## 2 Design, Modeling, and Control
### Simulation Model  
The simulation model of this legged-wheeled robot, built in the SimTwo simulator, was created by decomposing it into a set of rigid bodies having mass and size equal to the corresponding parts of the real robot.  
Leg links and the main body were represented as cuboids, while joints and wheels as cylinders (joints allow connection between different rigid bodies). Except for the joint for rotation of the omnidirectional wheel, all the others are actuated. Thus, and as shown in Figure 22, the model comprises four legs that connect to the main body on their uppermost joint, and eight wheels.  
The locomotion system of the robot was modeled by combining, per leg, three links with four joints (includes the wheels' joint). The \(z\)-axis is the actuation axis of the hip  
Figure 21: Complete robot control block diagram.  
Figure 22: Robot simulation model.  
joint, while the \(y\)-axis is that of the remaining ones. Non-rigidity of the two joints that follow the hip joint was modeled by assembling a compound joint: two separate joints connect the leg links to an intermediate cylindrical link; one of them is actuated and the other has a spring (simulating the elastic component of the joint). Omnidirectionality of each non-actuated wheel was established by defining two coefficients of friction onto their surface: one equal to less than one to allow lateral motion, and the other equal to one to hinder forward/backward motion.  
In the real robot, while in purely wheeled mode and performing in low power, its lowest height is kept due to mechanical constraints. Modeling this behavior required placing cuboid shells that restrict the robot position to remain at the lowest height as, upon collision, they stay fixed, avoiding rigid bodies from plunging into one another.  
As previously mentioned, the robot implements PID controllers for position control of each hip joint and speed control of the wheels, while state-space controllers were developed for control of the non-rigid joints. SimTwo has built-in position and speed PID controllers, which were thus used to control the hip joints and wheels of the model. The state-space controllers were fully implemented within SimTwo using its IDE (integrated development environment).  
Control of the vehicle in the \(x\) and \(y\) axes has been included in simulations by applying the equations of the robot kinematics described in Section 2.5. Consequently, a first approach to trajectory control has been developed.  
The implemented tracking method requires the robot to follow a line or curve from an initial point to a target point. A line is defined by a set of two points, while a curve by three. The omnidirectional motion ability of this robot enabled the use of controllers to correct its position with both lateral and angular motion, relative to the error in the distance to the reference line and in the angular position, respectively. When following a line, the reference is the line itself, while, when following a curve, it corresponds to the tangent to the curve in the nearest point between the curve and the robot's position. The linear speed of the robot is constant, up to a defined distance to the target, where it initiates a smooth, linear deceleration motion to reach and stop at the final position. Line tracking requires P controllers in \(v_{n}\), and \(w_{r}\), while curve tracking uses PD controllers.