can not be changed. N is the number of the group members, n is the position number in the group, Wn is the normalized weighting factor of the cyclist’s position in the group with

∑=

=N

nW1

n 1 . (13)

ε(HRn) is the error between the predicted HR and the reference value of the cyclist in position n. WP, WS and WG are weighting factors for the control signals ΔPos, ΔSpd, and ΔGrp respectively. The control signal of splitting the group ΔGrp is not considered for a homogenous small group and is not considered in our training tests either. If the cyclist position and the group speed are not necessary to be changed, both ΔPos and ΔSpd have the value 0. ΔPos = 1 when the cyclist position is to be changed. ΔSpd = 1 for the cases group speed will be increased or decreased. Thus we have four possibilities at each prediction step in total: changing position, speed increment, speed decrement and no changing. For the L prediction steps, we have 4L possible combinations of the control signals within the prediction horizon L*M*TA. Each combination of the control signals yields one value of J. There is one minimum among these 4L of J values. Then, the combination of the control signals that corresponds to the minimal J gives the optimized control sequence for the period of L*M*TA. However, only the first control signal of the optimized sequence is applied to the cyclist group. At the next sampling step, new sensor information is gained. Based on the new information, the entire control trajectories are recalculated and the optimization procedure is repeated.

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7th ISEA CONFERENCE 2008 A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

4- Evaluation of training results Several training sessions of two- or four-cyclist group were conducted on a circular stadium track to evaluate the MPC. As an example, we will show the training results of a two-cyclist group. The physiological characteristics and the HR model parameters of the cyclists participated in the test are listed in Table 1. The values of the HR model parameters were identified by least squares method of the interval training data (Le et al. in print 2008).

Weight [kg]

Height [cm]

PiAT [W]

HRiAT [bpm]

HRS [bpm]

K1 [bpm/W]

K3 [bpm/W] K4 [s-1] K5 [s-1] Cyclists K2

AC 97 191 254 151 108 0.0020 0.9856 0.0036 -0.000017 0.000010

DS 68 176 215 159 100 0.0028 0.9875 0.0016 -0.000304 0.000015

Table 1: Physiological characteristics and HR model parameters of the cyclists participated in the training tests

The environment temperature was between 14.5 °C and 15.3 °C during the training test with a sea level atmospheric pressure. Because the test was conducted on a circular track and the wind speed was very low, the influence of wind speed was neglected. The training was started after a warm up with the start HR of 108 and 100 bpm for cyclist AC and DS, respectively. The standard dropped racing posture was taken by the cyclists. The total projected frontal area of the cyclist and bicycle was calculated corresponding to Heil (2002) by the equation

594.0c )(04091.0 MAp ⋅= . (14)

According to our test bicycle type and track surface, we took the parameters listed in Table 2 to calculate the resistances and cost function. The parameters for the calculation of the resistance were selected by comparing with the data from Di Prampero (2000) and Olds (1998). The parameters for the cost function were determined by our training simulations according to the cyclist’s performance and training plan. From these parameters, we can see that the prediction horizon L*M*TA is 30 seconds. The minimal leading time M*TA is 6 seconds.

Parameter cd cR cm dw (m) WP WS W1 W2 L M TA(s)

value 0.8 0.006 0.95 0.5 25 50 0.55 0.45 5 6 1

Table 2: Parameters for the calculation of cycling resistances and cost function

During the training, the cyclists were required to ride with a wheel-to-wheel distance of 0.5 m between the leading and subsequent cyclists. While changing the position of cyclists, the lead cyclist was required to stop pedalling shortly and then drop to the end of the group. The cyclist behind him took the lead position automatically without varying the group speed. When the controller gave out the instruction of adjusting the group speed, the lead cyclist was instructed to adjust the group speed such that his power output might be changed by 20 watt. The training result is shown in Figure 2. The target HR of cyclist AC and DS are 135 and 130 bpm, respectively. The mean values of their training results are 135.06 and 130.13 bpm, respectively. The standard deviations are 3.38 and 4.66 bpm.

120

125

130

135

140

Hea

rtrat

e [b

pm]

Cyclist AC Cyclist DS

Figure 2: Training result of a two-cyclist group Time [min]

0 5 10 15 20 25

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7th ISEA CONFERENCE 2008 A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

The training result of cyclist AC is shown in Figure 3. The grey line is the distribution of his position during the training. Position 1 means the lead position and 2 the end of the group with drafting. We can see that the training is started when cyclist AC is in lead position. His HR rises when he is in lead position and drops when he is in non-lead position. During the 28 minutes training test, the cyclist AC is about 12 minutes in the leading position.

115

120

125

130

135

140

Hea

rt ra

te [b

pm]

0

1

2

3

Posi

tion

in g

roup

Heart rate Position in group

Time [min]

0 5 10 15 20 25

Figure 3: Training result of cyclist AC

The distribution of the control signals are shown in Figure 4. In order to present the control signals more obviously, they are shown in the figure with different values. Value 2 means position change, value 1 means group speed increment, value 0 means no change (keep going) and value -1 means group speed decrement. We can see that after the beginning of the training the controller requires the cyclists to increase the group speed. When the group speed has reached the desired value, the controller instructs the cyclist to hold the speed and the group formation. After about 1 minute, the cyclists are instructed to change the position. There are totally 29 times of position changes during a training period of 28 minutes. The position is changed nearly once per minute on average. Because of the cardiac drift effect, the cyclists are also instructed to decrease the group speed 6 times during the training. The frequency of changing the position and speed can be adjusted by the weighting factor WP and WS in the cost function according to the training plan.

-1

0

1

2

Con

trol s

igna

l

Figure 4: Distribution of control signals during training period

5- Conclusion In this study, we have presented a model predictive controller integrated in a sensor-based TCTS system for the optimization of the cycling group training. Based on a cycling performance model and a dynamic heart rate prediction model, the model predictive controller predicts the exercise intensity of each cyclist in the group and optimizes the group training by minimizing a predetermined cost function. Training test results of a two-cyclist group on a circular stadium track indicate that the TCTS with the model predictive controller is an effective aid for the group training in cycling

6- Acknowledgement This work was supported by the research centre Ambient Intelligence at the University of Kaiserslautern. The authors gratefully acknowledge Marcus Trapp for his assistance in the completion of this study, as well as Andreas Christmann and Daniel Schmidt for their enthusiastic participation in the training tests.

Time [min] 5 10 15 20 25

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7th ISEA CONFERENCE 2008 A Model Predictive Controller for Sensor-based Training Optimization of a Cyclist Group

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