## 1 Introduction  
The demand for autonomous mobile robots is growing rapidly due to their services. In recent years, many robots have been on the market delivering packages [1, 2]. They require some tasks to be solved, such as perceiving the environment [3, 4], planning a trajectory [5], controlling [6], and avoiding obstacles [7, 8]. The locomotion system of the robot is an important aspect to consider regarding the robot's design, which depends on the environment and technical criteria such as its mobility and stability. A robot's autonomous task also involves perceiving its environment to create a controller according to its pose and location. Wheeled robots have the particularity that their stability is guaranteed by maintaining the wheels' contact with the ground. However, due to the different shapes of the surfaces, such as uneven terrain and steps, the robot's wheels can lose contact with the ground and cause the robot to fall. Therefore, recognition of the geometry of the surface in real-time will allow the robot to act quickly towards maintaining its stability. The contact angle between the wheel and the ground is a critical parameter that determines the wheels' position concerning the surface, which specifies if the robot's wheel is climbing a vertical, inclined, or irregular surface. The research carried out to determine the wheel's point of contact is subject to using sensors that allow measuring or estimating this relationship. Force and torque sensors were installed in mobile robots to obtain the contact point; however, their installation is complex [9]. An array of pressure sensors also measure the normal stress distribution to accurately estimate the traction force, depending on the number of segments implemented in the sensor array [10]. Sensitive bumper skin also determines the contact with the surface, and it is possible to create controllers according to the deformation of the sensors, which incorporate a flexible substrate, allowing to determine the point of contact with precision. However, its effectiveness depends on the number of sensors in the wheel in contact with the surface [11]. Vision sensors such as mono-cameras are used to find the contact angle by installing a camera on each wheel; this method is verified only in a limited indoor environment [12]. Another method to estimate the contact angle derives from the robot's internal sensors and kinematic equations; this method is easy to install but depends on the robot speed being different from zero [13], and algorithms to reduce the noise produced by the sensors [14]. The terrain information given by the contact angle between the wheel-ground and the robot velocity permit adjusts the velocity of each wheel based on the kinematics model. The Jacobian matrices of each wheel are calculated and then their inverse [13; 15; 16]. Another method of controlling the robot's velocity is setting torque to each wheel according to dynamic analysis [17]. The performance of autonomous wheeled robots is also evaluated by robot designs [18; 19; 20], a complex task involving multiple design variables, objectives, constraints, and evaluation criteria [21; 22; 23]. The contributions of the paper are given as follows. * The design methodology finds optimal values of design parameters by considering the kinematic constraints of the multiwheeled mobile robot and resulting in the climbable environmental geometric parameters such as the maximum height and the minimum length of the step. * The contact angle is measured using laser scanner sensors, and the loss of contact between the wheel and the ground is monitored in real time, which activates the emergency controller that stabilizes the robot pose when climbing up and down stairs. * Simulation and experiments demonstrate the robot's performance using the design methodology, the contact angle method, and the emergency controller to climb up and down safely. The rest of the paper is organized as follows. Section 2 describes the kinematic constraints and the optimization of the robot based on the Taguchi method.