# Design of a New Bilayer Multipole Electromagnetic Brake System for a Haptic Interface
## 3 Multi-Objective Optimization  
An effective electromagnetic brake can provide large magnetic force that can generate large braking torque at the expense of small input power. It is also desirable to have faster response, particularly for the applications where time-delays adversely affect the efficiency of a task. The analytical models developed in Section II realize the fact that, for a constrained volume of the brake, the more which is the coil current, the more it generates magnetic force, but also increases the power consumption. Alternatively, a greater number of turns of the coil can limit the input power, but it also increases the inductive TC; hence, brake response is degraded.  
In order to come up with an optimum design that can best deal with these conflicting properties, a multi-objective optimization procedure is proposed. For a fixed volume of electromagnetic brake, the objective is to maximize the magnetic force \(F_{mug}\) while minimizing the inductive TC and the power consumption \(P_{cons}\). The optimization procedure is formulated as  
\[\begin{split} x=\begin{bmatrix}\theta_{C},R_{i_{\min}},R_{i_{ \max}},R_{o_{\min}},R_{o_{\max}},w_{i},w_{o}\end{bmatrix}^{T},\text{ with}\,x_{\min}\leq x\leq x_{\max}\\ \text{Maximize}:f(x)=\alpha\frac{F_{\max}}{F_{\max}}+\beta\frac{T_{\max}}{ \max(TC_{i},TC_{o})}+\gamma\frac{P_{\max}}{F_{\max}},\\ \text{Subject to}:r_{i}<R_{i_{\min}}<R_{i_{\max}}<R_{o_{\min}}<R_{o_{\max}}<r_{o },\\ \Phi_{i}\leq B_{sat}A_{gi}\,,\end{split} \tag{22}\]  
where \(x\) is the vector of design variables, shown in Figure 4, which are bounded by lower and upper values, as described by \(x_{\min}\) and \(x_{\max}\), respectively. The objective function is comprised of the scaled and normalized sum of magnetic force, inductive time constant and power consumption. \(F_{norm}\), \(TC_{norm}\) and \(P_{norm}\) are the normalized values obtained through the optimization of individual parameters in the given constraints, and they were chosen as 133 N, 20 ms and 1.5 W, respectively. The scaling terms \(\alpha\), \(\beta\) and \(\gamma\) are chosen according to the priority given to the brake properties, and their values were selected as 0.5, 0.3 and 0.2 respectively. The proposed design of magnetic brake has many tiny air-gaps in its magnetic circuit which allow it to operate even for greater range of input power. Because of the limitation of maximum allowable magnetic flux density \(B_{sat}\), the most important requirement is to achieve maximum possible magnetic force that can generate a large braking torque before the devicegets saturated. The brake was being designed focusing on its application in haptics; therefore, a faster response would have much more importance.  
A sequential quadratic programming (SQP) method was adopted to carry out an optimization procedure. SQP is a gradient-based optimization algorithm that leads the objective function towards the nearest minimum in the neighborhood of the chosen initial point. The optimization procedure was input with initial values of the design variables, taken from the parameters of initial design of BLMP brake, shown in Figure 6a. The procedure took five iterations to reach the final values, suggesting the optimal design shown in Figure 6b. For a fixed current supply to the coils of each layer, the cross-sectional area of the poles was adjusted in the optimal design to provide more space to solenoid coils and minimize the distance between the poles. A comparison of the initial and optimal design variables is shown in Table 1.