# Cycling Biomechanics and Its Relationship to Performance
## 2 Crank Power-Output  
A bicycle can be viewed as a device that converts the work produced at the crank by the cyclist into kinetic energy (\(=\frac{1}{2}\). \(m.v^{2}\), with \(m\) the mass of the bicycle plus cyclist and \(v\) the velocity of the bike). Thus, maximizing performance requires increasing the kinetic energy of the system (cyclist plus bicycle) and, consequently, the crank power over time. Maximal crank power and the ability to maintain a high power output over time are the primary performance parameters to assess in cyclists [17; 18]. It can be noted that forward acceleration is reduced for greater mass of the bike and cyclist and for greater resistances (aerodynamic forces, rolling friction and gravity during uphill cycling), but these later points will not be discussed in this paper. At each instant, the crank power output (in Watt) represents the power developed by the two pedals which are computed for each pedal using the scalar product (dot product) between the vector force applied at the pedal (N) and the pedal velocity (m/s). Crank power can be computed equivalently using the scalar product between the torque vector at the crank (N\(\cdot\)m) and its angular velocity (rad/s). Thus, only the forces applied orthogonally to the crank arm, named the effective forces, generate power. The other forces directions are not effective, and they might cost energy metabolically [19]. Moreover, effective forces can be in the direction of the movement (i.e., positive), increasing the total power output, or in the opposite direction (i.e., negative), decreasing the total power output. The efficacy of the pattern of force application is assessed by analyses of the pedal forces [20; 21; 22]. A classic measure is the ratio between the orthogonal force and the total force applied on the pedal, named the mechanical effectiveness (ME, [23]). When averaged over one cycle, ME is in the range of 0.15 to 0.8 [7; 24], with 1 indicating that only effective forces are created throughout the cycle. Previous studies showed that ME can be improved with practice [20; 25; 26; 27], notably by pulling on the pedal during the upstroke to compensate for the weight of the leg [28]. However, this resulted in more complex muscle activity patterns [27] and additional energy expenditure [29]. A previous study suggested that gross efficiency (the ratio between mechanical work and metabolic energy expenditure) could be improved as well with more practice (i.e., \(>\)15 min) [25]. Therefore, the relation between ME and cycling performance is still unclear, and further studies are needed to clarify this topic [30]. Using a simple static mechanical analysis using the maximal torque capabilities of the hip, knee and ankle joints, it can be shown that maximizing the power-output without producing ineffective forces is unlikely to be feasible (Figure 1). In order to produce this figure we used the relation \(\boldsymbol{\tau}=\mathbf{J}^{T}\times\mathbf{f}\) to identify the feasible forces that could be generated given the joint moments at the hip, knee and ankle [31; 32]. In this relation, \(\mathbf{f}\) is a 2-by-1 pedal forces vector, \(\mathbf{J}\) is a 2-by-3 Jacobian matrix relating the variations in hip, knee and ankle joint angles and the variations in foot tip positions, and \(\boldsymbol{\tau}\) is a 3-by-1 joint torques vector. The symbol T represents matrix transposition. In this model we neglected the changes in joint moment due to joint position and joint angular velocity and also gravity and inertial effects. The joint torques and kinematics were obtained from [PERSON] et al. [33]. the length of the thigh, leg and foot were 0.45 m, 0.38 m and 0.15 m, respectively. Figure 1 provides an illustration of the possible and limited force directions and amplitudes for different joint configurations during cycling. It shows that the forces that would provide the maximum effective force necessarily produce an ineffective force component (i.e., the force F\({}_{1}\) in Figure 1).