## 2 Kinematics Constraints and Robot Design Methodology
### Robot Design and Kinematics Constraints
 \tag{1}\]  
One of the conditions for the robot to be autonomous is recognizing the obstacles in front of it and determining if they can be scaled; otherwise, the robot must take another path. It is expected that the wheels maintain contact with the ground to avoid falls. However, in reality, the wheels can lose contact; therefore, they lose their stability and fall. To ensure the robot can climb, we calculate the maximum height and the minimum length of the step that the robot can climb up and down an obstacle or step as shown in Figure 3. The maximum height can vary according to the size of the robot. Five different maximum height steps are obtained based on the constraints and parameters of the robot Equation (2), where \(\gamma_{i}\) is the contact angle between each wheel and the ground, \(\alpha_{20}\) denotes the angle between \(l_{2}\) and \(l_{4}\), \(\alpha_{2}\) denotes the angle between \(l_{2}\) and the horizontal global axis, \(\theta_{b}\) denotes the maximum bogie rotation, \(\alpha_{30}\) denotes the initial angle between \(l_{3}\) and the horizontal global axis when the robot is in a flat surface, and \(\alpha_{3}\) denotes the angle between \(l_{3}\) and the horizontal that varies with the tilt of the chassis. \(b_{x}\) denotes the horizontal distance between the center of the bogie and the contact point of the middle wheel, and \(b_{y}\) denotes the vertical distance from the center of the bogie to the contact point of the middle wheel. Figure 2: 2D Schematic diagram of the wheeled robot. The maximum height the robot can climb up or down an obstacle is determined by Equation (3) after obtaining the maximum heights for each constraint. * \(h_{1}\) prevents the front and middle wheel from climbing up or down together, as shown in Figure 3a. The middle wheel maintains contact with the ground before starting to climb, and the front wheel must maintain a contact angle fewer than 65 degrees, based on experiments. * \(h_{2}\) is the maximum rotation of the bogie in the counterclockwise direction Figure 3a. * \(h_{3}\) is the maximum rotation of the bogie in a clockwise direction Figure 3b. * Similar to \(h_{1}\), \(h_{4}\) restricts the middle and rear wheels from climbing up or down simultaneously Figure 3b. * \(h_{5}\) is the maximum height step according to the maximum inclination of the chassis to avoid falls, obtained from Figure 3b. \[h_{1} =\sqrt{l_{4}^{2}-(r_{2}-r_{1}\sin(\gamma_{1})^{2}}-r_{1}\cos( \gamma_{1})+r_{2}\] \[h_{2} =l_{4}\sin(\alpha_{2}-\alpha_{20})-r_{1}\cos(\gamma_{1})+r_{2}\] \[h_{3} =l_{3}\sin(\theta_{B}+\alpha_{30}+\alpha_{2}-\alpha_{20})-b_{y}+r _{3} \tag{2}\] \[h_{4} =\sqrt{l_{3}{}^{2}-(r_{3}+b_{x})^{2}}-b_{y}+r_{3};\] \[h_{5} =l_{3}\sin\alpha_{3}-b_{y}+r_{3}; \tag{3}\]  
\[min(h_{i})=h_{max} \tag{4}\]  
The minimum length of the step required to climb up or down depends on the step height \(h\), which must be less than \(h_{max}\) Equation (3). The first case is when \(h\) is equal to \(h_{max}\), the minimum length of the step when the robot climbs up is the length of \(l_{4}\) plus \(r_{1}\), \(r_{2}\), and the horizontal distance between the middle and rear-wheel contact point. However, when the robot climbs down with \(h_{max}\), the minimum step length required is the total length of the robot. The second case occurs when the front and middle wheel maintain contact with the tread or the middle and rear wheel maintain contact with the tread. Then we include a parameter \(d\), which is the distance between the middle wheel and the step. If the parameter \(d\) is greater than zero, there is a space between the middle wheel and the step. Furthermore \(h\) is less than \(h_{max}\) and we can obtained \(t_{1}\) and \(t_{2}\). \(t_{1}\) denotes the horizontal distance between the front and middle wheel in contact with the tread, including both wheel's radius, and \(t2\) denotes the horizontal distance between the middle and rear wheel in contact with the tread, including both wheel's radius. The middle wheel can climb up or down with fewer than 65 degrees of contact angle.